The cross states of a composite quantum system: separability and entanglement in any Hilbert space dimension

The cross states of a comp osite quan tum system: separabilit y and en tanglemen t in an y Hilb ert space dimension P aolo Aniello 1 , 2 ∗ 1 Dip artimento di Fisic a “Ettor e Pancini”, Universit` a di Nap oli “F e deric o II”, Complesso Universitario di Monte S. A ngelo, via Cintia, I-80126 Nap oli, Italy 2 Istituto Nazionale di Fisic a Nucle ar e, Sezione di Nap oli, Complesso Universitario di Monte S. A ngelo, via Cintia, I-80126 Nap oli, Italy Abstract W e in tro duce a class of states of a composite quantum system — the so-called cr oss states — that turn to play a ma jor role in the theory of entanglemen t for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilb ert spaces of the bipartition is finite- dimensional, all states are cross states, whereas, in the gen uinely infinite-dimensional setting where the dimension of b oth Hilbert spaces is not finite, the cross states form a trace-norm dense, con vex, prop er subset of the set of all states. In the latter case, the cross states can b e regarded as those physical states that p ossess a finite amoun t of entanglemen t; accordingly , all separable states are of this kind. W e prov e that, for any Hilb ert space dimension, the separable states can b e c haracterized as those cross states that minimize a suitable norm, i.e., the pr oje ctive norm asso ciated with the pro jective tensor pro duct of tw o trace classes; all other cross states are density op erators b elonging to the pro jectiv e tensor pro duct space. This is a generalization of the classical cr oss norm criterion of separability . Finally , we define an extended real-v alued entanglement function and study its main properties. Coheren tly with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the pro jectiv e norm, precisely on the cross states of the system. 1 In tro duction En tanglement is one of the most distinguishing features of quan tum v ersus classical mechanics [1, 2]. Among the manifold consequences of this very fundamen tal asp ect of quantum theory , it is worth men tioning the applications in the con text of quan tum information science [3]. Ho wev er, to capture the in timate essence of entanglemen t is a highly non trivial task, and it turns out that actually , in order to obtain a profound understanding of this exquisitely quan tum phenomenon, a rigorous mathematical approach cannot b e disp ensed with. This fact, on the one hand, probably explains wh y a finite-dimensional setting — where sev eral in tricacies asso ciated with the machinery of infinite-dimensional Hilb ert spaces b ecome m uch milder, or simply disapp ear — is often assumed in the literature [4, 5]. On the other hand, whereas certain quan tum systems are sufficiently well describ ed b y resorting to an effectiv e finite-dimensional mo del, the understanding of entanglemen t of physical states in any Hilb ert space dimension is of cen tral imp ortance, e.g., in the context of quan tum measuremen t theory or in the study of op en quan tum systems [6–8]. Recall that, in the standard form ulation of quantum mechanics [6–11], the tw o main families of fundamen tal ph ysical en tities — i.e., states and observables — can be realized as suitable classes of ∗ Email: paolo.aniello@na.infn.it 1 linear op erators on a separable complex Hilb ert space H . Sp ecifically , the (normal, or σ -additive) quan tum states on H form a distinguished subset of the complex Banach space B 1 ( H ) of trace class op erators — namely , the c on vex set D ( H ) of density op er ators — whereas the elemen ts of the selfadjoin t part of the C ∗ -algebra B ( H ) of all b ounded op erators on H play the role of (b ounded) observ ables. The pairing b etw een states (density op erators) and observ ables (b ounded selfadjoin t op erators), whic h provides the relev ant measuremen t probability distributions and exp ectation v alues, is implemented b y the tr ac e functional. This picture b ecomes somewhat more complicated as so on as one considers the mathematical mo deling of two mutual ly inter acting quantum systems . In this setting, according to one of the fundamen tal axioms of quan tum mec hanics, the description of the resulting bip artite c omp osite quantum system inv olv es the tensor pr o duct H ⊗ J of tw o (separable, complex) Hilbert spaces H and J [6–9, 11], and it is precisely at this stage that entanglemen t enters in to the game. If one restricts to considering pur e states (i.e., rank-one pro jections) only , since these states can b e describ ed in terms of state ve ctors in the Hilb ert space H ⊗ J , then the intricacies of quan tum en tanglement are mainly c onc eptual rather than mathematic al , because the notion of tensor product of Hilb ert spaces is fairly simple and straightforw ard [8, 9, 11]. The sep ar able pur e states are precisely the rank-one pro jections asso ciated with (normalized nonzero) v ectors in H ⊗ J of the elemen tary factorized form ϕ ⊗ ψ , whereas all other pure states are entangle d . If, instead, one is in terested in considering the mixe d states of the comp osite system as w ell (e.g., in the case where the de c oher enc e [12] effects cannot be neglected), then the relev ant op erator spaces asso ciated with the pro duct Hilb ert space H ⊗ J must b e prop erly defined and studied. It is an in teresting fact, inciden tally , that a precise and complete definition of separabilit y/entanglemen t, in the general case of mixed states, seems to ha ve app eared rather late in the literature [13]. According to this (ph ysically grounded) definition, the sep ar able mixe d states are either pro duct states of the form ρ ⊗ σ — where ρ , σ are densit y op erators — or suitable (generalized) statistical mixtur es of suc h states; all other states are entangled. This definition is also coherent with the central role pla yed b y c onvexity [14] in the context of quantum theory . Once again, one ma y av oid most mathematical complications by simply assuming that the Hilb ert spaces H and J are finite-dimensional. Ho wev er, as previously observed, there are cases where one is forced to leav e this relatively safe playground and face the most general situation. The main aim of the present contribution is to achiev e a complete characterization of the conv ex set of all sep ar able (i.e., non-en tangled) states of a bipartite quan tum system — for an y Hilb ert space dimension of the subsystems — in terms of the most fundamental structur es in volv ed in the mathematical formalization of a comp osite quan tum system, i.e., the tensor product of the relev ant op erator (Banach) spaces and the canonically asso ciated norms [15–20]. As a byproduct, we will also obtain a natural ‘measure of en tanglement’. Let us briefly outline the fundamental problem and the main ideas at the basis of our work. Our primary scop e is to c haracterize the con vex set D ( H ⊗ J ) se of all separable states of a bipartite quantum system, whose carrier Hilb ert space H ⊗ J is the tensor pro duct of the ‘lo cal’ Hilb ert spaces H and J . This set is defined as the close d c onvex hul l of the set of pro duct states, i.e., D ( H ⊗ J ) se : = co { ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J ) } . (1) Here, the conv ex hull is supp osed to b e closed wrt the trace norm ∥ · ∥ ⊗ 1 of the am bient Banach space B 1 ( H ⊗ J ) — the trace class (or Sc hatten 1-class) on H ⊗ J — the bipartite states D ( H ⊗ J ) b elong to. A t this p oint, it is natural to consider, at first, the algebr aic tensor pr o duct B 1 ( H ) ˘ ⊗ B 1 ( J ) of the ‘lo cal’ (or factor) trace classes B 1 ( H ) and B 1 ( J ), that can b e iden tified with the complex linear span of the elementary tensor pro ducts ρ ⊗ σ of ‘lo cal’ density op erators. It can b e easily shown 2 that, by taking the completion, wrt the norm ∥ · ∥ ⊗ 1 , of this linear space one obtains precisely the Banac h space B 1 ( H ⊗ J ), i.e., the bip artite tr ac e class . It should b e noted, how ev er, that, if one regards the factor trace classes B 1 ( H ) and B 1 ( J ) as abstr act Banach spaces, then completing the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ) wrt the trace norm ∥ · ∥ ⊗ 1 is not a very natural choice. In fact, in order to obtain a Banach space out of the algebraic tensor pro duct of t w o abstract Banac h spaces the most natural choices turn out to be the pr oje ctive tensor pr o duct and the inje ctive tensor pr o duct [15–20]. Both types of tensor products will pla y some role in the following; ho wev er, in some sense, the first of these t wo options is a particularly natural choice when dealing with trace class op erators, b ecause the trace class B 1 ( H ) itself can b e regarded as — i.e., it is isomorphic to — the pro jectiv e tensor pro duct of tw o copies of the Hilb ert space H , that we will denote by H b ⊗ H . Let us then fo cus on the pro jective tensor pro duct B 1 ( H ) b ⊗ B 1 ( J ), that w e will call the cr oss tr ac e class ; i.e., the Banach space completion of the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ) wrt the pr oje ctive norm — or gr e atest cr oss norm — ∥ · ∥ b ⊗ 1 (not to be confused with the trace norm ∥ · ∥ ⊗ 1 of B 1 ( H ⊗ J )) asso ciated with the trace norm ∥ · ∥ 1 of the ‘lo cal’ trace classes B 1 ( H ) and B 1 ( J ). It has been first discov ered by Rudolph [21, 22], who considered the case where b oth the Hilb ert spaces of the bipartition H ⊗ J are finite-dimensional, that a density op erator D ∈ D ( H ⊗ J ) is separable iff ∥ D ∥ b ⊗ 1 = 1; otherwise stated, that the conv ex set D ( H ⊗ J ) se is the intersection of the unit sphere in B 1 ( H ) b ⊗ B 1 ( J ) with the set D ( H ⊗ J ) of all densit y op erators on H ⊗ J . This imp ortan t discov ery has attracted a certain interest in the c haracterization of entanglemen t b y means of the pro jectiv e norm [23, 24]. A fundamental step forw ard in the understanding of the central role of the pro jective norm in the study of entanglemen t is due to Arv eson [25], who, using elegant metho ds of conv ex analysis and of the theory of op erator algebras, has pro ved — among other things — that Rudolph’s c haracterization of separabilit y remains true in the more general case where at least one of the Hilb ert spaces of the bipartition H ⊗ J (or, more generally , of a multipartite Hilb ert space) is finite-dimensional. It is then natural to w onder what happ ens in the most general case where no restriction on the Hilb ert space dimension is assumed. This is a non trivial issue, b ecause it turns out that, actually , there is a sharp difference b et w een the case where the at least one of the Hilbert spaces of the bipartition is finite-dimensional and the genuinely infinite-dimensional c ase . In fact, the follo wing observ ation is of cen tral importance: Whereas, if at least one of the Hilb ert spaces H and J is finite-dimensional, the cr oss trace class B 1 ( H ) b ⊗ B 1 ( J ) can b e iden tified — as a set , or as a linear space — with the trace class B 1 ( H ⊗ J ), in the case where dim( H ) = dim( J ) = ∞ , instead, B 1 ( H ) b ⊗ B 1 ( J ) can be realized as a pr op er subset (or, more precisely , as a prop er linear subspace) of B 1 ( H ⊗ J ). Sp ecifically , one can prov e (and we will actually pro vide pro ofs of ) the following facts: 1. In the case where dim( H ) < ∞ and/or dim( J ) < ∞ , the norms ∥ · ∥ ⊗ 1 (trace norm) and ∥ · ∥ b ⊗ 1 (pro jective norm) are mutual ly e quivalent on the linear space B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) (set equalit y , or isomorphism of linear spaces, only). Otherwise stated, the Banach space B 1 ( H ) b ⊗ B 1 ( J ) can b e regarded as a r enorming of the trace class B 1 ( H ⊗ J ). Ho wev er, whereas ‘all bipartite states are equal’ as far as the trace norm is concerned — namely , for ev ery densit y op erator D ∈ D ( H ⊗ J ), ∥ D ∥ ⊗ 1 = 1 — the pro jectiv e norm is able to efficiently dete ct entanglement ; precisely , ∥ D ∥ b ⊗ 1 ≥ 1, and D is en tangled iff ∥ D ∥ b ⊗ 1 > 1. 2. In the case where dim( H ) = dim( J ) = ∞ (i.e., in the genuinely infinite-dimensional setting), instead, ev en if the pro jective tensor product B 1 ( H ) b ⊗ B 1 ( J ) admits a natural em b edding in the trace class B 1 ( H ⊗ J ) — and, actually , is a dense (wrt the trace norm) linear subspace of B 1 ( H ⊗ J ) — the pro jective norm ∥ · ∥ b ⊗ 1 and the restriction of the trace norm ∥ · ∥ ⊗ 1 to the 3 cross trace class B 1 ( H ) b ⊗ B 1 ( J ) are not equiv alent (the latter is simply p oint wise dominated b y the former). In particular, focusing on the set D ( H ⊗ J ) b of all cr oss states — namely , the con v ex set of bipartite states defined b y D ( H ⊗ J ) b : = D ( H ⊗ J ) ∩ B 1 ( H ) b ⊗ B 1 ( J ) — one can prov e that, in the gen uinely infinite-dimensional setting (only), D ( H ⊗ J ) b ⊊ D ( H ⊗ J ), and the pro jective norm ∥ · ∥ b ⊗ 1 is unb ounde d on cross states. In spite of our last remark, and of th e consequent tec hnical subtleties, the pro jectiv e norm turns out to b e still a v enerable to ol in the gen uinely infinite-dimensional case, b ecause: 3. The set D ( H ⊗ J ) se of all separable states is (strictly) con tained in the conv ex set D ( H ⊗ J ) b of all cross states, of which, actually , it is a distinguished conv ex subset. 4. In fact, the pro jective norm ∥ · ∥ b ⊗ 1 is able to characterize the separability of bipartite states, for any dimension of the ‘lo cal’ Hilb ert spaces H and J ; i.e., D ∈ D ( H ⊗ J ) se iff ∥ D ∥ b ⊗ 1 = 1. This is the core fact of the Extende d Cr oss Norm Criterion of separabilit y (ECNC), whic h is one of the main results of our present con tribution. As a byproduct of our inv estigation, we also get sev eral insights in to the relations b et ween the bip artite trace class B 1 ( H ⊗ J ) and the cr oss trace class B 1 ( H ) b ⊗ B 1 ( J ), and we obtain some imp ortan t duality relations inv olving the Banach space B 1 ( H ) b ⊗ B 1 ( J ). 5. The norm ∥ · ∥ b ⊗ 1 (as well as the strictly related Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 , see b elow) can b e regarded as a me asur e of entanglement . Accordingly , the cross states D ( H ⊗ J ) b can b e regarded as those bipartite states that p ossess a finite amount of entanglement . In the case where dim( H ) < ∞ and/or dim( J ) < ∞ , actually all states enjoy this prop ert y — i.e., D ( H ⊗ J ) b = D ( H ⊗ J ) — whereas, in the genuinely infinite-dimensional setting, this is no longer true. The picture of cross states as finitely entangled states is supp orted b y the following further imp ortan t p oin ts: 6. W e can define an extended real-v alued entanglement function E : D ( H ⊗ J ) → [1 , + ∞ ]. It turns out that, if restricted to the conv ex set D ( H ⊗ J ) b of all cross states, it coincides with the pro jective norm ∥ · ∥ b ⊗ 1 ; hence, it is able to characterize quan tum entanglemen t. W e will also prov e that — in the genuinely infinite-dimensional case where dim( H ) = dim( J ) = ∞ , and in that c ase only — the v alue + ∞ is actually attained by E . Precisely , E ( D ) = + ∞ iff D ∈ D ( H ⊗ J ) is not a cross state. W e also sho w that, as one should exp ect, the en tanglement function E has a re gular b eha vior wrt to restriction to subspaces, is inv arian t wrt ‘lo cal’ unitary transformations and is well-behav ed wrt the action of suitable ‘local quan tum maps’; i.e., such a map, by acting on a cross state, cannot increase the en tanglement of this state. 7. W e next prov e that if E ( D ) < + ∞ — i.e., in the case where D is a cross state — then the v alue E ( D ) of the entanglemen t function can b e approximated, with arbitrary precision, by measuring a suitable kind of quantum observ able, a so-called entanglement witness (in this regard, how ev er, w e w arn the reader that our definition of an en tanglement witness differs, ev en if in a nonessen tial wa y , from the con ven tion usually adopted in the literature [2, 26, 27]). In order to pro v e our main results, after deriving sev eral important properties of the cross trace class (e.g., we prov e that B 1 ( H ) b ⊗ B 1 ( J ) is a Banach algebra), we exploit tw o different kinds of mathematical tools: certain t yp es of b aryc entric de c omp ositions of separable densit y op erators, and some metho ds of functional analysis and of the theory of op erator algebras. In the first case, w e 4 ha ve b een influenced b y ideas and results of Holevo, Shiroko v and W erner [28–30]; in the second case, b y Arveson’s fundamen tal pap er [25]. It is w orth mentioning, how ever, that Grothendieck’s groundbreaking contributions — in particular, his celebrated R ´ esum ´ e [15] — lie at the foundations of the mo dern metric theory of tensor pro ducts, hence, of the applications w e are studying here. Hoping to hav e con vey ed some of the key p oin ts — or, at least, the flavor — of the issues w e are going to inv estigate, it is now w orth sp ending a few w ords ab out the meaning of our w ork in the context of our present understanding of quan tum theory . Even if the characterization of separabilit y/entanglemen t in terms of the pro jective norm ma y not b e of muc h practical use, b ecause, in man y cases of in terest, this quan tity is not analytically computable — and, accordingly , v arious related computable quantities ha ve b een prop osed that give rise to p artial separability criteria [31–34] (namely , sufficien t conditions for a bipartite quantum state to b e en tangled or, equiv alently , necessary conditions for separability) — w e believe that our work may shed new light on the very essenc e of en tanglement. In fact, to the best of our knowledge, the pro jective norm is the only mathematical tool stemming dir e ctly (and in a natural w ay) from the tensor pro duct structure that is able to provide us with a simple characterization of separable versus entangled states, for an y dimension of the Hilb ert spaces forming a comp osite quan tum system. In our presen t contribution, mainly for the sake of simplicit y , w e will consider the bipartite en tanglement only; we will consider the general multipartite case elsewhere. The pap er is organized as follo ws. In Section 2, we collect some of the main notations and some basic facts that will b e used in the rest of the pap er. Next, in Section 3, w e in tro duce some mathematical to ols that will b e fundamental for the characterization of separable states; in particular, the pro jective tensor pro duct of trace classes and t wo ‘canonical decompositions’ of a cross trace class op erator. The following three sections con tain our main results. In particular, in Section 4, we fo cus on the notion of separabilit y of quantum states, and discuss the barycentric decomp ositions that characterize separable states. In Section 5, we complete the characterization of separable states in terms of the pro jective norms. In particular, in Subsection 5.6, w e condense in a unique statemen t — the Extended Cross Norm Criterion — some of our main findings. Since the c haracterization of separabilit y in terms of the pro jective norms crucially dep ends on the dimension of the Hilbert spaces inv olved in the bipartition H ⊗ J , this summary should help the reader to ha ve a synopsis of the v arious (essentially three) p ossible cases: dim( H ⊗ J ) < ∞ , the case where precisely one of the Hilb ert spaces H and J is infinite-dimensional, and dim( H ) = dim( J ) = ∞ . Finally , in Section 6, we in tro duce the entanglemen t function and study its main prop erties. The pap er is addressed to a readership from div erse bac kgrounds, that ideally may include both ph ysicists and mathematicians. Therefore, throughout the pap er, the reader will find some claims — we refer to as facts , b ecause they are mainly well known — that are particularly relev ant for our purp oses, and hence deserve to b e prop erly emphasized. These facts, that are stated without pro ofs (that can b e found in standard references, or easily reconstructed by the reader), ma y b e more or less familiar to physicists, or to mathematicians, depending on their backgrounds. F or the sak e of conciseness, we will also omit the pro ofs of all those results that can reasonably b e prov ed b y the reader, without muc h effort, using the information provided in the pap er. 2 Preliminaries and main notations In this section, we will establish our main conv entions and notations, and recall some preliminary facts. All technical facts recalled in the follo wing can b e found by the reader in standard references on operator theory and functional-analytic asp ects of quantum mechanics [8–11, 35–41], the theory of op erator algebras [11,42–44], Banac h space theory [39, 40, 45–49], con v ex analysis [50, 51], measure theory [52–54] (esp ecially on metric and Polish spaces [55–57], and the theory of standard Borel 5 spaces [56–58]). Everywhere in the pap er, w e will denote by R ∗ , R + , R + ∗ , R − ∗ the sets of nonzero, non-negativ e, strictly p ositiv e and strictly negativ e real num b ers, resp ectiv ely . 2.1 Op erators on Hilb ert spaces Coheren tly with the notation adopted in the in tro duction, we will denote b y H — or b y J , K — a separable complex Hilb ert space. The scalar pro duct ⟨ · , · ⟩ ≡ ⟨ · , · ⟩ H on H is supp osed to b e conjugate-linear in its first argument. Given a subspace V of H , we denote by V ⊥ its ortho gonal c omplement , i.e., the closed subspace { ϕ ∈ H : ⟨ ϕ, ψ ⟩ = 0 , ∀ ψ ∈ V } . W e adopt the symbol I for the identity op er ator on H (or J , K ). W e denote b y B 1 ( H ) the (separable) c omplex Banach space of all tr ac e class op er ators on H and by B 1 ( H ) R the r e al Banac h space of all selfadjoint trace class op erators. W e denote by tr( · ) : B 1 ( H ) → C the tr ac e functional, and by ∥ · ∥ 1 ≡ ∥ · ∥ tr the tr ac e norm (on B 1 ( H ) or B 1 ( H ) R ); i.e., for every S ∈ B 1 ( H ), ∥ S ∥ 1 : = tr( | S | ) = ∥ S ∗ ∥ 1 , with | S | ∈ B 1 ( H ) denoting the absolute value of S ( | S | ≡ √ S ∗ S ). The set B 1 ( H ) + ⊂ B 1 ( H ) R of all p ositive trace class op erators is a norm-closed con vex subset of B 1 ( H ). The Banach space dual B 1 ( H ) ∗ of B 1 ( H ) is identified — via the pairing B 1 ( H ) × B ( H ) ∋ ( S, B ) 7→ tr( S B ) — with B ( H ), i.e., the complex Banach space of all b ounde d op er ators on H , endo wed with the standard op er ator norm ∥ · ∥ ∞ . The conv ex set of all p ositive b ounded op erators will b e denoted b y B ( H ) + . The trace class B 1 ( H ) is a two-side d ∗ -ide al in B ( H ) and, for any B ∈ B ( H ) and S ∈ B 1 ( H ), w e hav e that ∥ B S ∥ 1 , ∥ S B ∥ 1 ≤ ∥ B ∥ ∞ ∥ S ∥ 1 ; moreo ver, for every S ∈ B 1 ( H ) ⊂ B ( H ), ∥ S ∥ ∞ ≤ ∥ S ∥ 1 . Let us collect a few more facts concerning these norms and the Banach space B 1 ( H ). F act 2.1. F or the op erator norm ∥ · ∥ ∞ on B ( H ), w e ha ve: ∥ B ∥ ∞ : = sup {∥ B ψ ∥ : ∥ ψ ∥ = 1 } = sup {|⟨ ϕ, B ψ ⟩| : ∥ ϕ ∥ = ∥ ψ ∥ = 1 } , ∀ B ∈ B ( H ) . (2) In particular, if B ∈ B ( H ) R , then ∥ B ∥ ∞ = sup {|⟨ ψ , B ψ ⟩| : ∥ ψ ∥ = 1 } . F act 2.2. The trace norm ∥ · ∥ 1 is submultiplic ative ; i.e., for an y S, T ∈ B 1 ( H ), ∥ S T ∥ 1 ≤ ∥ S ∥ 1 ∥ T ∥ 1 . F act 2.3. Ev ery elemen t S of B 1 ( H ) can be expressed as a linear com bination of (at most) four p ositiv e trace class operators S 1 , . . . , S 4 ∈ B 1 ( H ) + : S = S 1 − S 2 + i ( S 3 − S 4 ), where, for S ∈ B 1 ( H ) R , S 3 − S 4 = 0. In particular, the op erators S 1 , . . . , S 4 ∈ B 1 ( H ) + can b e taken of the form S 1 = 1 4 ( | S + S ∗ | + S + S ∗ ) , S 2 = 1 4 ( | S + S ∗ | − S − S ∗ ) , (3) S 3 = 1 4 ( | S − S ∗ | − i ( S − S ∗ )) , S 4 = 1 4 ( | S − S ∗ | + i ( S − S ∗ )) , (4) and, in this case, w e hav e that S 1 S 2 = S 2 S 1 = 0 = S 3 S 4 = S 4 S 3 . Thus, if S = S 1 − S 2 ∈ B 1 ( H ) R , then, with suc h a c hoice of the positive trace class op erators S 1 and S 2 , we ha ve that | S | = S 1 + S 2 , ∥ S ∥ 1 : = tr( | S | ) = tr( S 1 ) + tr( S 2 ) and S 1 S 2 = S 2 S 1 = 0. F act 2.4. Every trace class op erator S ∈ B 1 ( H ) admits a decomp osition of the form S = X n s n | η n ⟩ ⟨ ϕ n | , ( | η n ⟩ ⟨ ϕ n | ) ψ : = ⟨ ϕ n , ψ ⟩ η n , ψ ∈ H , (5) where s n ≥ 0, for every n , and { η n } , { ϕ n } are orthonormal systems in H ; if dim( H ) = ∞ , the sum — whenever not finite — is absolutely con vergen t wrt the trace norm ∥ · ∥ 1 , and, in fact, P n s n = tr( | S | ) = ∥ S ∥ 1 . 6 The set of all trace class op erators on H admitting a finite decomp osition of the form (5) is a dense linear subspace of B 1 ( H ), that coincides with the set F ( H ) of finite r ank op er ators on H , and F ( H ) = span {| η ⟩ ⟨ ϕ | : η , ϕ ∈ H} ; see, e.g., Chapter 6 of [35]. F ( H ) is a ∥ · ∥ ∞ -dense linear subspace of the Banach space C ( H ) of c omp act op er ators on H ; endow ed with the op erator norm, C ( H ) is a C ∗ -subalgebra of B ( H ) [42]. R emark 2.1 (The singular v alue decomp osition) . By F act 2.4, every S ∈ B 1 ( H ), with S  = 0, admits a singular value de c omp osition (SVD) of the form S = P k s k | η k ⟩ ⟨ ϕ k | , where s k > 0, for ev ery v alue of the index k , and { η k } , { ϕ k } are orthonormal systems in H ; the sum, whenever not finite, is absolutely conv ergent wrt the trace norm, and P k s k = tr( | S | ) = ∥ S ∥ 1 (whereas, tr( S ) = P k s k ⟨ ϕ k , η k ⟩ ). The set { s k } of all singular values of S coincides with the set of the nonzer o eigenvalues of the p ositive op er ator | S |  = 0, each eigenv alue being coun ted according to its multiplicit y . W e stress that, compared to decomp osition (5), the SVD is a she er de c omp osition of the trace class op erator S  = 0; i.e., it do es not c ontain any zer o term . F act 2.5. The linear spaces F ( H ), C ( H ) of finite rank op erators and of compact op erators on H — F ( H ) ⊂ C ( H ) ⊂ B ( H ) = B 1 ( H ) ∗ — are norming subsp ac es of B ( H ) = B 1 ( H ) ∗ for B 1 ( H ); i.e., for every S ∈ B 1 ( H ), ∥ S ∥ 1 = sup {| tr( S F ) | : F ∈ F ( H ) , ∥ F ∥ ∞ = 1 } = sup {| tr( S K ) | : K ∈ C ( H ) , ∥ K ∥ ∞ = 1 } . (6) The set of density op er ators on H — the unit trace, p ositive trace class op erators, here denoted b y D ( H ) — is a norm-closed conv ex s ubset of B 1 ( H ), the in tersection of B 1 ( H ) + with the closed set of unit trace op erators. W e will iden tify D ( H ) with the set of all the (normal, or σ -additive) states of a quan tum system with Hilb ert space H [6, 7, 10, 11]. The conv ex set D ( H ) contains the set P ( H ) of all rank-one pro jections on H , i.e., the so-called pur e states. The dual space of B 1 ( H ) R — i.e., B ( H ) R the r e al Banach space of all selfadjoint bounded operators — will b e regarded as the space of (b ounded) quan tum observables . F act 2.6. F or ev ery B ∈ B ( H ) R , ∥ B ∥ ∞ = sup {| tr( P B ) | : P ∈ P ( H ) } = sup {| tr( D B ) | : D ∈ D ( H ) } . F act 2.7 (The standard top ology of D ( H ) [59]) . The w eak and the strong op erator top ologies on D ( H ) (inherited from the space of b ounded op erators B ( H )), as well as the top ologies induced on D ( H ) by the metrics asso ciated with the Schatten p -norms ∥ · ∥ p , 1 ≤ p ≤ ∞ , all coincide. This unique top ology will b e called the standar d top olo gy of D ( H ). 2.2 A few facts and notations concerning normed and Banach spaces A norm on a, real or complex, vector space V — in particular, a Banac h or a Hilb ert space — will b e usually denoted by the symbols ∥ · ∥ or | | | · | | | (with or without some complement, in order to further sp ecialize their meaning), and, whenev er no risk of confusion ma y o ccur, the normed space ( V , ∥ · ∥ ) will b e simply denoted by V . The sym b ols B ( V ), S ( V ) will denote, resp ectiv ely , the close d unit b al l and the unit spher e centered at the origin of the normed space V ; i.e., we put B ( V ) : = { x ∈ V : ∥ x ∥ ≤ 1 } and S ( V ) : = { x ∈ V : ∥ x ∥ = 1 } . W e say that ( V , | | | · | | | ) is a r enorming of ( V , ∥ · ∥ ) if the norms ∥ · ∥ and | | | · | | | are equiv alen t. The (contin uous) dual sp ac e of a normed space V — which is alwa ys a Banac h space — will b e denoted b y V ∗ ≡ ( V , ∥ · ∥ ) ∗ (unless another symbol b e more suitable for the o ccasion), and, for the time b eing, w e will denote by ∥ · ∥ V ∗ the norm of the Banach space V ∗ ; precisely , setting ∥ ξ ∥ V ∗ : = sup {∥ ξ ( x ) ∥ : x ∈ V , ∥ x ∥ ≤ 1 } , the norm ∥ · ∥ V ∗ is called the dual norm of ∥ · ∥ . If V is a Banac h space itself, then V ∗ is also called the Banach sp ac e dual of V . 7 If S is a linear subspace of a normed space V , then, for every b ounded functional ξ ∈ V ∗ , ξ 0 ≡ ξ | S ∈ S ∗ (i.e., the restriction ξ 0 of ξ to S is a b ounded functional), and ∥ ξ 0 ∥ S ∗ ≤ ∥ ξ ∥ V ∗ . Here, k eeping ξ 0 ∈ S ∗ fixed, the inequalit y is saturated b y some elemen t ξ of V ∗ suc h that ξ 0 = ξ | S . Indeed, conv ersely , by the Hahn-Banac h (norm-preserving, con tinuous extension) theorem — see, e.g., Theorem 1.9.6 of [46], or Corollary 2.3 of [47] — ev ery element ξ 0 of S ∗ admits a Hahn-Banach extension ξ ∈ V ∗ ; i.e., there is a functional ξ ∈ S ∗ suc h that ξ | S = ξ 0 and ∥ ξ ∥ V ∗ = ∥ ξ 0 ∥ S ∗ . An elemen t of S ∗ ma y admit more than one Hahn-Banac h extension. How ever, if S is a norm-dense linear subspace of V , then every elemen t of S ∗ will admit exactly one contin uous extension, so that S ∗ is isomorphic to the Banac h space V ∗ , where the isomorphism of Banach spaces is implemen ted b y the map V ∗ ∋ ξ 7→ ξ | S ∈ S ∗ , and w e will simply write S ∗ = V ∗ . The w ∗ -top olo gy (namely , the weak ∗ top ology) on the dual V ∗ of a normed space V — also called the σ ( V ∗ , V ) -top olo gy — is defined as the initial top olo gy (or inv erse image top ology) induced b y the family of maps { V ∗ ∋ ξ 7→ ξ ( x ) ∈ C : x ∈ V } ; i.e., the weak est top ology on V ∗ suc h that all these maps are contin uous [37–40, 46–48]. Equiv alen tly , the σ ( V ∗ , V )-top ology can b e defined as the initial top ology induced by the family of semi-norms { V ∗ ∋ ξ 7→ | ξ ( x ) | ∈ R + : x ∈ V } , and V ∗ is a lo cally con vex top ological vector space wrt this top ology (see Chapter 2 of [46] and Chapter 8 of [48]). The σ ( V ∗ , V )-top ology , by Theorem 2.6.2 of [46], is a completely regular — i.e., T ychonoff or T 3 1 2 — and locally con vex subtop ology of the w eak top ology (hence, of the norm top ology) of V ∗ , that is, of the σ ( V ∗ , V ∗∗ )-top ology . A net { ξ i } ⊂ V ∗ con verges to some ξ ∈ V ∗ wrt the σ ( V ∗ , V )-top ology iff lim i ξ i ( x ) = ξ ( x ), for all x ∈ V . F act 2.8. Let V 0 b e a norm-dense subset of the normed space V , or, also, a norm-dense subset of S ( V ). Then, a norm-b ounde d net { ξ i } ⊂ V ∗ con verges to ξ ∈ V ∗ wrt the σ ( V ∗ , V )-top ology iff lim i ξ i ( x 0 ) = ξ ( x 0 ), for all x 0 ∈ V 0 . F act 2.9. Let S b e a norm-dense linear subspace of the normed space V , and let B a norm- b ounde d subset of S ∗ = V ∗ (ev ery elemen t of S ∗ b eing iden tified with its contin uous extension to V ∗ ). Then, the relative top ology of B wrt the σ ( S ∗ , S )-top ology coincides with the relative top ology of B wrt the σ ( V ∗ , V )-top ology . Giv en norms ∥ · ∥ , | | | · | | | : V → R + on the v ector space V , w e will say that the norm ∥ · ∥ is dominate d — or majorize d — by the norm | | | · | | | if, for every x ∈ V , ∥ x ∥ ≤ | | | x | | | . F act 2.10. Let ∥ · ∥ , | | | · | | | : V → R + b e norms on the v ector space V , and suppose that the norm ∥ · ∥ is dominated b y the norm | | | · | | | . Then, the dual space V ∗ ≡ ( V ∗ , ∥ · ∥ ∗ ) = ( V , ∥ · ∥ ) ∗ of ( V , ∥ · ∥ ), with ∥ · ∥ ∗ ≡ ∥ · ∥ V ∗ , is a linear subspace of V ⊛ ≡  V ⊛ , | | | · | | | ⊛  = ( V , | | | · | | | ) ∗ , and the σ ( V ∗ , V )-top ology coincides with the subspace top ology of V ∗ ⊂ V ⊛ wrt the σ ( V ⊛ , V )-top ology of V ⊛ . In the follo wing, w e will deal with Banach spaces, denoted b y the symbols V , Z and S . Giv en a real or complex Banach space V , a countable — i.e., at most denumerable — and ordered set of v ectors { e i } i ∈I ⊂ V is called a Schauder b asis if, for ev ery x ∈ V , there is a unique set of scalars { c i ( x ) } i ∈I , such that x = P i ∈I c i ( x ) e i , where the sum, whenever not finite, con verges wrt the norm of V . Every Banac h space admitting a Sc hauder basis is sep ar able , b ecause, for every countable subset V 0 of V , the norm-closed linear span of V 0 is separable (see, e.g., Prop osition 1.12.1 of [46]). Assuming that the reader is familiar with the basic facts concerning the absolutely c onver gent series in a Banac h space (see, e.g., Chapter 1 of [46]), and the double sequences and series of real n umbers (see, e.g., Chapter 7 of [60]), we recall a few facts about the double series in a Banac h space V . W e say that a double se quenc e { x j l } ⊂ V is absolutely summable — or that the double series P j l x j l is absolutely c onver gent — if an y of the follo wing e quivalent c onditions is satisfied: 8 • ev ery ‘ro w series’ P l x j l is absolutely con vergen t — P l ∥ x j l ∥ < ∞ — and P j P l ∥ x j l ∥ < ∞ ; • ev ery ‘column series’ P j x j l is absolutely con vergen t and P l P j ∥ x j l ∥ < ∞ ; • the double (real) series P j l ∥ x j l ∥ conv erges ` a la Pringsheim ; • P n ∥ x j ( n ) l ( n ) ∥ < ∞ , for some bijection N ∋ n 7→ ( j ( n ) , l ( n )) ∈ N × N ; • P n ∥ x j ( n ) l ( n ) ∥ < ∞ , for every bijection N ∋ n 7→ ( j ( n ) , l ( n )) ∈ N × N . F act 2.11. Let the double sequence { x j l } ⊂ V b e absolutely summable. Then, the double series P j l x j l is conv ergent ` a la Pringsheim to some vector v ∈ V — in sym b ols, v = P j l x j l — namely , for ev ery ϵ > 0 there is some N ϵ ∈ N suc h that m, n > N ϵ = ⇒ ∥ v − P m j =1 P n l =1 x j l ∥ < ϵ . Moreov er, w e ha ve that v = X j l x j l = X j X l x j l = X l X j x j l = X n x j ( n ) l ( n ) , (7) for every bijection N ∋ n 7→ ( j ( n ) , l ( n )) ∈ N × N . The previous notions and results ab out double series extend, in a straightforw ard w a y , to multiple series in a Banach space. F rom this point onw ards, the standard norm of b ounded linear or bilinear maps betw een generic (real or complex) Banach spaces will b e usually denoted by ∥ · ∥ B , unless another symbol b e more appropriate. Clearly , this is a Banac h space norm itself, but using the sym b ol ∥ · ∥ B — rather than the bare symbol ∥ · ∥ (or ∥ · ∥ ∗ for the asso ciated dual norm) — is a wa y to emphasize the fact that w e are dealing precisely with the standard norm of a Banac h space of b ounde d line ar maps . The meaning, how ever, should b e alwa ys clear from the con text. E.g., B ( V ; S ) denotes the Banac h space of all b ounded linear maps from the Banac h space V in to the Banach space S , endow ed with the norm ∥ · ∥ B defined in the usual wa y . Analogously , giv en (real or complex) Banach spaces V , Z and S , and a b ounded bilinear map λ : V × Z → S , w e set: ∥ λ ∥ B : = sup {∥ λ ( x, y ) ∥ : x ∈ V , y ∈ Z , ∥ x ∥ , ∥ y ∥ ≤ 1 } , λ ∈ B ( V , Z ; S ) . (8) Clearly , here the symbol B ( V , Z ; S ) denotes the Banach space of all b ounded bilinear maps from V × Z in to S , endo w ed with the norm ∥ · ∥ B defined abov e. In particular, w e will consider the case where V = H , Z = J (Hilb ert spaces) and S = C , i.e., the complex Banach space B ( H , J ; C ) of all b ounded bilinear forms on H × J . W e sa y that a bilinear map λ : V × Z → S is a biline ar isometry if ∥ λ ( x, y ) ∥ = ∥ x ∥ ∥ y ∥ , for all x ∈ V and y ∈ Z . Clearly , if λ is a bilinear isometry , then it is b ounded and ∥ λ ∥ B = 1. Apart from the previously in tro duced operator norm ∥ · ∥ ∞ on B ( H ) ≡ B ( H ; H ) (Subsection 2.1), w e will sp ecialize the symbol ∥ · ∥ B only in the remark able case where all Banac h spaces inv olved are trace classes of Hilb ert space op erators; see b elo w. F urther notations, generalizing in a straigh tforward w ay the previously in tro duced symbols, are summarized in the following table (here, V , W and Z are real or complex v ector spaces): Sym b ol Meaning L 1 ( H ) ≡ B ( B 1 ( H )) ≡ B ( B 1 ( H ); B 1 ( H )) b ounded linear op erators on B 1 ( H ) F 1 ( H , J ) ≡ B ( B 1 ( H ) , B 1 ( J ); C ) b ounded bilinear forms on B 1 ( H ) × B 1 ( J ) L 1 ( H , J ; K ) ≡ B ( B 1 ( H ) , B 1 ( J ); B 1 ( K )) b ounded bil. maps from B 1 ( H ) × B 1 ( J ) in to B 1 ( K ) B ( B 1 ( H ) , B 1 ( H ); S ) b ounded bilinear maps from B 1 ( H ) × B 1 ( H ) into S L ( V ), L ( V , W ; C ) linear maps on V , bilinear forms on V × W L ( V , W ; Z ) bilinear maps from V × W into Z 9 The norm of the Banac h space L 1 ( H ) will denoted by ∥ · ∥ [1] (defined as the norm ∥ · ∥ B ). The norms of the Banach spaces of biline ar forms F 1 ( H , J ) and of biline ar maps L 1 ( H , J ; K ) will all b e denoted by ∥ · ∥ (1) . A (real or complex) Banach space S is said to ha ve the appr oximation pr op erty if, for ev ery compact subset K of S and ev ery ϵ > 0, there is a finite rank op erator F : S → S such that ∥ x − F x ∥ < ϵ , for all x ∈ K ; see, Chapter 4 of [18]. S is said to hav e the R adon-Niko d´ ym pr op erty if the Radon-Nik o d´ ym theorem holds for suitable S -v alued measures; see Chapter 5 of [18]. F act 2.12. The Banac h space B 1 ( H ) has b oth the approximation prop erty and the Radon-Nikod ´ ym prop ert y . 2.3 Linear spans, conv ex cones, con vex hulls A subset C of a real or complex v ector space V is called a c onvex c one if, for any p, q ∈ R + ∗ and v , w ∈ C , p v + q w ∈ C . A conv ex cone C is said to b e p ointe d if C ∩ ( − C ) = { 0 } . E.g., the set B 1 ( H ) + of all p ositive trace class op erators on H is a p oin ted con vex cone in B 1 ( H ); analogously , B ( H ) + is a p oin ted conv ex cone in B ( H ). Giv en a nonempty subset X of a complex Banac h space ( S , ∥ · ∥ ), w e will denote by co ( X ) the c onvex hul l of X , and by co ( X ) the close d c onvex hul l , i.e., the smallest norm-closed conv ex set con taining X . As is well kno wn, co ( X ) = cl (co ( X )) ≡ cl ∥ · ∥ (co ( X )). Moreov er, it turns out that co ( X ) = co  cl ∥ · ∥ ( X )  . (9) R emark 2.2 . Given a dual Banach space S ∗ , endow ed with the w ∗ -top ology , w e will also consider the w ∗ -closed con vex hull of a nonempty subset Y of S ∗ . Since S ∗ , endo wed with the w ∗ -top ology , is a top olo gic al ve ctor sp ac e [46–48], then, as it happ ens for the norm-closed conv ex h ull, we hav e that co w ∗ ( Y ) = w ∗ - cl (co ( Y )), i.e., the w ∗ -closed con vex hull of Y coincides with the w ∗ -closure of the conv ex hull of Y (see, e.g., Theorem 2.2.9 -(i) of [46]). Analogously , span ( X ) denotes the complex line ar sp an of X ⊂ S , and span ( X ) the close d line ar sp an , namely , the smallest norm-closed linear subspace of S containing X ; moreo ver, span ( X ) = cl (span ( X )), and span ( X ) = span (cl ( X )). F urther useful relations are the following: span ( X ) = co ( C X ); span (co ( X )) = span ( X ) = co (span ( X )), and hence span (co ( X )) = span (co ( X )) = span ( X ) = co (span ( X )) = co (span ( X )). F act 2.13. If C is a closed con vex subset of a complex Banac h space S , then every norm-con vergen t, coun tably infinite con vex com bination P ∞ n =1 p n x n of elements { x n } n ∈ N of C ( p n > 0, P ∞ n =1 p n = 1) is contained in C . Hence, giv en a subset X of S , every ∥ · ∥ -conv ergent, coun tably infinite conv ex com bination P ∞ n =1 p n x n of elements { x n } n ∈ N of X b elongs to co ( X ). F act 2.14. D ( H ) = co ( P ( H )). Recall that an elemen t x of a con vex subset C of a real or complex v ector space V (in particular, of a Banac h space S ) is said to b e an extr eme p oint of C if there do not exist y , z ∈ C , with y  = z , and t ∈ (0 , 1) such that x = ty + (1 − t ) z ; namely , if x do es not lie b etwe en any two distinct p oints of V . W e will denote the set of all extreme p oints of C b y ext( C ). F act 2.15. If V is a, real or complex, normed space, then the extreme p oin ts of its closed unit ball b elong to its unit sphere: ext( B ( V )) ⊂ S ( V ) : = { x ∈ V : ∥ x ∥ = 1 } . F act 2.16. The extreme p oin ts of the conv ex set D ( H ) form precisely the set P ( H ) of all rank-one pro jections on H (pure states); i.e., ext( D ( H )) = P ( H ). F act 2.17. Let X , Y b e nonempty con v ex subsets of V . Then, X ⊂ Y = ⇒ ext( Y ) ∩ X ⊂ ext( X ). 10 2.4 The algebraic tensor pro duct There are tw o main equiv alent approac hes to the definition of the algebraic tensor pro duct of t wo v ector spaces; see [18] and [20], resp ectiv ely (also see Chapter 4 of [20], for an analysis of v arious approac hes). In the approac h adopted, e.g., in [18] — i.e., the line ar-biline ar appr o ach which will no w briefly sk etched — the so-called universal pr op erty is a consequence of the definition rather than part of it (as it happ ens in the approach adopted in [20]). F or the sake of definiteness, we consider the case of complex vector spaces; the case of real vector spaces is analogous. Let V , W b e (complex) vector spaces, and let us denote b y V ′ , W ′ the algebr aic dual spaces of V and W , resp ectiv ely . W e will denote b y L ( V , W ; C ) the vector space of all bilinear forms on V × W . Giv en any pair of vectors v ∈ V and w ∈ W , we can define a linear functional v ˘ ⊗ w : L ( V , W ; C ) → C , by putting  v ˘ ⊗ w  ( ω ) : = ω ( v , w ), for all ω ∈ L ( V , W ; C ). Then, the (standard realization of the) algebr aic tensor pr o duct of V and W is the linear subspace V ˘ ⊗ W of L ( V , W ; C ) ′ — i.e., of the algebraic dual of L ( V , W ; C ) — defined as follows V ˘ ⊗ W : = span  v ˘ ⊗ w  ≡ span  v ˘ ⊗ w ∈ L ( V , W ; C ) ′ : v ∈ V , w ∈ W  , (10) together with the bilinear map ˘ θ : V × W ∋ ( v , w ) 7→ v ˘ ⊗ w ∈ V ˘ ⊗ W , the so-called (standard realization of the) natur al biline ar map on V × W . T aking into account that, for every c ∈ C , c ( v ˘ ⊗ w ) = ( c v ) ˘ ⊗ w , ev ery element of V ˘ ⊗ W can b e expressed as a finite sum P n k =1 v k ˘ ⊗ w k of elementary tensors . By the universal pr op erty of algebr aic tensor pr o ducts — see Prop osition 1.4 of [18] — for ev ery bilinear map θ ∈ L ( V , W ; Z ) from V × W into a complex v ector space Z , there is a unique linear map Θ : V ˘ ⊗ W → Z such that θ = Θ ◦ ˘ θ (universalit y relation) — i.e., θ ( v , w ) = Θ  v ˘ ⊗ w  , ∀ v ∈ V , ∀ w ∈ W (11) — and, moreov er, the one-to-one corresp ondence θ ↔ Θ is an isomorphism b et ween the v ector spaces L ( V , W ; Z ) and L  V ˘ ⊗ W ; Z  . If span (ran ( θ )) = Z and, moreo ver, Θ is a linear isomorphism of V ˘ ⊗ W on to Z — i.e., since, b y (11), ran (Θ) = span (ran ( θ )) = Z , if Θ is injective — then θ ∈ L ( V , W ; Z ) automatically enjo ys the crucial universal pr op erty ; i.e., giv en any bilinear map λ : V × W → e Z (where e Z is a complex v ector space), there is a unique linear map Λ θ : Z → e Z such that λ = Λ θ ◦ θ . Precisely , denoting by Λ : V ˘ ⊗ W → e Z the unique linear map such that λ ( v , w ) = Λ  v ˘ ⊗ w  , for all v ∈ V and w ∈ W , we hav e that Λ θ = Λ ◦ Θ − 1 ◦ θ . Con versely , if θ ∈ L ( V , W ; Z ) enjoys the univ ersal property (including the uniqueness condition of the linear map asso ciated with a bilinear map on V × W ), then span (ran ( θ )) = Z and, moreo v er, Θ is a linear isomorphism of V ˘ ⊗ W onto Z . In fact, in this case, we ha ve that ˘ θ = ˘ Θ ◦ θ = ˘ Θ ◦ Θ ◦ ˘ θ and θ = Θ ◦ ˘ θ = Θ ◦ ˘ Θ ◦ θ , (12) where Θ : V ˘ ⊗ W → Z and ˘ Θ : Z → V ˘ ⊗ W are uniquely determined linear maps such that dom(Θ) = V ˘ ⊗ W = span  ran  ˘ θ  = ran  ˘ Θ  , dom  ˘ Θ  = Z ⊃ span (ran ( θ )) = ran (Θ) . (13) Here, by the uniqueness of the linear map ˘ Θ, we conclude that, actually , Z = span (ran ( θ )). Hence, Θ is a linear isomorphism of V ˘ ⊗ W onto Z and ˘ Θ = Θ − 1 . Therefore, it is natural to set the following: Definition 2.1. If θ ∈ L ( V , W ; Z ) is such that span (ran ( θ )) = Z and, moreov er, the unique linear map Θ : V ˘ ⊗ W → Z such that θ = Θ ◦ ˘ θ is injectiv e (hence, a linear isomorphism of V ˘ ⊗ W on to Z ), we sa y that the pair ( Z , θ ∈ L ( V , W ; Z )) is (a r e alization of ) the algebr aic tensor pr o duct of V and W , and w e will usually put v ⊗ w ≡ θ ( v , w ) and V ˘ ⊗ W ≡ Z . The bilinear map θ : V × W ∋ ( v , w ) 7→ v ⊗ w ∈ Z ≡ V ˘ ⊗ W (14) 11 is called (a r e alization of ) the natur al biline ar map on V × W . R emark 2.3 . W e will not adhere to a standard usage, in the mathematical literature, of denoting b y V ⊗ W the algebraic tensor pro duct space, b ecause, when considering the case where V and W are Banac h or Hilb ert spaces, this notation will b e used for indicating the completion of (any realization of ) the algebraic tensor product space V ˘ ⊗ W wrt to some natural norm. This is the case, e.g., of the tensor pro duct H ⊗ J of tw o Hilb ert spaces. Another relev ant fact, for our purp oses, concerns a suitable realization of the algebraic tensor pro duct of spaces of linear op erators. F act 2.18. Let H , J and K b e complex vector spaces, and supp ose that the pair  K ≡ H ˘ ⊗ J , H × J ∋ ( ϕ, ψ ) 7→ ϑ ( ϕ, ψ ) ≡ ϕ ⊗ ψ ∈ K  , (15) — with K ≡ H ˘ ⊗ J = span (ran ( ϑ )) = span  ϕ ⊗ ψ : ϕ ∈ H , ψ ∈ J  — b e a realization of the algebraic tensor pro duct of H and J . Then, the follo wing claims hold true: • F or every pair ( A ∈ L ( H ) , B ∈ L ( J )), there is a unique linear map A ⊗ B ∈ L ( K ) suc h that  A ⊗ B  ( ϕ ⊗ ψ ) = ( Aϕ ) ⊗ ( B ψ ), for all ϕ ∈ H and ψ ∈ J . • F or ev ery pair ( V ⊂ L ( H ) , W ⊂ L ( J )) of linear subspaces V , W of L ( H ) and L ( J ), the pair  Z , θ : V × W ∋ ( A, B ) 7→ A ⊗ B ∈ Z  (16) — with Z = span (ran ( θ )) = span { A ⊗ B ∈ L ( K ) : A ∈ V ⊂ L ( H ) , B ∈ W ⊂ L ( J ) } — is a realization of the algebraic tensor pro duct of V and W . 2.5 Probabilit y measures and standard Borel spaces Let ( X , B ( X ) , µ ) b e a Bor el pr ob ability me asur e sp ac e ; i.e., let X b e a topological space, B ( X ) the asso ciated Borel σ -algebra and µ a probability measure on B ( X ). W e will denote by P ( X ) the collection of all suc h measures. Consider the set N µ : =  N ∈ B ( X ) : µ  N  = 0  of all µ -nul l subsets in B ( X ). Then, the set B ( X ) µ ⊃ B ( X ) defined by B ( X ) µ : =  G = E ∪ F : E ∈ B ( X ) , F ⊂ N for some N ∈ N µ  is a σ -algebra, and ev ery element G ∈ B ( X ) µ is said to b e a µ -me asur able set. Moreov er, there is a unique extension of µ to a probability measure µ on B ( X ) µ , which is called the c ompletion of µ ; see Theorem 1.9 of [52] (also see [56], Chapter I I, Section 17.A). W e sa y that a µ -measurable set G ∈ B ( X ) µ is of ful l me asur e if µ ( G ) = 1. The supp ort of the measure µ ∈ P ( X ) is defined as the set supp ( µ ) : = { ξ ∈ X : µ ( O ) > 0, for every open neigh b orho o d O of ξ } . (17) Otherwise stated, supp ( µ ) is the complement of the union of all p oin ts admitting a µ -null open neigh b orho od. It is a close d subset of the top ological space X , as it is clear from the following: F act 2.19. The set supp ( µ ) c = X \ supp ( µ ) is the union of all µ -n ull open subsets of X . Therefore, supp ( µ ) is the in tersection of all closed subsets of X of full measure. R emark 2.4 . Let δ ξ ∈ P ( X ) b e the Dirac (p oin t mass) me asure concentrated at ξ ∈ X . If the top ological space X is T 2 (Hausdorff ), then, for ev ery other p oin t ξ ′  = ξ in X , there are op en neigh b orho ods O ξ , O ξ ′ of ξ and ξ ′ , respectively , such that O ξ ∩ O ξ ′ = ∅ ; hence, δ ξ ( O ξ ) = 1 and δ ξ ( O ξ ′ ) = 0 ( O ξ ′ is contained in the complement of O ξ ), so that supp ( δ ξ ) = { ξ } . Moreo ver, as is w ell known, a top ological space X is T 1 (F rech ´ et) iff ev ery singleton { ξ } ⊂ X is closed. Therefore, if, for every ξ ∈ X , there exists a Borel probability measure µ ξ suc h that supp ( µ ξ ) = { ξ } , then the top ological space X m ust b e T 1 . 12 Note that supp ( µ ) ma y be empt y . How ever, this will not b e our case, because w e will deal with se c ond c ountable top ological spaces. F act 2.20. Let the top ological space X b e second coun table. Then, the probabilit y measure µ is not lo c al ly me asur e zer o ; i.e., supp ( µ )  = ∅ . In fact, µ (supp ( µ ) c ) = 0, so that supp ( µ )  = ∅ and µ (supp ( µ )) = 1. Precisely , in this case supp ( µ ) is the smallest closed subset of X of full measure. In particular, if supp ( µ ) is a singleton — i.e., if supp ( µ ) = { ξ } , for some ξ ∈ X — then µ is the Dirac measure δ ξ at ξ . Recall that a Polish sp ac e P is a separable, completely metrizable top ological space (see [56], Chapter I, Section 3); namely , a topological space homeomorphic to a complete metric space ha ving a countable dense subset. Note that, in particular, a P olish space is second countable. W e will consider probability measures on the Borel space ( P , B ( P )), where B ( P ) is the natural Borel structure on P (i.e., the Borel structure asso ciated with the norm top ology). More generally , a standar d Bor el sp ac e — see [56], Chapter I I, Section 12.B, Chapter 3 of [57], or [58], Chapter V, Section 1 — is a measurable space ( X , A ) that is isomorphic to ( P , B ( P )), for some Polish space P ; equiv alently , ( X , A ) is a standard Borel space if there is a Polish top ology on X and A = B ( X ) is the Borel σ -algebra asso ciated with this top ology . E.g., consider a Borel space of the form  X ⊂ P , B P ( X ) : = {E ∩ X : E ∈ B ( P ) }  , where P is a P olish space. Note that B P ( X ) is precisely the Borel σ -algebra asso ciated with the subspace top ology of X as a subset of P . It turns out that  X , B ( X ) ≡ B P ( X )  is a standard Borel space iff X is a Bor el subset of P ([58], Chapter V, Section 1); actually , every standard Borel space is, up to isomorphisms, of this form (this is the so-called R amsey-Mackey the or em ; see Theorem 3.3.22 of [57], where a standard Borel space is defined precisely as a measurable space isomorphic to a Borel subset of a Polish space). The motiv ation for using the term standar d rests on the following result: F act 2.21. A standar d Borel space ( X , A = B ( X )) is completely characterized — up to Borel isomorphisms — by its cardinality; i.e., tw o standard Borel spaces are isomorphic iff they ha ve the same cardinality (Kuratowski’s theorem). In particular, the set X is finite, coun table or has the p o w er of the con tinuum; in the latter case, it is Borel isomorphic to the unit in terv al [0 , 1] (endo w ed with the Borel σ -algebra asso ciated with the standard top ology). If ( X , B ( X )), ( Y , B ( Y )) are standard Borel spaces and f : X → Y is an injectiv e Borel map, then, f maps Borel sets to Borel sets; i.e., for every E ∈ B ( X ), f ( E ) ∈ B ( Y ). Therefore, the mapping X ∋ ξ 7→ f ( ξ ) ∈ f ( X ) — where f ( X ) is endow ed with the Borel σ -algebra B Y ( f ( X )) asso ciated with the subspace top ology — is a Borel isomorphism. 3 T o ols: tensor pro ducts, cross trace classes and cross states W e will now recall some salient facts ab out the tensor pro ducts of Hilb ert spaces and of Hilb ert space op erators [9, 11, 20]. Next, we will introduce the pro jectiv e tensor pro duct [15–20] of trace classes, and the related notions of cross trace class operator and of cross state of a bipartite quan tum system. The notion of cross state will allo w us to obtain, in the subsequent sections, remark able c haracterizations of separability and entanglemen t for a bipartite quantum system. 3.1 In tro ducing the relev ant tensor pro ducts: Hilb ert spaces and op erators Let us first recall that the the tensor product H ⊗ J of tw o separable complex Hilb ert spaces H and J can b e introduced as follows [11, 20]. F or every pair of vectors ϕ ∈ H and ψ ∈ J , we can 13 define a bilinear form ϕ ⊗ ψ on H ∗ × J ∗ , where H ∗ , J ∗ are the (top ological) dual sp ac es of H and J , resp ectiv ely; i.e., we set ϕ ⊗ ψ : H ∗ × J ∗ ∋  ⟨ η , · ⟩ H , ⟨ χ, · ⟩ J  7→ ⟨ η , ϕ ⟩ H ⟨ χ, ψ ⟩ J ∈ C , η ∈ H , χ ∈ J . (18) Clearly , the Hilb ert space H (or J ) is iden tified with its dual space H ∗ (resp ectiv ely , with J ∗ ) via the usual an tilinear isomorphism J H : H ∋ η 7→ η ∗ ∈ H ∗ , where η ∗ ≡ ⟨ η , · ⟩ H : H → C (resp ectiv ely , J J : J ∋ χ 7→ χ ∗ ≡ ⟨ χ, · ⟩ J ∈ J ∗ ), and w e ha ve:  ϕ ⊗ ψ  ( η ∗ , χ ∗ ) = η ∗ ( ϕ ) χ ∗ ( ψ ) = ⟨ η , ϕ ⟩ H ⟨ χ, ψ ⟩ J . Note that putting span  ϕ ⊗ ψ  ≡ span  ϕ ⊗ ψ ∈ L ( H ∗ , J ∗ ; C ) : ϕ ∈ H , ψ ∈ J  , where L ( H ∗ , J ∗ ; C ) is the complex vector space of all bilinear forms on H ∗ × J ∗ , the map ϑ : H × J ∋ ( ϕ, ψ ) 7→ ϕ ⊗ ψ ∈ span  ϕ ⊗ ψ  ≡ span  ϕ ⊗ ψ ∈ L ( H ∗ , J ∗ ; C ) : ϕ ∈ H , ψ ∈ J  (19) is a (bounded) bilinear map. This observ ation allows us iden tify the pair  span  ϕ ⊗ ψ  , ϑ  with the algebr aic tensor pr o duct of the vector spaces H and J , as defined in the theory of tensor pro ducts of abstract vector spaces — in particular, normed and Banac h spaces — recall Subsection 2.4, and see, e.g., Chapter 1 of [18]. In fact, for any ϕ ∈ H and ψ ∈ J , let ϕ ˘ ⊗ ψ : L ( H , J ; C ) → C b e the linear form — on the complex vector space L ( H , J ; C ) of all bilinear forms on H × J — defined b y  ϕ ˘ ⊗ ψ  ( ω ) : = ω ( ϕ, ψ ) , ∀ ω ∈ L ( H , J ; C ) . (20) Then, the pair formed b y the (complex) linear span span  ϕ ˘ ⊗ ψ  ≡ span  ϕ ˘ ⊗ ψ ∈ L ( H , J ; C ) ′ : ϕ ∈ H , ψ ∈ J  (21) and by the bilinear map ˘ ϑ : H × J ∋ ( ϕ, ψ ) 7→ ϕ ˘ ⊗ ψ ∈ span  ϕ ˘ ⊗ ψ  is the standard realization of the algebraic tensor pro duct of H and J . Then, recalling Definition 2.1, we ha ve: Prop osition 3.1. The p air  span  ϕ ⊗ ψ  , ϑ : H × J → span  ϕ ⊗ ψ  is a r e alization of the algebr aic tensor pr o duct of H and J . In fact, the mapping  ϕ ˘ ⊗ ψ : ϕ ∈ H , ψ ∈ J  ∋ η ˘ ⊗ χ 7→ η ⊗ χ ∈  ϕ ⊗ ψ : ϕ ∈ H , ψ ∈ J  (22) extends to a line ar isomorphism Θ : span  ran  ˘ ϑ  = span  ϕ ˘ ⊗ ψ  → span  ϕ ⊗ ψ  = span  ran ( ϑ )  that satisfies the universality r elation ϑ = Θ ◦ ˘ ϑ . By Prop osition 3.1, we can define the algebraic tensor pro duct of the Hilb ert spaces H and J as the pair  H ˘ ⊗ J : = span  ϕ ⊗ ψ ∈ L ( H ∗ , J ∗ ; C ) : ϕ ∈ H , ψ ∈ J  , ϑ : H × J → H ˘ ⊗ J  . (23) A t this p oin t, the complex vector space H ˘ ⊗ J can b e endow ed with a (well defined) natural scalar pro duct b y setting ⟨ ϕ 1 ⊗ ψ 1 , ϕ 2 ⊗ ψ 2 ⟩ H ˘ ⊗ J = ⟨ ϕ 1 , ϕ 2 ⟩ H ⟨ ψ 1 , ψ 2 ⟩ J , and then extending the mapping ⟨ · , · ⟩ H ˘ ⊗ J : ( H ˘ ⊗ J ) × ( H ˘ ⊗ J ) → C — by antilinearit y in the first argumen t and b y linearit y in the second one — so obtaining a sesquilinear form on H ˘ ⊗ J . Finally , the tensor pro duct Hilb ert space H ⊗ J ≡ ( H ⊗ J , ⟨ · , · ⟩ H⊗J ) is obtained as the Hilb ert space completion of this space; see Section 10.2.1 of [11] (or, Section 9.2 of [20]) for all technical details. Clearly , the application ϑ : H × J ∋ ( ϕ, ψ ) 7→ ϕ ⊗ ψ ∈ H ⊗ J is a b ounded bilinear map — whic h, with a slight abuse, we denote b y the same sym b ol adopted for the map (19) — and can be regarded, again with a slight abuse, as a (realization of ) the natural bilinear map on H × J . 14 R emark 3.1 . An alternativ e (yet equiv alent) approach to the tensor pro duct of Hilb ert spaces — relying on the introduction of suitable equiv alence classes in the Cartesian pro duct of vector spaces (which yields a definition of the algebraic tensor pro duct of these spaces) — is discussed in Chapter I I of [9]. F urther equiv alent approaches are described in Section 2.9 of [8] and in Section 2.6 of [43]. F act 3.1 (The Sc hmidt decomposition of a bipartite v ector) . Every nonzer o vector a ∈ H ⊗ J can b e expressed as a sum (norm-con vergen t, if infinite) of the form a = s X n =1 a n ( ϕ n ⊗ ψ n ) , 1 ≤ s ≡ srank ( a ) ≤ M ≡ min { dim( H ) , dim( J ) } ≤ ∞ , (24) where { ϕ n } s n =1 , { ψ n } s n =1 are orthonormal systems in the Hilb ert spaces H and J , resp ectively , and { a n } s n =1 ≡ scfs ( a ) is a set of strictly p ositive r e al numb ers — the so-called Schmidt c o efficients of a — that is uniquely determined when arranged in a weakly decreasing order a 1 ≥ a 2 ≥ · · · . In particular, the cardinalit y s ≡ srank ( a ) ≤ min { dim( H ) , dim( J ) } of the set scfs ( a ) — the so-called Schmidt r ank of a — is uniquely determined. R emark 3.2 . By F act 3.1, e v ery vector a ∈ H ⊗ J admits an expansion of the form a = M X m =1 a m ( ϕ m ⊗ ψ m ) , M ≡ min { dim( H ) , dim( J ) } ≤ ∞ , (25) where a m ≥ 0, and { ϕ m } M m =1 , { ψ m } M m =1 are orthonormal systems in H and J , resp ectiv ely . W e will call an y expansion of the form (25) an extende d Schmidt de c omp osition of the vector a . A t this p oin t, the tensor pro duct of t wo b ounded op erators A ∈ B ( H ), B ∈ B ( J ) can b e defined in the standard wa y (see Section 10.2.2 of [11], Section 9.2 of [20] and Section 6.3 of [42]), i.e., A ⊗ B is the unique b ounded linear op erator on the Hilb ert space ( H ⊗ J , ⟨ · , · ⟩ H⊗J ) suc h that  A ⊗ B  ( ϕ ⊗ ψ ) = ( Aϕ ) ⊗ ( B ψ ), for every pair of v ectors ϕ ∈ H and ψ ∈ J , and — denoting b y ∥ · ∥ ⊗ ∞ the norm of the Banac h space B ( H ⊗ J ) — ∥ A ⊗ B ∥ ⊗ ∞ = ∥ A ∥ ∞ ∥ B ∥ ∞ (the op erator norm ∥ · ∥ ⊗ ∞ is a cr oss norm ); moreo ver, ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ and ( A ⊗ B )( E ⊗ F ) = ( AE ) ⊗ ( B F ). R emark 3.3 . Regarding B ( H ), B ( J ) as linear subspaces of L ( H ) and L ( J ), resp ectively , then, b y F act 2.18, putting B ( H ) ˘ ⊗ B ( J ) : = span { A ⊗ B : A ∈ B ( H ) , B ∈ B ( J ) } , (26) the pair  B ( H ) ˘ ⊗ B ( J ) , B ( H ) × B ( J ) ∋ ( A, B ) 7→ A ⊗ B ∈ B ( H ) ˘ ⊗ B ( J )) (27) is a realization of the algebraic tensor pro duct of B ( H ) and B ( J ). Here, with a sligh t abuse, w e are iden tifying the linear op erator A ⊗ B ∈ L ( H ) ˘ ⊗ L ( J ) — uniquely defined on the algebraic tensor pro duct H ˘ ⊗ J , according to the first claim in F act 2.18 — with its contin uous extension to H ⊗ J (see Section 9.2 of [20], and in particular Theorem 9.8 therein, for a more precise, but also more cum b ersome, notation). By the second claim in F act 2.18, the algebraic tensor product of an y pair of linear subspaces of L ( H ) and L ( J ), resp ectiv ely , can b e realized analogously . Let us now fo cus on the tensor pro duct of trace class op erators. By the last assertion in Remark 3.3, setting B 1 ( H ) ˘ ⊗ B 1 ( J ) : = span { S ⊗ T : S ∈ B 1 ( H ) , T ∈ B 1 ( J ) } , (28) 15 the pair  B 1 ( H ) ˘ ⊗ B 1 ( J ) , B 1 ( H ) × B 1 ( J ) ∋ ( S, T ) 7→ S ⊗ T ∈ B 1 ( H ) ˘ ⊗ B 1 ( J )  is (a realization of ) the algebr aic tensor pr o duct of the trace classes B 1 ( H ) and B 1 ( J ). With a standard abuse, in the following w e will usually denote the algebraic tensor pro duct of B 1 ( H ) and B 1 ( J ) b y B 1 ( H ) ˘ ⊗ B 1 ( J ) tout c ourt ; i.e., by the sym b ol of the tensor pro duct space. Analogous considerations will b e understo o d for the algebraic tensor pro duct B 1 ( H ) R ˘ ⊗ B 1 ( J ) R : = span R { S ⊗ T : S ∈ B 1 ( H ) R , T ∈ B 1 ( J ) R } (29) of the r e al Banach spaces B 1 ( H ) R and B 1 ( J ) R ( selfadjoint trace class op erators). F act 3.2. If S ∈ B 1 ( H ) and T ∈ B 1 ( J ), then S ⊗ T ∈ B 1 ( H ⊗ J ) — hence, B 1 ( H ) ˘ ⊗ B 1 ( J ) is a linear subspace of B 1 ( H ⊗ J ) — and tr( S ⊗ T ) = tr( S ) tr( T ); moreo ver, | S ⊗ T | = | S | ⊗ | T | so that the trace norm ∥ · ∥ ⊗ 1 of the Banac h space B 1 ( H ⊗ J ) is a cross norm, namely , ∥ S ⊗ T ∥ ⊗ 1 : = tr( | S ⊗ T | ) = tr( | S | ⊗ | T | ) = ∥ S ∥ 1 ∥ T ∥ 1 , ∀ S ∈ B 1 ( H ) , ∀ T ∈ B 1 ( J ) . (30) F act 3.3. Given trace class op erators X ∈ B 1 ( H ) and Y ∈ B 1 ( J ), if X ⊗ Y ∈ B 1 ( H ⊗ J ) + , then X ⊗ Y = S ⊗ T , for some S ∈ B 1 ( H ) + and T ∈ B 1 ( J ) + . F act 3.4. Giv en nonzero p ositiv e trace class op erators S, X ∈ B 1 ( H ) + and T , Y ∈ B 1 ( J ) + , we ha ve: S ⊗ T = X ⊗ Y iff S = r X and T = r − 1 Y , where r = tr( Y ) / tr( T ) = tr( S ) / tr( X ) > 0. Then, giv en an y densit y op erators ρ, σ ∈ D ( H ) and τ , ω ∈ D ( J ), ρ ⊗ σ = τ ⊗ ω iff ρ = τ and σ = ω . As previously noted, since, for S ∈ B 1 ( H ) and T ∈ B 1 ( J ), the bounded op erator S ⊗ T is contained in the trace class B 1 ( H ⊗ J ) of the Hilb ert space H ⊗ J , then B 1 ( H ) ˘ ⊗ B 1 ( J ) is (iden tified with) a linear subspace of B 1 ( H ⊗ J ), and then can b e endow ed with the norm ∥ · ∥ ⊗ 1 of this space, i.e., ∥ P k S k ⊗ T k ∥ ⊗ 1 : = tr( | P k S k ⊗ T k | ); namely , with the natur al or sp atial norm of the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ) (these terms refer to the fact that this is a restriction of the norm ∥ · ∥ ⊗ 1 of the natur al ambient sp ac e B 1 ( H ⊗ J ) con taining B 1 ( H ) ˘ ⊗ B 1 ( J )). Clearly , for S ∈ B 1 ( H ) R and T ∈ B 1 ( J ) R , S ⊗ T ∈ B 1 ( H ⊗ J ) R , and B 1 ( H ) R ˘ ⊗ B 1 ( J ) R is a linear subspace of B 1 ( H ⊗ J ) R . 3.2 The natural tensor pro duct of trace classes By F act 3.2, the norm ∥ · ∥ ⊗ 1 of the Banac h space B 1 ( H ⊗ J ) is a cr oss norm , and the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ) can b e endow ed with the natural restriction of this norm (the natural, or spatial, norm); i.e., ∥ S ⊗ T ∥ ⊗ 1 = ∥ S ∥ 1 ∥ T ∥ 1 , for all S ∈ B 1 ( H ) and T ∈ B 1 ( J ). W e now consider the Banach space completion of the normed space ( B 1 ( H ) ˘ ⊗ B 1 ( J ) , ∥ · ∥ ⊗ 1 ); namely , we put B 1 ( H ) ⊗ B 1 ( J ) : = span { S ⊗ T : S ∈ B 1 ( H ) , T ∈ B 1 ( J ) } = cl ∥ · ∥ ⊗ 1  B 1 ( H ) ˘ ⊗ B 1 ( J )  , (31) where span ≡ span ⊗ ≡ span ∥ · ∥ ⊗ 1 ( ∥ · ∥ ⊗ 1 -closed linear span). Clearly , B 1 ( H ) ⊗ B 1 ( J ) is a Banac h subspace of the the bip artite tr ac e class B 1 ( H ⊗ J ), i.e., with the trace class of the bipartite Hilb ert space H ⊗ J (but also see Theorem 3.1 b elow, for a more precise c haracterization). Analogously , w e can define the following subspace of the real Banach space B 1 ( H ⊗ J ) R : B 1 ( H ) R ⊗ B 1 ( J ) R : = span R { S ⊗ T : S ∈ B 1 ( H ) R , T ∈ B 1 ( J ) R } = cl ∥ · ∥ ⊗ 1  B 1 ( H ) R ˘ ⊗ B 1 ( J ) R  . (32) Definition 3.1. W e call the Banach space ( B 1 ( H ) ⊗ B 1 ( J ) , ∥ · ∥ ⊗ 1 ) defined by (31) the natur al — or sp atial — tensor product of the trace classes B 1 ( H ) and B 1 ( J ). Analogously , B 1 ( H ) R ⊗ B 1 ( J ) R , endo wed with (the restriction of ) the trace norm ∥ · ∥ ⊗ 1 , is called the natur al (or sp atial ) tensor pro duct of B 1 ( H ) R and B 1 ( J ) R . 16 Giv en any nonempt y subset Z of B 1 ( H ) ⊗ B 1 ( J ), coherently with the previously introduced notation, w e denote by span ( Z ) ≡ span ⊗ ( Z ) the ∥ · ∥ ⊗ 1 -closed linear span of Z ; i.e., the closed subspace cl ∥ · ∥ ⊗ 1 (span ( Z )) of B 1 ( H ) ⊗ B 1 ( J ). Note that the application θ : B 1 ( H ) × B 1 ( J ) ∋ ( S, T ) 7→ S ⊗ T ∈ B 1 ( H ⊗ J ) (33) is a b ounded bilinear map — i.e., θ ∈ L 1 ( H , J ; H ⊗ J ) ≡ B ( B 1 ( H ) , B 1 ( J ); B 1 ( H ⊗ J )) — the natur al biline ar map on B 1 ( H ) × B 1 ( J ); precisely , by relation (30), θ is a bilinear isometry — ∥ θ ( S, T ) ∥ ⊗ 1 = ∥ S ⊗ T ∥ ⊗ 1 = ∥ S ∥ 1 ∥ T ∥ 1 , for all ( S, T ) ∈ B 1 ( H ) × B 1 ( J ), so that ∥ θ ∥ (1) : = sup 0  = S ∈B 1 ( H ) 0  = T ∈B 1 ( J ) ∥ θ ( S, T ) ∥ ⊗ 1 ∥ S ∥ 1 ∥ T ∥ 1 = 1 . (34) Notation 3.1. Given an y pair X , Y of nonempt y subsets of B 1 ( H ) and B 1 ( J ), resp ectiv ely , in the follo wing w e set θ ( X , Y ) : = { θ ( S, T ) = S ⊗ T : S ∈ X , T ∈ Y } , (35) whereas we will r efr ain fr om denoting this set by X ⊗ Y (see Notation 3.2 b elo w). R emark 3.4 . Recall that b y a w ell-known result — see, e.g., Theorem 6.2 of [20] — the b oundedness of a bilinear map λ : V × Z → S , where V , Z , S are complex Banach spaces, is equiv alen t to its joint con tinuit y wrt the metric d V , Z (( x, w ) , ( y , z )) : = max {∥ x − y ∥ , ∥ w − z ∥} , x, y ∈ V , w , z ∈ Z (36) — i.e., wrt the pro duct top ology — or, also, to is sep ar ate con tinuit y . Therefore, the natural bilinear map on B 1 ( H ) × B 1 ( J ) is contin uous wrt the pro duct top ology; i.e., wrt the metric d 1 , 1 (( X , W ) , ( Y , Z )) : = max {∥ X − Y ∥ 1 , ∥ W − Z ∥ 1 } , X, Y ∈ B 1 ( H ) , W, Z ∈ B 1 ( J ) . (37) Notation 3.2. Let X , Y (nonempty) subsets of B 1 ( H ) and B 1 ( J ), resp ectiv ely . W e put X ˘ ⊗ Y : = co ( θ ( X , Y )) , X ⊗ Y : = co ( θ ( X , Y )) = cl ∥ · ∥ ⊗ 1  X ˘ ⊗ Y  ; (38) i.e., X ⊗ Y is the ∥ · ∥ ⊗ 1 -closed conv ex hull co ( θ ( X , Y )) ≡ co ⊗ ( θ ( X , Y )) of the set θ ( X , Y ). Privileging the c onvex structures wrt to the linear ones in Notation 3.2 is essentially motiv ated b y the applications we hav e in mind. Ho wev er, it is worth observing that, in the case where X = B 1 ( H ) and Y = B 1 ( J ), Notation 3.2 is consisten t with our previous definition of the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ) and of the natural tensor pro duct B 1 ( H ) ⊗B 1 ( J ) as span  θ ( B 1 ( H ) , B 1 ( J ))  and span  θ ( B 1 ( H ) , B 1 ( J ))  , resp ectiv ely; see Corollary 3.1 b elo w. Prop osition 3.2. F or every p air X , Y of (nonempty) subsets of B 1 ( H ) and B 1 ( J ) , r esp e ctively, span ( θ ( X , Y )) = span  θ (span ( X ) , span ( Y ))  , (39) span ( θ ( X , Y )) = co  θ (span ( X ) , span ( Y ))  = : span ( X ) ˘ ⊗ span ( Y ) (40) and X ˘ ⊗ Y : = co ( θ ( X , Y )) = co  θ (co ( X ) , co ( Y ))  = : co ( X ) ˘ ⊗ co ( Y ) , (41) 17 so that span ( θ ( X , Y )) = span  θ (span ( X ) , span ( Y ))  , (42) span ( θ ( X , Y )) = co  θ (span ( X ) , span ( Y ))  = : span ( X ) ⊗ span ( Y ) (43) and X ⊗ Y : = co ( θ ( X , Y )) = cl ∥ · ∥ ⊗ 1  co  θ (co ( X ) , co ( Y ))  = co  θ (co ( X ) , co ( Y ))  = : co ( X ) ⊗ co ( Y ) . (44) Mor e over, we have that X ⊗ Y : = co ( θ ( X , Y )) = co  θ (co ( X ) , co ( Y ))  = : co ( X ) ⊗ co ( Y ) (45) — wher e co ( X ) = cl ∥ · ∥ 1 (co ( X )) , co ( Y ) = cl ∥ · ∥ 1 (co ( Y )) — and span  θ ( X , Y )  = span  θ (co ( X ) , co ( Y ))  = span ( X ⊗ Y ) . (46) Corollary 3.1. L et X , Y b e line ar subsp ac es of B 1 ( H ) and B 1 ( J ) , r esp e ctively. Then, we have: X ⊗ Y : = co  θ ( X , Y )  = span ( θ ( X , Y )) . (47) Corollary 3.2. Every element of B 1 ( H ) ˘ ⊗ B 1 ( J ) c an b e expr esse d as a (finite) line ar c ombination of states of the form ρ ⊗ σ ; i.e., B 1 ( H ) ˘ ⊗ B 1 ( J ) = span  θ ( D ( H ) , D ( J ))  = span { ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J ) } . (48) Ther efor e, we have: B 1 ( H ) ⊗ B 1 ( J ) = span  θ ( D ( H ) , D ( J ))  . (49) Corollary 3.3. Sinc e D ( H ) = co ( P ( H )) and D ( J ) = co ( P ( J )) , then D ( H ) ⊗ D ( J ) : = co  θ ( D ( H ) , D ( J ))  = co  θ ( P ( H ) , P ( J ))  = : P ( H ) ⊗ P ( J ) (50) and B 1 ( H ) ⊗ B 1 ( J ) = span  θ ( D ( H ) , D ( J ))  = span  θ ( P ( H ) , P ( J ))  , (51) B 1 ( H ) ⊗ B 1 ( J ) = span ( D ( H ) ⊗ D ( J )) = span ( P ( H ) ⊗ P ( J )) . (52) W e are now almost ready to deriv e the — previously announced — more precise c haracterization of the natural tensor pro duct B 1 ( H ) ⊗ B 1 ( J ). W e only need to note a further tec hnical fact. Lemma 3.1. Every op er ator of the form c ac : = | a ⟩ ⟨ c | , for some ve ctors a , c ∈ H ⊗ J , b elongs to the natur al tensor pr o duct B 1 ( H ) ⊗ B 1 ( J ) . Pr e cisely, we have that c ac ∈ span n c η ϕ ⊗ c χψ : η , ϕ ∈ H , χ, ψ ∈ J o ⊂ B 1 ( H ) ⊗ B 1 ( J ) . (53) Mor e over, for the line ar sp ac e F ( H ⊗ J ) = span  c ac : a , c ∈ H ⊗ J  of al l finite rank op erators on H ⊗ J , we have: cl ∥ · ∥ ⊗ 1 ( F ( H ⊗ J )) = span n c η ϕ ⊗ c χψ : η , ϕ ∈ H , χ, ψ ∈ J o . (54) In p articular, putting b a ≡ c aa : = | a ⟩ ⟨ a | , a ∈ H ⊗ J , we have that b a ∈ span R n b ϕ ⊗ b ψ : ϕ ∈ H , ψ ∈ J o ⊂ B 1 ( H ) R ⊗ B 1 ( J ) R , (55) with b ϕ ≡ c ϕϕ , b ψ ≡ c ψ ψ , and span R ( P ( H ⊗ J )) = span R n b ϕ ⊗ b ψ : ϕ ∈ H , ψ ∈ J o , (56) wher e P ( H ⊗ J ) is the set of al l rank-one pro jections on H ⊗ J . 18 Theorem 3.1. F or every p air of (sep ar able, c omplex) Hilb ert sp ac es H and J , the natur al tensor pr o duct B 1 ( H ) ⊗ B 1 ( J ) c oincides with the whole bip artite tr ac e class B 1 ( H ⊗ J ) . Inde e d, we have that B 1 ( H ) ⊗ B 1 ( J ) = span { ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J ) } = span { π ⊗ ϖ : π ∈ P ( H ) , ϖ ∈ P ( J ) } = span n c η ϕ ⊗ c χψ : η , ϕ ∈ H , χ, ψ ∈ J o = B 1 ( H ⊗ J ) , (57) wher e c η ϕ ≡ | η ⟩ ⟨ ϕ | . Analo gously, we have: B 1 ( H ) R ⊗ B 1 ( J ) R = span R { π ⊗ ϖ : π ∈ P ( H ) , ϖ ∈ P ( J ) } = span R { ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J ) } = B 1 ( H ⊗ J ) R . (58) Pr o of. The first tw o equalities in (57) hold b y relation (51) in Corollary 3.3. Moreov er, since — as a consequence, sa y , of the singular value de c omp osition (Remark 2.1) of a trace class op erator — B 1 ( H ⊗ J ) = cl ∥ · ∥ ⊗ 1 ( F ( H ⊗ J )) , (59) b y relation (54) in Lemma 3.1 one m ust conclude that B 1 ( H ⊗ J ) = span  c η ϕ ⊗ c χψ : η , ϕ ∈ H , χ, ψ ∈ J  ⊂ cl ∥ · ∥ ⊗ 1 ( B 1 ( H ) ˘ ⊗ B 1 ( J )) , (60) and hence cl ∥ · ∥ ⊗ 1 ( B 1 ( H ) ˘ ⊗ B 1 ( J )) = B 1 ( H ) ⊗ B 1 ( J ) ⊂ B 1 ( H ⊗ J ) ⊂ cl ∥ · ∥ ⊗ 1 ( B 1 ( H ) ˘ ⊗ B 1 ( J )); so that, actually , B 1 ( H ⊗ J ) = span  c η ϕ ⊗ c χψ : η , ϕ ∈ H , χ, ψ ∈ J  = B 1 ( H ) ⊗ B 1 ( J ) . (61) Analogously , by the sp ectral decomposition of a selfadjoint trace class op erator (that con verges wrt the trace norm) and b y relation (56), w e ha ve: B 1 ( H ⊗ J ) R = span R ( P ( H ⊗ J )) = span R { π ⊗ ϖ : π ∈ P ( H ) , ϖ ∈ P ( J ) } ⊂ cl ∥ · ∥ ⊗ 1 ( B 1 ( H ) R ˘ ⊗ B 1 ( J ) R ) = B 1 ( H ) R ⊗ B 1 ( J ) R . (62) Then, since B 1 ( H ) R ⊗ B 1 ( J ) R ⊂ B 1 ( H ⊗ J ) R , the inclusion in relation (62) is actually an equality . It only remains to pro ve that the second equalit y in (58) holds to o. In fact, this equality is an immediate consequence of the relation span R { ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J ) } ⊂ span R { π ⊗ ϖ : π ∈ P ( H ) , ϖ ∈ P ( J ) } ⊂ span R { ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J ) } , (63) where the first inclusion holds by the sp ectral decomp osition of a densit y op erator and by the con tinuit y of the natural bilinear map (33). R emark 3.5 . The fact that B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ⊗ J ), for any dimension of H and J , is non trivial. Indeed, it is known (see [44], Example 11.1.5, Example 11.1.6 and Exercise 11.5.7) that the natur al, or sp atial, tensor pr o duct B ( H ) ⊗ B ( J ) — i.e., the Banac h space completion of the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) wrt the norm ∥ · ∥ ⊗ ∞ of B ( H ⊗ J ), also called the tensor pr o duct of the (c oncr ete, or r epr esente d) C ∗ -algebr as B ( H ) and B ( J ) (see, e.g., Section 11.1 of [44]) — coincides with the whole Banac h space B ( H ⊗ J ) iff min { dim( H ) , dim( J ) } < ∞ , and in such a case we actually hav e that B ( H ) ˘ ⊗ B ( J ) = B ( H ) ⊗ B ( J ) = B ( H ⊗ J ); if dim( H ) = dim( J ) = ∞ , instead, 19 B ( H ) ⊗ B ( J ) is a pr op er Banach subspace of B ( H ⊗ J ). In the case where dim( H ) = dim( J ) = ∞ , to obtain the full Banac h space B ( H ⊗ J ), one needs to consider the so-called von Neumann tensor pr o duct of the ‘lo cal’ von Neumann algebras B ( H ) and B ( J ) (see Definition 10.34 of [11], and the subsequent discussion; also see Section 11.1 of [44]); i.e., the closure of the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) wrt the strong (equiv alen tly , the weak) op erator top ology of B ( H ⊗ J ). If min { dim( H ) , dim( J ) } < ∞ , the spatial tensor pro duct and the von Neumann tensor pro duct coincide (Exercise 11.2.2 of [44]). No w, let S b e a complex Banac h space, and let Λ ⊗ : B 1 ( H ) ⊗ B 1 ( J ) → S b e a b ounded linear map. Then, λ : = Λ ⊗ ◦ θ : B 1 ( H ) × B 1 ( J ) → S (64) is a b ounded bilinear map, the bilinear map induc e d by Λ ⊗ . Definition 3.2. A bounded bilinear map λ : B 1 ( H ) × B 1 ( J ) → S is said to b e natur al ly line arizable if is of the form (64), for some b ounde d linear map Λ ⊗ : B 1 ( H ) ⊗ B 1 ( J ) → S . Prop osition 3.3. L et S b e a c omplex Banach sp ac e, and let λ : B 1 ( H ) × B 1 ( J ) → S b e a b ounde d biline ar map. Then, ther e exists a unique line ar map Λ ˘ ⊗ : B 1 ( H ) ˘ ⊗ B 1 ( J ) → S such that λ ( S, T ) = Λ ˘ ⊗ ( S ⊗ T ) , ∀ S ∈ B 1 ( H ) , ∀ T ∈ B 1 ( J ) . (65) The b ounde d biline ar map λ is natur al ly line arizable iff the line ar map Λ ˘ ⊗ : B 1 ( H ) ˘ ⊗ B 1 ( J ) → S is b ounde d, B 1 ( H ) ˘ ⊗ B 1 ( J ) b eing endowe d with (the r estriction of ) the norm ∥ · ∥ ⊗ 1 . Mor e over, if λ is natur al ly line arizable, then the b ounde d line ar map Λ ⊗ : B 1 ( H ) ⊗ B 1 ( J ) → S — the so-c al le d natural linearization of (or natural linear map inducing) λ — is unique (i.e., the unique b ounde d extension of Λ ˘ ⊗ ) and ∥ λ ∥ B ≤ ∥ Λ ⊗ ∥ B . 3.3 The pro jective tensor pro ducts W e will denote b y B 1 ( H ) b ⊗ B 1 ( J ) the pr oje ctive tensor pr o duct [16–20] of the Banach spaces B 1 ( H ) and B 1 ( J ); i.e., the Banach space completion of the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ) wrt the norm ∥ C ∥ b ⊗ 1 : = inf  P k ∥ S k ∥ 1 ∥ T k ∥ 1 : P k S k ⊗ T k = C  , C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) . (66) Here, the infimum is tak en ov er all (finite) decomp ositions of C into elemen tary tensors. Note that, b y its definition, the norm (66) is the largest p ossible (sub-)cross norm on B 1 ( H ) ˘ ⊗ B 1 ( J ), and extends by contin uit y to a norm on B 1 ( H ) b ⊗ B 1 ( J ) — the so-called pr oje ctive norm — that will b e still denoted b y ∥ · ∥ b ⊗ 1 . Since ∥ S ∥ 1 = ∥ S ∗ ∥ 1 , for ev ery S ∈ B 1 ( H ), it is clear that ∥ C ∥ b ⊗ 1 = ∥ C ∗ ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) . (67) This prop ert y actually extends to the full Banach space B 1 ( H ) b ⊗ B 1 ( J ) (see Corollary 3.5 b elo w). R emark 3.6 . As is w ell known, the Hilb ert space tensor pro duct H ⊗ J is isomorphic to the Hilbert space B 2 ( J ; H ) of all Hilb ert-Sc hmidt op erators from J into H , and this isomorphism can b e implemen ted b y extending — b y linearit y , and, next, by con tinuit y — the mapping ϕ ⊗ ψ 7→ | ϕ ⟩ ⟨ J ψ | , J : J → J , (68) where J is any c omplex c onjugation (an inv olutive an tiunitary op erator) on J . Notice that the mapping (68) admits a unique linear extension b y the univ ersal prop ert y of the algebraic tensor 20 pro duct H ˘ ⊗ J , b ecause the application H × J ∋ ( ϕ, ψ ) 7→ | ϕ ⟩ ⟨ J ψ | ∈ B 2 ( J ; H ) is a bilinear map. It is known [16–20] — also see [63], Theorem 5.12 in Section 10 of Chapter V — that, analogously , the Banac h space completion H b ⊗ J of the algebraic tensor pro duct H ˘ ⊗ J wrt the pro jectiv e norm | | a | | b ⊗ : = inf  P k ∥ ϕ k ∥ 1 ∥ ψ k ∥ 1 : P k ϕ k ⊗ ψ k = a  , a ∈ H ˘ ⊗ J , (69) is isomorphic to B 1 ( J ; H ), the Banach space of trace class op erators from J into H . Again, the isomorphism of Banach spaces H b ⊗ J ∼ = B 1 ( J ; H ) can b e implemen ted by extending (by linearit y , and, next, b y con tinuit y wrt the appropriate norms) the mapping (68). Analogously , w e will denote b y B 1 ( H ) R b ⊗ B 1 ( J ) R the pro jective tensor pro duct of the r e al Ba- nac h spaces B 1 ( H ) R and B 1 ( J ) R , i.e., the Banach space completion of B 1 ( H ) R ˘ ⊗ B 1 ( J ) R wrt the cross norm | | | C | | | b ⊗ 1 : = inf  P k ∥ S k ∥ 1 ∥ T k ∥ 1 : P k S k ⊗ T k = C, S k ⊗ T k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R )  , (70) for ev ery C ∈ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , whic h is the largest p ossible (sub-)cross norm on B 1 ( H ) R ˘ ⊗ B 1 ( J ) R . Here, recall that θ ( B 1 ( H ) R , B 1 ( J ) R ) : =  S ⊗ T : S ∈ B 1 ( H ) R , T ∈ B 1 ( J ) R  . Clearly , since the infim um in (70) is extended ov er all (finite) decomp ositions of C via selfadjoint elementary tensors only , we ha ve that ∥ C ∥ b ⊗ 1 ≤ | | | C | | | b ⊗ 1 , ∀ C ∈ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R . (71) W e will call the norm | | | · | | | b ⊗ 1 on B 1 ( H ) R b ⊗ B 1 ( J ) R — obtained extending b y con tinuit y the norm (70) defined on the algebraic tensor pro duct B 1 ( H ) R ˘ ⊗ B 1 ( J ) R — the Hermitian pr oje ctive norm . R emark 3.7 . By what previously observed, it follows easily that the cross norms ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 ma jorize an y (sub-)cross norm on B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R , resp ectiv ely . It turns out that, actually , the norms ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 are e quivalent on B 1 ( H ) R ˘ ⊗ B 1 ( J ) R . Prop osition 3.4. F or every tr ac e class op er ator C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) , ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 . Mor e over, ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ | | | C | | | b ⊗ 1 ≤ 2 ∥ C ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R . (72) 3.4 Immersion maps, cross trace class op erators and cross states Since b oth the Banac h spaces B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) ⊗ B 1 ( J ) are completions of the linear space B 1 ( H ) ˘ ⊗ B 1 ( J ) (wrt the norms ∥ · ∥ b ⊗ 1 and ∥ · ∥ ⊗ 1 , resp ectiv ely) and, moreo ver, the pro jective norm ∥ · ∥ b ⊗ 1 ma jorizes the trace norm ∥ · ∥ ⊗ 1 on B 1 ( H ) ˘ ⊗ B 1 ( J ), there is a natural (linear, injective) immersion map j : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ⊗ J ) . (73) Sp ecifically , the immersion map j is (well) defined by j ( C ) : = C , for all C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ), and { C n } ⊂ B 1 ( H ) ˘ ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1 – lim n C n = C ∈ B 1 ( H ) b ⊗ B 1 ( J ) = ⇒ j ( C ) : = ∥ · ∥ ⊗ 1 – lim n C n . (74) W e stress that the limit on the rhs of (74) does exist, b ecause, b y the inequalit y ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 , for ev ery C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) (Prop osition 3.4), the sequence { C n } is ∥ · ∥ ⊗ 1 -Cauc hy . Moreo ver, b y the same inequality and by the con tinuit y of the norm(s), w e ha ve: ∥ j ( C ) ∥ ⊗ 1 = lim n ∥ C n ∥ ⊗ 1 ≤ lim n ∥ C n ∥ b ⊗ 1 = ∥ C ∥ b ⊗ 1 . (75) Analogously , taking in to account that the Hermitian pro jectiv e norm | | | · | | | b ⊗ 1 ma jorizes the trace norm ∥ · ∥ ⊗ 1 on B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , there is a natural ( R -linear, injectiv e) immersion map j R : B 1 ( H ) R b ⊗ B 1 ( J ) R → B 1 ( H ) R ⊗ B 1 ( J ) R = B 1 ( H ⊗ J ) R , (76) 21 defined by j R ( C ) : = C , for all C ∈ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , and { C n } ⊂ B 1 ( H ⊗ J ) R , | | | · | | | b ⊗ 1 – lim n C n = R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R = ⇒ j R ( R ) : = ∥ · ∥ ⊗ 1 – lim n C n . (77) Here, since the antilinear map B 1 ( H ⊗ J ) ∋ A 7→ A ∗ ∈ B 1 ( H ⊗ J ) is an isometry (in particular, it is con tinuous), the limit C ≡ ∥ · ∥ ⊗ 1 – lim n C n is indeed contained in B 1 ( H ⊗ J ) R ; i.e., C = C ∗ . Then, b y the first t wo inequalities in (72), ∥ j R ( R ) ∥ ⊗ 1 = lim n ∥ C n ∥ ⊗ 1 ≤ lim n ∥ C n ∥ b ⊗ 1 ≤ lim n | | | C n | | | b ⊗ 1 = | | | R | | | b ⊗ 1 . (78) Finally , since the norm | | | · | | | b ⊗ 1 dominates the norm ∥ · ∥ b ⊗ 1 on B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , there is a natural ( R -linear, injective) immersion map k : B 1 ( H ) R b ⊗ B 1 ( J ) R → B 1 ( H ) b ⊗ B 1 ( J ) , (79) defined by k ( C ) : = C , for all C ∈ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , and { C n } ⊂ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , | | | · | | | b ⊗ 1 – lim n C n = R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R = ⇒ k ( R ) : = ∥ · ∥ b ⊗ 1 – lim n C n ; (80) hence, by the second inequality in (72), ∥ k ( R ) ∥ b ⊗ 1 = lim n ∥ C n ∥ b ⊗ 1 ≤ lim n | | | C n | | | b ⊗ 1 = | | | R | | | b ⊗ 1 . (81) Moreo ver, b y the first inequality in (78), we also ha ve that ∥ j R ( R ) ∥ ⊗ 1 = lim n ∥ C n ∥ ⊗ 1 ≤ lim n ∥ C n ∥ b ⊗ 1 = ∥ k ( R ) ∥ b ⊗ 1 . (82) R emark 3.8 . With a sligh t abuse, w e are considering a R -linear map — i.e., k (as well as the maps K : B 1 ( H ⊗ J ) R → B 1 ( H ⊗ J ) and (88) defined in Prop osition 3.5 b elow) — from a r e al vector space to a c omplex one. F ull mathematical rigor is restored b y simply regarding the latter as a real v ector space (by field restriction). The previously in tro duced immersion maps allo w us to ‘compare’ the norms ∥ · ∥ ⊗ 1 , ∥ · ∥ b ⊗ 1 , | | | · | | | b ⊗ 1 . Prop osition 3.5. The line ar maps j : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ⊗ J ) , j R : B 1 ( H ) R b ⊗ B 1 ( J ) R → B 1 ( H ⊗ J ) R (83) and k : B 1 ( H ) R b ⊗ B 1 ( J ) R → B 1 ( H ) b ⊗ B 1 ( J ) (84) ar e b ounde d, and ∥ j ∥ B = ∥ j R ∥ B = ∥ k ∥ B = 1 ; ther efor e, in p articular, we have: ∥ j ( C ) ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) b ⊗ B 1 ( J ) . (85) The norm | | | · | | | b ⊗ 1 dominates the norm ∥ · ∥ b ⊗ 1 and is dominated by 2 ∥ · ∥ b ⊗ 1 on B 1 ( H ) R b ⊗ B 1 ( J ) R . Pr e cisely, denoting by K : B 1 ( H ⊗ J ) R → B 1 ( H ⊗ J ) (86) the natur al (isometric) immersion of B 1 ( H ⊗ J ) R into B 1 ( H ⊗ J ) , we have: ∥ j R ( R ) ∥ ⊗ 1 = ∥ K ◦ j R ( R ) ∥ ⊗ 1 ≤ ∥ k ( R ) ∥ b ⊗ 1 ≤ | | | R | | | b ⊗ 1 ≤ 2 ∥ k ( R ) ∥ b ⊗ 1 , ∀ R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R . (87) Ther efor e, the norms ∥ · ∥ b ⊗ 1 ◦ k and | | | · | | | b ⊗ 1 on B 1 ( H ) R b ⊗ B 1 ( J ) R ar e equiv alen t . Mor e over, we have that j ◦ k = K ◦ j R : B 1 ( H ) R b ⊗ B 1 ( J ) R → B 1 ( H ⊗ J ) . (88) 22 The range of the immersion map j is a dense linear subspace (containing the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J )) of the Banach space B 1 ( H ⊗ J ), and we put B 1 ( H ⊗ J ) b : = j  B 1 ( H ) b ⊗ B 1 ( J )  ⊂ B 1 ( H ⊗ J ) . (89) Analogously , the range of the immersion map j R is a dense linear subspace (con taining the R -linear span B 1 ( H ) R ˘ ⊗ B 1 ( J ) R ) of the real Banac h space B 1 ( H ⊗ J ) R ; we put B 1 ( H ⊗ J ) b R : = j R  B 1 ( H ) R b ⊗ B 1 ( J ) R  ⊂ B 1 ( H ⊗ J ) R . (90) Clearly , B 1 ( H ⊗ J ) b R can b e identified with a subset of B 1 ( H ⊗ J ) b , b ecause B 1 ( H ⊗ J ) b R : = j R  B 1 ( H ) R b ⊗ B 1 ( J ) R  = K ◦ j R  B 1 ( H ) R b ⊗ B 1 ( J ) R  = j ◦ k  B 1 ( H ) R b ⊗ B 1 ( J ) R  ⊂ j  B 1 ( H ) b ⊗ B 1 ( J )  = : B 1 ( H ⊗ J ) b , (91) where the second equalit y is just a natural identification obtained via the isometric immersion map K : B 1 ( H ⊗ J ) R → B 1 ( H ⊗ J ), and next we ha ve used relation (88). W e also introduce the set D ( H ⊗ J ) b : = D ( H ⊗ J ) ∩ B 1 ( H ⊗ J ) b , (92) whic h is a con vex subset of the linear space B 1 ( H ⊗ J ) b containing, in particular, all (finite) conv ex com binations of elementary tensors of the form ρ ⊗ σ , with ρ ∈ D ( H ) and σ ∈ D ( J ). Definition 3.3. W e will call the trace class op erators in B 1 ( H ⊗ J ) b , the selfadjoint trace class op erators in B 1 ( H ⊗ J ) b R and the densit y op erators in D ( H ⊗ J ) b , resp ectiv ely , the cr oss tr ac e class op er ators , the selfadjoint cr oss tr ac e class op er ators and the cr oss states (or cr oss density op er ators ) wrt the bipartition H ⊗ J . By its definition (90) and b y the inclusion relation B 1 ( H ⊗ J ) b R ⊂ B 1 ( H ⊗ J ) b (see (91)), we kno w that the set of selfadjoin t cross trace class op erators must satisfy the relation B 1 ( H ⊗ J ) b R ⊂ B 1 ( H ⊗ J ) R ∩ B 1 ( H ⊗ J ) b =  C ∈ B 1 ( H ⊗ J ) b : C = C ∗  , (93) and, hence, for the set of cross states w e ha ve: D ( H ⊗ J ) ∩ B 1 ( H ⊗ J ) b R ⊂ D ( H ⊗ J ) ∩ B 1 ( H ⊗ J ) b = : D ( H ⊗ J ) b . (94) W e will so on pro ve (see equation (120) in Corollary 3.5 b elo w) that in (93) — and, hence, in (94) — the inclusion relation is actually an equality . T o this end, and for our later purp oses, w e need to establish some useful decomp ositions of cross trace class op erators, whic h is our next task. 3.5 The standard decomp osition of a cross trace class op erator In the following, we will consider suitable decomp ositions of an element C of the complex Banac h space B 1 ( H ) b ⊗ B 1 ( J ) — or of the real Banac h space B 1 ( H ) R b ⊗ B 1 ( J ) R — of the general form of a finite, or coun tably infinite, sum C = X k c k ( S k ⊗ T k ) , (95) with c k ∈ R , 0  = S k ⊗ T k ∈ θ ( B 1 ( H ) , B 1 ( J )) (resp ectiv ely , 0  = S k ⊗ T k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R )). The sum, if not finite, is supp osed to b e absolutely c onver gent wrt the pro jectiv e norm ∥ · ∥ b ⊗ 1 23 (for C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , wrt the Hermitian pro jective norm | | | · | | | b ⊗ 1 , or wrt the equiv alent norm ∥ · ∥ b ⊗ 1 ◦ k ); i.e., P k | c k | ∥ S k ⊗ T k ∥ b ⊗ 1 = P k | c k | ∥ S k ∥ 1 ∥ T k ∥ 1 < ∞ , so that ∥ C ∥ b ⊗ 1 ≤ X k | c k | ∥ S k ⊗ T k ∥ b ⊗ 1 = X k | c k | ∥ S k ∥ 1 ∥ T k ∥ 1 < ∞ . (96) Clearly , if C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R and, for ev ery v alue of the index k , S k ⊗ T k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) in (95), then an analogous inequality holds for | | | C | | | b ⊗ 1 to o. W e will call an expansion of C ∈ B 1 ( H ) b ⊗ B 1 ( J ) (or of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R ) of the form (95) a simple-tensor de c omp osition . R emark 3.9 . Since, for ev ery S ∈ B 1 ( H ) and T ∈ B 1 ( J ), ∥ S ⊗ T ∥ b ⊗ 1 = ∥ S ∥ 1 ∥ T ∥ 1 = ∥ S ⊗ T ∥ ⊗ 1 , it is clear that a series of the form P k c k ( S k ⊗ T k ), with c k ∈ R and S k ⊗ T k ∈ θ ( B 1 ( H ) , B 1 ( J )), is absolutely con vergen t wrt the pro jective norm ∥ · ∥ b ⊗ 1 iff it is absolutely con vergen t wrt the trace norm ∥ · ∥ ⊗ 1 . Moreo ver, since the immersion map j : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ⊗ J ) is con tinuous (Prop osition 3.5), we ha ve: C = ∥ · ∥ b ⊗ 1 – X k c k ( S k ⊗ T k ) = ⇒ ∥ · ∥ ⊗ 1 – X k c k ( S k ⊗ T k ) = j ( C ) ∈ B 1 ( H ⊗ J ) . (97) Analogously , the series P k c k ( S k ⊗ T k ), with c k ∈ R and S k ⊗ T k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ), is absolutely con vergen t wrt the Hermitian pro jective norm | | | · | | | b ⊗ 1 iff it is absolutely con vergen t wrt the trace norm ∥ · ∥ ⊗ 1 , and R = | | | · | | | b ⊗ 1 – P k c k ( S k ⊗ T k ) = ⇒ ∥ · ∥ ⊗ 1 – P k c k ( S k ⊗ T k ) = j R ( R ) ∈ B 1 ( H ⊗ J ) R . R emark 3.10 . Some of the co efficients { c k } in (95) may well b e zero, ev en if C  = 0, but the corresp onding zero terms in { c k ( S k ⊗ T k ) } are still to b e en visioned as part of the decomp osition, b ecause they may play some role in those argumen ts where this decomp osition is inv olved. A simple-tensor decomp osition of C  = 0, of the general form (95), will b e said to b e she er if it do es not contain an y zero term. Definition 3.4. A decomp osition of C ∈ B 1 ( H ) b ⊗ B 1 ( J ) of the form (95) is called ∥ · ∥ b ⊗ 1 - optimal (or, for the sak e of conciseness, simply optimal ) — alternativ ely , in the case where, for ev ery v alue of the index k , S k ⊗ T k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) in (95) (hence, C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R ), | | | · | | | b ⊗ 1 - optimal — if the norm ∥ C ∥ b ⊗ 1 (resp ectiv ely , | | | C | | | b ⊗ 1 ) is giv en precisely by X k | c k | ∥ S k ∥ 1 ∥ T k ∥ 1 ∈ R + ; (98) i.e., if the first inequality in (96) (resp ectively , the inequality | | | C | | | b ⊗ 1 ≤ P k | c k | ∥ S k ∥ 1 ∥ T k ∥ 1 ) is satu- rated. In this case, C is said to b e ∥ · ∥ b ⊗ 1 - optimal ly de c omp osable , or simply optimal ly de c omp osable (resp ectiv ely , C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is said to b e | | | · | | | b ⊗ 1 - optimal ly de c omp osable ). Definition 3.5. A simple-tensor decomp osition of the form (95) — with the scalar co efficien ts { c k } and the op erators { S k ⊗ T k } subje ct to sp e cific r e quir ements — of a generic element C of a certain subset S of B 1 ( H ) b ⊗ B 1 ( J ) (or of B 1 ( H ) R b ⊗ B 1 ( J ) R ), will b e said to b e a ∥ · ∥ b ⊗ 1 -norming de c omp osition (resp ectively , a | | | · | | | b ⊗ 1 -norming de c omp osition ), if the norm ∥ C ∥ b ⊗ 1 (or | | | C | | | b ⊗ 1 ) of C ∈ S can b e obtained as inf  P k | c k | ∥ S k ∥ 1 ∥ T k ∥ 1 : P k c k ( S k ⊗ T k ) = C  , (99) where the infim um is extended o ver all expansions of C of the given sp ecified form. 24 Theorem 3.2. Every element C of the c omplex Banach sp ac e  B 1 ( H ) b ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1  admits a simple-tensor de c omp osition of the form C = X k r k ( X k ⊗ Y k ) , r k ≥ 0 , X k ⊗ Y k ∈ θ ( B 1 ( H ) , B 1 ( J )) , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 , (100) wher e the sum is either finite or — in the c ase wher e the Hilb ert sp ac e H ⊗ J is infinite-dimensional — p ossibly c ountably infinite. In the latter c ase, the series is supp ose d to c onver ge absolutely wrt to the pr oje ctive norm ∥ · ∥ b ⊗ 1 , and, in fact, the non-ne gative r e al se quenc e { r k } is assume d to b e summable, so that ∥ C ∥ b ⊗ 1 ≤ X k ∥ r k ( X k ⊗ Y k ) ∥ b ⊗ 1 = X k r k ∥ X k ∥ 1 ∥ Y k ∥ 1 = X k r k < ∞ . (101) Mor e over, for every C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , we have that ∥ C ∥ b ⊗ 1 = inf  P k r k : P k r k ( X k ⊗ Y k ) = C  , (102) wher e the infimum is taken over al l p ossible exp ansions of C of the gener al form (100) ; namely, the de c omp osition (100) is ∥ · ∥ b ⊗ 1 -norming. A nalo gously, every element R of the r e al Banach sp ac e ( B 1 ( H ) R b ⊗ B 1 ( J ) R , | | | · | | | b ⊗ 1 ) admits a simple-tensor de c omp osition of the form R = X k r k ( X k ⊗ Y k ) , r k ≥ 0 , X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 , (103) absolute c onver genc e — that holds b e c ause P k r k < ∞ — and al l other pr op erties as ab ove, b eing understo o d wrt to the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 ; in p articular, the de c omp osition (103) is | | | · | | | b ⊗ 1 -norming. Pr o of. In the case where dim( H ⊗ J ) < ∞ and/or C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ), the statemen t is clear. Let us then assume that dim( H ⊗ J ) = ∞ and C ∈ ( B 1 ( H ) b ⊗ B 1 ( J )) \ ( B 1 ( H ) ˘ ⊗ B 1 ( J )). By a classical result of Grothendiec k [15] — also see Prop osition 2.8 of [18], Prop osition 1.1.4 of [19], or Theorem 7.9 of [20] — for every C of this kind there is a sequence of the form { S k ⊗ T k } , with S k ∈ B 1 ( H ) and T k ∈ B 1 ( J ), suc h that X k ∥ S k ∥ 1 ∥ T k ∥ 1 < ∞ and C = X k S k ⊗ T k , (104) where, b y the first of these relations, the series must con verge wrt to the pro jective norm ∥ · ∥ b ⊗ 1 . Moreo ver, it turns out that ∥ C ∥ b ⊗ 1 : = inf  P k ∥ S k ∥ 1 ∥ T k ∥ 1 : P k S k ⊗ T k = C  , (105) where the infimum is taken o v er all decomp ositions of C of the form (104). Finally , by suitably re-expressing the sequence { S k ⊗ T k } in the form { r k ( X k ⊗ Y k ) } , as sp ecified in (100) (so that r k = ∥ S k ∥ 1 ∥ T k ∥ 1 ), the statement follows. An analogous result holds for the decomp osition (103), b ecause the previously cited result of Grothendiec k applies to the pro jectiv e tensor pro duct of any pair of real or complex Banach spaces. R emark 3.11 . In decomposition (100) — or (103) — w e do not distinguish, in general, b et ween the v arious representations of the elemen tary tensor product X k ⊗ Y k . Ho wev er, w e can alwa ys assume, without loss of generalit y (wrt the condition that ∥ X k ⊗ Y k ∥ ⊗ 1 = ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1), that in (100) — or in (103) — ∥ X k ∥ 1 = ∥ Y k ∥ 1 = 1; moreov er, in decomp osition (103), we can alwa ys supp ose that X k ∈ B 1 ( H ) R and Y k ∈ B 1 ( J ) R . 25 R emark 3.12 . W e stress that any series P k r k ( X k ⊗ Y k ) of the form sp ecified in (100) m ust con verge in the Banach space B 1 ( H ) b ⊗ B 1 ( J ) (assuming that P k r k < ∞ ), b ecause in such a case it is absolutely conv ergent wrt the pro jective norm ∥ · ∥ b ⊗ 1 , so that the sequence of the partial sums is ∥ · ∥ b ⊗ 1 -Cauc hy . As noted in Remark 3.9, it is also absolutely con vergen t wrt the trace norm ∥ · ∥ ⊗ 1 , and j ( C ) = j ( P k r k ( X k ⊗ Y k )) = P k r k ( X k ⊗ Y k ), where the latter sum con verges in B 1 ( H ⊗ J ). Analogous facts hold true in the case of the real Banach spaces B 1 ( H ) R b ⊗ B 1 ( J ) R and B 1 ( H ⊗ J ) R . Corollary 3.4. The Banach sp ac es B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R ar e sep ar able. Pr o of. By Theorem 3.2, a generic elemen t C of the Banach space B 1 ( H ) b ⊗ B 1 ( J ) admits a de- comp osition of the form (100), which, for mere economy of this pro of, in the following will b e assumed to b e c ountably infinite , p ossibly putting r k = 0 and choosing a whatever pair of trace class op erators X k ∈ B 1 ( H ), Y k ∈ B 1 ( J ) (with ∥ X k ∥ 1 = ∥ Y k ∥ 1 = 1), for k ≥ N , for some suitable N ∈ N . No w, let B 1 ( H ) 0 , B 1 ( J ) 0 b e an y pair of c ountable dense subsets of the separable Banac h spaces B 1 ( H ) and B 1 ( J ), resp ectiv ely . F or every n ∈ N , we set C n = ∞ X k =1 r k ; n ( X k ; n ⊗ Y k ; n ) , 0 ≤ r k ; n ∈ Q , X k ; n ∈ B 1 ( H ) 0 , Y k ; n ∈ B 1 ( J ) 0 , (106) where the non-negative sequence { r k ; n ∥ X k ; n ∥ 1 ∥ Y k ; n ∥ 1 : k ∈ N } is supp osed to b e summable — so that the series will conv erge, wrt to the pro jective norm ∥ · ∥ b ⊗ 1 , to an elemen t of B 1 ( H ) b ⊗ B 1 ( J ) — but this time we are not assuming that ∥ X k ; n ∥ 1 ∥ Y k ; n ∥ 1 = 1. By construction, the v ector C n b elongs to the ∥ · ∥ b ⊗ 1 -closure of a c ountable subset of B 1 ( H ) b ⊗ B 1 ( J ); i.e., of the (coun table) Q -linear span span Q { θ ( B 1 ( H ) 0 , B 1 ( J ) 0 ) } = span Q { S ⊗ T : S ∈ B 1 ( H ) 0 , T ∈ B 1 ( J ) 0 } . (107) Let us sho w that the sequence { C n } — sub ject to the sp ecified conditions — can be constructed in such a wa y as to con verge to a generic element C (expressed b y a countably infinite expansion of the form (100), with ∥ X k ∥ 1 = ∥ Y k ∥ 1 = 1) of B 1 ( H ) b ⊗ B 1 ( J ). T o this end, w e will assume that | r k − r k ; n | < s n 2 k +1  = ⇒ r k ; n < r k + s n 2 k +1  , (108) with s ≡ P k r k < ∞ , and ∥ X k − X k ; n ∥ 1 < 3 7 n 2 k +1  = ⇒ ∥ X k ; n ∥ 1 < ∥ X k ∥ 1 + 3 7 n 2 k +1  , ∥ Y k − Y k ; n ∥ 1 < 3 7 n 2 k +1 , (109) where w e recall once again that ∥ X k ∥ 1 = ∥ Y k ∥ 1 = 1. Clearly , these assumptions are legitimate by the density of the rationals Q in R , and of coun table the sets B 1 ( H ) 0 , B 1 ( J ) 0 in B 1 ( H ) and B 1 ( J ), resp ectiv ely . Note moreov er that, since 0 ≤ r k ; n ∥ X k ; n ∥ 1 ∥ Y k ; n ∥ 1 <  r k + s n 2 k +1   1 + 3 7 n 2 k +1  2 , (110) where w e ha ve used the inequalities in brack ets in (108) and (109), then, b y an elementary estimate, ∞ X k =1 ∥ r k ; n ( X k ; n ⊗ Y k ; n ) ∥ b ⊗ 1 = ∞ X k =1 r k ; n ∥ X k ; n ∥ 1 ∥ Y k ; n ∥ 1 < 2 s ( n + 1) n ; (111) 26 hence, for every n ∈ N , the series in (106) is indeed absolutely con vergen t wrt to the pro jective norm ∥ · ∥ b ⊗ 1 . At this p oin t, observ e that ∥ C − C n ∥ b ⊗ 1 ≤ ∥ P k ( r k − r k ; n ) ( X k ⊗ Y k ) ∥ b ⊗ 1 + ∥ P k r k ; n (( X k − X k ; n ) ⊗ Y k ) ∥ b ⊗ 1 + ∥ P k r k ; n ( X k ; n ⊗ ( Y k − Y k ; n )) ∥ b ⊗ 1 ≤ P k | r k − r k ; n | ∥ X k ∥ 1 ∥ Y k ∥ 1 + P k r k ; n ∥ X k − X k ; n ∥ 1 ∥ Y k ∥ 1 + P k r k ; n ∥ X k ∥ 1 ∥ Y k − Y k ; n ∥ 1 < s n . (112) Therefore, even tually we see that ∥ · ∥ b ⊗ 1 – lim n C n = C , and hence C ∈ cl ∥ · ∥ b ⊗ 1  span Q { θ ( B 1 ( H ) 0 , B 1 ( J ) 0 ) }  , (113) whic h pro ves that the Banach space B 1 ( H ) b ⊗ B 1 ( J ) is separable, as claimed. The pro of in case of the real Banac h space B 1 ( H ) R b ⊗ B 1 ( J ) R is analogous. Another relev ant consequence of Theorem 3.2 is that the cross trace class op erators B 1 ( H ⊗ J ) b and the selfadjoint cross trace class op erators B 1 ( H ⊗ J ) b R admit an elemen tary c haracterization in terms of simple tensors; moreov er, they are, in natural wa y , Banach spaces isomorphic to — and, therefore, will be iden tified with — B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R , respectively . F or the sak e of simplicity , with a sligh t abuse, we will denote the norms of the t w o pairs of isomorphic Banac h spaces by the same symbols ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 . Corollary 3.5 (The standard decomp osition) . The set B 1 ( H ⊗ J ) b of al l cr oss tr ac e class op er ators wrt the bip artition H ⊗ J — i.e., the r ange of the immersion map j : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ⊗ J ) — is char acterize d as the c omplex line ar sp ac e c onsisting of every tr ac e class op er ator C on H ⊗ J that admits an absolutely ∥ · ∥ ⊗ 1 -c onver gent simple-tensor de c omp osition of the form C = X k r k ( X k ⊗ Y k ) , (114) with r k ≥ 0 ( P k r k < ∞ ) , X k ⊗ Y k ∈ θ ( B 1 ( H ) , B 1 ( J )) and ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 . Mor e over, her e one c an always assume that X k and Y k ar e r ank-one p artial isometries, i.e., X k = | η k ⟩ ⟨ ϕ k | , Y k = | ψ k ⟩ ⟨ χ k | , for some normalize d ve ctors η k , ϕ k ∈ H and ψ k , χ k ∈ J . (115) The line ar sp ac e B 1 ( H ⊗ J ) b is a selfadjoint subset of the bip artite tr ac e class B 1 ( H ⊗ J ) , i.e., C = P k r k ( X k ⊗ Y k ) ∈ B 1 ( H ⊗ J ) b = ⇒ C ∗ = P k r k ( X ∗ k ⊗ Y ∗ k ) ∈ B 1 ( H ⊗ J ) b , (116) and b e c omes a sep ar able Banach sp ac e — isomorphic to B 1 ( H ) b ⊗ B 1 ( J ) — if endowe d with the norm ∥ C ∥ b ⊗ 1 : = inf  P k r k : P k r k ( X k ⊗ Y k ) = C  , C ∈ B 1 ( H ⊗ J ) b , (117) wher e the infimum is extende d over al l absolutely ∥ · ∥ ⊗ 1 -c onver gent de c omp ositions of C of the form (114) ; to evaluate the norm ∥ C ∥ b ⊗ 1 , we c an assume that X k and Y k ther ein ar e, in p articular, r ank-one p artial isometries. Any such a simple-tensor de c omp osition wil l c onver ge absolutely to C wrt the norm ∥ · ∥ b ⊗ 1 of B 1 ( H ⊗ J ) b , as wel l. Mor e over, for every C ∈ B 1 ( H ⊗ J ) b , ∥ C ∥ b ⊗ 1 = ∥ C ∗ ∥ b ⊗ 1 . The set B 1 ( H ⊗ J ) b R of al l selfadjoin t cr oss tr ac e class op er ators wrt the bip artition H ⊗ J — i.e., the r ange of the immersion map j R : B 1 ( H ) R b ⊗ B 1 ( J ) R → B 1 ( H ⊗ J ) R — c oincides with 27 the r e al line ar sp ac e of al l selfadjoint tr ac e class op er ators C on H ⊗ J that admit an absolutely ∥ · ∥ ⊗ 1 -c onver gent de c omp osition of the form C = X k r k ( X k ⊗ Y k ) , r k ≥ 0 , X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 , (118) wher e the absolute c onver genc e holds b e c ause P k r k < ∞ . The line ar sp ac e B 1 ( H ⊗ J ) b R b e c omes a sep ar able (r e al) Banach sp ac e — isomorphic to B 1 ( H ) R b ⊗ B 1 ( J ) R — if endowe d with the norm | | | C | | | b ⊗ 1 : = inf  P k r k : P k r k ( X k ⊗ Y k ) = C, X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R )  , (119) wher e the infimum is extende d over al l absolutely ∥ · ∥ ⊗ 1 -c onver gent de c omp ositions of C of the sp e cifie d form, and any such a simple-tensor de c omp osition wil l c onver ge absolutely to C wrt the norm | | | · | | | b ⊗ 1 of B 1 ( H ⊗ J ) b R , as wel l. Mor e over, the fol lowing r elation holds: B 1 ( H ⊗ J ) b R = B 1 ( H ⊗ J ) R ∩ B 1 ( H ⊗ J ) b =  C ∈ B 1 ( H ⊗ J ) b : C = C ∗  . (120) The c omplex Banach sp ac e ( B 1 ( H ⊗ J ) b , ∥ · ∥ b ⊗ 1 ) is close d wrt the standar d pr o duct of op er ators (i.e., c omp osition), and — onc e endowe d with this pr o duct and with the adjoining map C 7→ C ∗ — b e c omes a Banach ∗ -algebr a. Pr o of. The first assertion follo ws immediately from Theorem 3.2, taking into accoun t Remark 3.12. Note that, b y the singular v alue decomp osition of a trace class op erator (Remark 2.1), we can expand the op erators X k ∈ B 1 ( H ), Y k ∈ B 1 ( J ) of decomp osition (114), where we put ∥ X k ∥ 1 = ∥ Y k ∥ 1 = 1, in terms of rank-one partial isometries; precisely , X k = X j s ( k ) j   η ( k ) j   ϕ ( k ) j   , Y k = X l t ( k ) l   ψ ( k ) l   χ ( k ) l   , (121) where — for e ach fixe d value of the index k — we hav e: P j s ( k ) j = ∥ X k ∥ 1 = 1 = ∥ Y k ∥ 1 = P l t ( k ) l , with s ( k ) j , t ( k ) l > 0, and  η ( k ) j  ,  ϕ ( k ) j  ⊂ H and  ψ ( k ) l  ,  χ ( k ) l  ⊂ J are orthonormal systems. Thus, the decomp osition (114) of the cross trace class op erator C ∈ B 1 ( H ⊗ J ) b can be re-formulated as follo ws: C = X k r k ( X k ⊗ Y k ) = X k X j X l r k s ( k ) j t ( k ) l    η ( k ) j   ϕ ( k ) j   ⊗   ψ ( k ) l   χ ( k ) l    = X k X m r k s ( k ) j ( m ) t ( k ) l ( m )    η ( k ) j ( m )   ϕ ( k ) j ( m )   ⊗   ψ ( k ) l ( m )   χ ( k ) l ( m )    . = X n r k ( n ) s ( k ( n )) j ( m ( n )) t ( k ( n )) l ( m ( n ))    η ( k ( n )) j ( m ( n ))   ϕ ( k ( n )) j ( m ( n ))   ⊗   ψ ( k ( n )) l ( m ( n ))   χ ( k ( n )) l ( m ( n ))    . (122) Here, a few comments are in order: • The second equality in (122) is obtained by considering the (p ossibly countably infinite) sum in decomp osition (114) as conv ergent wrt the trace norm ∥ · ∥ ⊗ 1 , and then exploiting the singular v alue decomp ositions (121) of X k ∈ B 1 ( H ) and Y k ∈ B 1 ( J ) and the contin uit y of the natural bilinear form θ : B 1 ( H ) × B 1 ( J ) → B 1 ( H ⊗ J ). Therefore, the iterated sum P k P j P l . . . conv erges wrt the norm ∥ · ∥ ⊗ 1 . 28 • Note that, for eac h fixed v alue of the index k , the double sequence (wrt the indices j, l )  s ( k ) j t ( k ) l    η ( k ) j   ϕ ( k ) j   ⊗   ψ ( k ) l   χ ( k ) l    (123) is absolutely summable, b ecause s ( k ) j t ( k ) l > 0, P j P l s ( k ) j t ( k ) l = 1 and      η ( k ) j   ϕ ( k ) j   ⊗   ψ ( k ) l   χ ( k ) l      ⊗ 1 = 1 =      η ( k ) j   ϕ ( k ) j   ⊗   ψ ( k ) l   χ ( k ) l      b ⊗ 1 . (124) Therefore, we can con v ert the iterated sum P j P l . . . on the second line of (122) into a single sum P m . . . by taking any bijection m 7→ ( j ( m ) , l ( m )) (F act 2.11). • Next, since r k s ( k ) j ( m ) t ( k ) l ( m ) ≥ 0 and P k P m r k s ( k ) j ( m ) t ( k ) l ( m ) = P k r k P j P l s ( k ) j t ( k ) l = P k r k < ∞ , according to F act 2.11 w e can con v ert the iterated sum P k P m . . . on the third line of (122) in to a single sum P n . . . by c ho osing any bijection n 7→ ( k ( n ) , m ( n )). Ev entually , by suitably renaming the co efficien ts and the indices in the expansion on the last line of (122), we get a decomp osition of the form C = X n c n ( | η n ⟩ ⟨ ϕ n | ⊗ | ψ n ⟩ ⟨ χ n | ) , c n = r k ( n ) s ( k ( n )) j ( m ( n )) t ( k ( n )) l ( m ( n )) ≥ 0 , ∥ η n ∥ = · · · = ∥ χ n ∥ = 1 , (125) where the sum conv erges absolutely wrt to the trace norm ∥ · ∥ ⊗ 1 and, in fact, X n c n = X n r k ( n ) s ( k ( n )) j ( m ( n )) t ( k ( n )) l ( m ( n )) = X k X m r k s ( k ) j ( m ) t ( k ) l ( m ) = X k X j X l r k s ( k ) j t ( k ) l = X k r k < ∞ . (126) Let us prov e the second assertion of the theorem. By (102), the norm on B 1 ( H ⊗ J ) b defined b y (117) coincides with the image, via the immersion map j : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ⊗ J ), of the pro jective norm ∥ · ∥ b ⊗ 1 . Hence, the linear space B 1 ( H ⊗ J ) b , endow ed with the norm (117), becomes a Banach space isomorphic to B 1 ( H ) b ⊗ B 1 ( J ). Moreo ver, since, for every C ∈ B 1 ( H ⊗ J ) b , the decomp osition (114) can alw ays b e transformed into a decomp osition of the sp ecial form (125), whose coefficients { c n } are related to the co efficien ts { r k } of the more general decomp osition (114) in such a w ay that relation (126) holds — i.e., P n c n = P k r k — we can assume that the trace class op erators X k , Y k in (117) are, in particular, rank-one partial isometries; namely , we ha ve: ∥ C ∥ b ⊗ 1 = inf  P k r k : P k r k ( | η k ⟩ ⟨ ϕ k | ⊗ | ψ k ⟩ ⟨ χ k | ) = C, r k ≥ 0 , ∥ η k ∥ = · · · = ∥ χ k ∥ = 1  . (127) As noted in Remark 3.9 (and next in Remark 3.12) — since, for ev ery S ∈ B 1 ( H ) and T ∈ B 1 ( J ), ∥ S ⊗ T ∥ b ⊗ 1 = ∥ S ∥ 1 ∥ T ∥ 1 = ∥ S ⊗ T ∥ ⊗ 1 — a series of the general form P k c k ( S k ⊗ T k ), with c k ∈ R and S k ⊗ T k ∈ θ ( B 1 ( H ) , B 1 ( J )), is absolutely con vergen t wrt the pro jectiv e norm ∥ · ∥ b ⊗ 1 iff it is absolutely con vergen t wrt the trace norm ∥ · ∥ ⊗ 1 , and, if C = ∥ · ∥ b ⊗ 1 – P k c k ( S k ⊗ T k ), then ∥ · ∥ ⊗ 1 – P k c k ( S k ⊗ T k ) = C . Therefore, in particular, the ∥ · ∥ ⊗ 1 -con vergen t simple-tensor decomposition (114) m ust conv erge absolutely to C wrt the norm ∥ · ∥ b ⊗ 1 of B 1 ( H ⊗ J ) b , as w ell. Moreo ver, taking in to accoun t that the adjoining map B 1 ( H ⊗ J ) ∋ C 7→ C ∗ ∈ B 1 ( H ⊗ J ) is an isometry (in particular, it is contin uous), for any ∥ · ∥ ⊗ 1 -con vergen t expansion of C ∈ B 1 ( H ⊗ J ) b of the form (114), C ∗ = P k r k ( X k ⊗ Y k ) ∗ = P k r k ( X ∗ k ⊗ Y ∗ k ), with ∥ X ∗ k ∥ 1 ∥ Y ∗ k ∥ 1 = ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1. Th us, C ∗ b elongs to B 1 ( H ⊗ J ) b , as w ell, and, by definition (117) of the norm ∥ · ∥ b ⊗ 1 , it is clear that ∥ C ∥ b ⊗ 1 = ∥ C ∗ ∥ b ⊗ 1 , for ev ery C ∈ B 1 ( H ⊗ J ) b . 29 The assertion of the theorem regarding the characterization of the Banach space B 1 ( H ⊗ J ) b R follo ws, once again, from Theorem 3.2 and Remark 3.12. W e now pro ve relation (120). W e already know that B 1 ( H ⊗ J ) b R ⊂ B 1 ( H ⊗ J ) R ∩ B 1 ( H ⊗ J ) b (recall relation (93)). It is then sufficien t to show that the rev erse inclusion holds too; i.e., that C ∈ B 1 ( H ⊗ J ) b , C = C ∗ = ⇒ C ∈ B 1 ( H ⊗ J ) b R . Let P k r k ( X k ⊗ Y k ) b e an expansion of C as sp ecified in (114). Note that eac h term of the form X ⊗ Y in this expansion (for notational con venience, here we are dropping the index k ) can b e expressed as ( X ℜ + i X ℑ ) ⊗ ( Y ℜ + i Y ℑ ) = X ℜ ⊗ Y ℜ − X ℑ ⊗ Y ℑ + i ( X ℜ ⊗ Y ℑ + X ℑ ⊗ Y ℜ ) , (128) where X ℜ : = 1 2 ( X + X ∗ ), X ℑ : = 1 2 i ( X − X ∗ ) ∈ B 1 ( H ) R , and Y ℜ , Y ℑ ∈ B 1 ( J ) R are defined analogously . Th us, C can b e expressed as the sum of tw o series, the first one containing all terms of the form X ℜ ⊗ Y ℜ − X ℑ ⊗ Y ℑ — i.e., restoring the index k , the (absolutely con vergen t wrt eac h of the norms ∥ · ∥ ⊗ 1 , ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 ) series X k r k ( X k ; ℜ ⊗ Y k ; ℜ − X k ; ℑ ⊗ Y k ; ℑ ) , (129) where ( X k ; ℜ ⊗ Y k ; ℜ ) , ( X k ; ℑ ⊗ Y k ; ℑ ) ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) — while the second one all terms of the form i ( X ℜ ⊗ Y ℑ + X ℑ ⊗ Y ℜ ). By imp osing the condition that C = C ∗ , we see that the second sum m ust b e zero, so that C can b e expressed by the series (129) alone (where, of course, the trace class op erators X k ; ℜ ⊗ Y k ; ℜ , X k ; ℑ ⊗ Y k ; ℑ can b e suitably renormalized b y mo difying the asso ciated scalar co efficien ts), and hence, by the previously obtained characterization of the Banach space B 1 ( H ⊗ J ) b R , we conclude that indeed C ∈ B 1 ( H ⊗ J ) b R , as w e wished to pro ve. Let us even tually pro ve that ( B 1 ( H ⊗ J ) b , ∥ · ∥ b ⊗ 1 ) is, in natural wa y , a Banach algebra. W e only need to sho w that C 1 , C 2 ∈ B 1 ( H ⊗ J ) b = ⇒ C 1 C 2 ≡ C 1 ◦ C 2 ∈ B 1 ( H ⊗ J ) b and ∥ C 1 C 2 ∥ b ⊗ 1 ≤ ∥ C 1 ∥ b ⊗ 1 ∥ C 2 ∥ b ⊗ 1 ; (130) i.e., that the Banach space ( B 1 ( H ⊗ J ) b , ∥ · ∥ b ⊗ 1 ) is closed wrt to comp osition of op erators and its norm is subm ultiplicativ e. T o this end, for an y ϵ > 0, let us pic k decomp ositions (of the form (114)) C j = X k r ϵ k ; j ( X ϵ k ; j ⊗ Y ϵ k ; j ) , j = 1 , 2 , (131) of the cross trace class operators C 1 and C 2 (con verging wrt b oth the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 ), suc h that X k r ϵ k ; j ≤ ∥ C j ∥ b ⊗ 1 + ϵ , j = 1 , 2 , (132) condition that can alwa ys b e satisfied according to definition (117). Note that the double series X kl r ϵ k ; 1 r ϵ l ; 2  ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 )  (133) is absolutely con vergen t — wrt the b oth the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 — to some cross trace class op erator C ϵ ∈ B 1 ( H ⊗ J ) b ; indeed, P kl r ϵ k ; 1 r ϵ l ; 2 = P k r ϵ k ; 1 P l r ϵ l ; 2 ≤ ( ∥ C 1 ∥ b ⊗ 1 + ϵ )( ∥ C 2 ∥ b ⊗ 1 + ϵ ) and ∥ ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 ) ∥ b ⊗ 1 = ∥ ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 ) ∥ ⊗ 1 = ∥ X ϵ k ; 1 X ϵ l ; 2 ∥ 1 ∥ Y ϵ k ; 1 Y ϵ l ; 2 ∥ 1 ≤ ∥ X ϵ k ; 1 ∥ 1 ∥ X ϵ l ; 2 ∥ 1 ∥ Y ϵ k ; 1 ∥ 1 ∥ Y ϵ l ; 2 ∥ 1 = 1 . (134) W e claim that C ≡ C ϵ = C 1 C 2 (for any ϵ > 0). 30 Indeed, since (by F act 2.2) ∥ A 1 A 2 ∥ ⊗ 1 ≤ ∥ A 1 ∥ ⊗ 1 ∥ A 2 ∥ ⊗ 1 , for all A 1 , A 2 ∈ B 1 ( H ⊗ J ), then the linear maps (of righ t and left m ultiplication b y a fixed trace class op erator) B 1 ( H ⊗ J ) ∋ A 1 7→ A 1 A 2 ∈ B 1 ( H ⊗ J ) and B 1 ( H ⊗ J ) ∋ A 2 7→ A 1 A 2 ∈ B 1 ( H ⊗ J ) (135) are bounded. By the con tinuit y of the first linear map in (135), for any C 1 , C 2 ∈ B 1 ( H ⊗ J ) b , and exploiting the expansion (131) with j = 1, we obtain that C 1 C 2 =  ∥ · ∥ ⊗ 1 – X k r ϵ k ; 1  X ϵ k ; 1 ⊗ Y ϵ k ; 1   C 2 = ∥ · ∥ ⊗ 1 – X k r ϵ k ; 1  ( X ϵ k ; 1 ⊗ Y ϵ k ; 1 ) C 2  , (136) where, b y the contin uity of the se c ond linear map and by the expansion (131) with j = 2, w e ha ve: ( X ϵ k ; 1 ⊗ Y ϵ k ; 1 ) C 2 = ( X ϵ k ; 1 ⊗ Y ϵ k ; 1 ) ∥ · ∥ b ⊗ 1 – X l r ϵ l ; 2  X ϵ l ; 2 ⊗ Y ϵ l ; 2  = ∥ · ∥ ⊗ 1 – X l r ϵ l ; 2  ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 )  . (137) Next, by relations (136) and (137), we find that C 1 C 2 = ∥ · ∥ ⊗ 1 – X k r ϵ k ; 1  ( X ϵ k ; 1 ⊗ Y ϵ k ; 1 ) C 2  = ∥ · ∥ ⊗ 1 – X k r ϵ k ; 1 ∥ · ∥ ⊗ 1 – X l r ϵ l ; 2  ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 )  = ∥ · ∥ ⊗ 1 – X kl r ϵ k ; 1 r ϵ l ; 2  ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 )  = ∥ · ∥ b ⊗ 1 – X kl r ϵ k ; 1 r ϵ l ; 2  ( X ϵ k ; 1 X ϵ l ; 2 ) ⊗ ( Y ϵ k ; 1 Y ϵ l ; 2 )  ∈ B 1 ( H ⊗ J ) b . (138) Here, the iterated series on the second line is equal to the double series on the third line by the absolute conv ergence of the latter (recall F act 2.11), actually wrt b oth the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 . Therefore, w e conclude that C 1 C 2 ∈ B 1 ( H ⊗ J ) b and, for ev ery ϵ > 0, ∥ C 1 C 2 ∥ b ⊗ 1 ≤ P k r ϵ k ; 1 P l r ϵ l ; 2 ≤ ( ∥ C 1 ∥ b ⊗ 1 + ϵ )( ∥ C 2 ∥ b ⊗ 1 + ϵ ). Hence, ∥ C 1 C 2 ∥ b ⊗ 1 ≤ ∥ C 1 ∥ b ⊗ 1 ∥ C 2 ∥ b ⊗ 1 , and the pro of is no w complete. R emark 3.13 . In the ligh t of Corollary 3.5, the ‘abstract’ complex Banach space B 1 ( H ) b ⊗ B 1 ( J ) will b e henceforth iden tified with the range B 1 ( H ⊗ J ) b of the immersion map j . Precisely , from no w on w e will denote by B 1 ( H ) b ⊗ B 1 ( J ) — rather than by B 1 ( H ⊗ J ) b — the Banach space of all trace class op erators of the form (114), the cr oss tr ac e class op er ators , endow ed with the norm ∥ · ∥ b ⊗ 1 defined b y (117), that will b e called the pr oje ctive (cr oss) norm . Once again, w e stress that, in the simple-tensor decomp osition (114), absolute conv ergence is understo o d wrt the norm ∥ · ∥ b ⊗ 1 or, equiv alently , wrt the trace norm ∥ · ∥ ⊗ 1 , b ecause the (absolute) summabilit y and the sum itself of the series do not dep end on which if the tw o norms is considered (Remark 3.12). With an analogous iden tification, from now on we will denote by B 1 ( H ) R b ⊗ B 1 ( J ) R the real vector space B 1 ( H ⊗ J ) b R (the range of the immersion map j R ) — i.e., the real vector space (recall relation (120))  B 1 ( H ) b ⊗ B 1 ( J )  R : =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b : C = C ∗  (139) of all selfadjoint cross trace class op erators — endo wed with the Hermitian pr oje ctive (cr oss) norm | | | · | | | b ⊗ 1 , defined b y (119). Therefore, skipping from no w on all natural immersion maps, the 31 inequalities (85) and (87) relating the v arious norms on B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R can b e rewritten, resp ectiv ely , as ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b , (140) and ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ | | | C | | | b ⊗ 1 ≤ 2 ∥ C ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R ≡ B 1 ( H ⊗ J ) b R . (141) Definition 3.6. Any expansion of a cross trace class operator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b of the form (114) will b e called a standar d de c omp osition of C . An y expansion of a selfadjoint cross trace class op erator C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R ≡ B 1 ( H ⊗ J ) b R of the form (114) — with r k ≥ 0 and X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) ( P k r k < ∞ , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1) — will b e called a Hermitian standar d de c omp osition of C . A standard decomp osition of C ∈ B 1 ( H ) b ⊗ B 1 ( J ) (or a Hermitian standard decomp osition of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R ) of the form (114), with C  = 0, is said to b e she er if it does not contain zero summands; i.e., if r k > 0, for ev ery k . 3.6 The cross trace class and optimal decomp ositions T aking into accoun t the imp ortan t Remark 3.13, we set the following: Definition 3.7. The Banach spaces  B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b , ∥ · ∥ b ⊗ 1  and  B 1 ( H ) R b ⊗ B 1 ( J ) R ≡ B 1 ( H ⊗ J ) b R , | | | · | | | b ⊗ 1  (142) where the norms ∥ · ∥ b ⊗ 1 , | | | · | | | b ⊗ 1 are defined b y (117) and (119), will be called the cr oss tr ac e class and the selfadjoint cr oss tr ac e class of the bipartite Hilb ert space H ⊗ J , resp ectiv ely . Prop osition 3.6. The Banach sp ac es B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b and B 1 ( J ) b ⊗ B 1 ( H ) ≡ B 1 ( J ⊗ H ) b (143) ar e isomorphic. This isomorphism is implemente d by the (wel l define d) tr ansp osition map T : B 1 ( H ) b ⊗ B 1 ( J ) ∋ C = X k r k ( X k ⊗ Y k ) 7→ X k r k ( Y k ⊗ X k ) = : T ( C ) ∈ B 1 ( J ) b ⊗ B 1 ( H ) , (144) wher e C = P k r k ( X k ⊗ Y k ) is any standar d de c omp osition of the cr oss tr ac e class op er ator C (and the mapping in (144) do es not dep end on the choic e of this de c omp osition). A n analo gous isomorphism holds for the Banach sp ac es B 1 ( H ) R b ⊗ B 1 ( J ) R ≡ B 1 ( H ⊗ J ) b R and B 1 ( J ) R b ⊗ B 1 ( H ) R ≡ B 1 ( J ⊗ H ) b R . Let us no w deriv e some direct consequences of Corollary 3.5. Prop osition 3.7. Every cr oss tr ac e class op er ator — wrt the bip artition H ⊗ J — c an b e expr esse d as a line ar c ombination of (at most) two selfadjoint cr oss tr ac e class op er ators; i.e., the fol lowing r elation holds: B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) R b ⊗ B 1 ( J ) R + i  B 1 ( H ) R b ⊗ B 1 ( J ) R  =  C 1 + i C 2 : C 1 , C 2 ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R  . (145) 32 Prop osition 3.8. F or the set D ( H ⊗ J ) b of al l cr oss states wrt the bip artition H ⊗ J , the fol lowing r elation holds: D ( H ⊗ J ) b : = D ( H ⊗ J ) ∩ B 1 ( H ) b ⊗ B 1 ( J ) = D ( H ⊗ J ) ∩ B 1 ( H ) R b ⊗ B 1 ( J ) R . (146) Mor e over, D ( H ⊗ J ) b is a ∥ · ∥ b ⊗ 1 -close d c onvex subset of B 1 ( H ) b ⊗ B 1 ( J ) and a | | | · | | | b ⊗ 1 -close d c onvex subset of B 1 ( H ) R b ⊗ B 1 ( J ) R . Pr o of. In fact, by relation (120), w e ha ve that D ( H ⊗ J ) ∩ B 1 ( H ⊗ J ) b R = D ( H ⊗ J ) ∩ B 1 ( H ⊗ J ) R ∩ B 1 ( H ⊗ J ) b = D ( H ⊗ J ) ∩ B 1 ( H ⊗ J ) b = : D ( H ⊗ J ) b . (147) Th us, with the iden tifications B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b and B 1 ( H ) R b ⊗ B 1 ( J ) R ≡ B 1 ( H ⊗ J ) b R , the first assertion follo ws. Now, let { D n } n ∈ N b e a ∥ · ∥ b ⊗ 1 -con vergen t sequence in D ( H ⊗ J ) b . Clearly , b y its definition, D ( H ⊗ J ) b is a con vex subset of B 1 ( H ) b ⊗ B 1 ( J ), and — since the trace norm ∥ · ∥ ⊗ 1 is dominated b y the pro jective norm ∥ · ∥ b ⊗ 1 on B 1 ( H ) b ⊗ B 1 ( J ), and, moreov er, D ( H ⊗ J ) is a ∥ · ∥ ⊗ 1 -closed subset of B 1 ( H ⊗ J ) — we hav e that ∥ · ∥ ⊗ 1 – lim D n = ∥ · ∥ b ⊗ 1 – lim D n = D ∈ D ( H ⊗ J ) ∩ B 1 ( H ) b ⊗ B 1 ( J ) = : D ( H ⊗ J ) b . (148) Therefore, D ( H ⊗ J ) b is a ∥ · ∥ b ⊗ 1 -closed subset of B 1 ( H ) b ⊗ B 1 ( J ). Recall that the pro jective norm and the Hermitian pro jective norm are equiv alent on B 1 ( H ) R b ⊗ B 1 ( J ) R , hence, a subset of B 1 ( H ) R b ⊗ B 1 ( J ) R is | | | · | | | b ⊗ 1 -closed iff it is ∥ · ∥ b ⊗ 1 -closed. Another noteworth y fact about the con vex set D ( H ⊗ J ) b is that it is ∥ · ∥ ⊗ 1 -dense in D ( H ⊗ J ). T o clarify this p oint, let us consider the following subset of the Hilb ert space H ⊗ J : ( H ⊗ J ) ⋆ : = { a ∈ H ⊗ J \ { 0 } : 1 ≤ srank ( a ) < ∞} , (149) where srank ( a ) is the Sc hmidt rank of a  = 0 (F act 3.1). Clearly , b y the Schmidt decomposition of a v ector in H ⊗ J , it is clear that ( H ⊗ J ) ⋆ is norm-dense in H ⊗ J . W e also consider the asso ciated subset P ( H ⊗ J ) ⋆ of P ( H ⊗ J ), defined b y P ( H ⊗ J ) ⋆ : = {| a ⟩ ⟨ a | ∈ P ( H ⊗ J ) : a ∈ ( H ⊗ J ) ⋆ , ∥ a ∥ = 1 } . (150) Prop osition 3.9. D ( H ⊗ J ) = co ( P ( H ⊗ J ) ⋆ ) = cl ∥ · ∥ ⊗ 1 ( D ( H ⊗ J ) b ) . According to a well-kno wn result [36], the bipartite trace class B 1 ( H ⊗ J ) is a tw o-sided ∗ -ideal in B ( H ⊗ J ): K ∈ B 1 ( H ⊗ J ), L, M ∈ B ( H ⊗ J ) = ⇒ K ∗ ∈ B 1 ( H ⊗ J ) and LK M ∈ B 1 ( H ⊗ J ). F or the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b ⊂ B 1 ( H ⊗ J ) ⊂ B ( H ⊗ J ), a weak er — yet imp ortan t — prop ert y holds; namely , C ∈ B 1 ( H ) b ⊗ B 1 ( J ) = ⇒ C ∗ ∈ B 1 ( H ) b ⊗ B 1 ( J ) (Corollary 3.5) and, moreo v er, one can pro ve the follo wing result (and the consequent Corollary 3.6): Prop osition 3.10. Given any cr oss tr ac e class op er ator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b , and any b ounde d op er ators A, E ∈ B ( H ) and B , F ∈ B ( J ) , the tr ac e class op er ator ( A ⊗ B ) C ( E ⊗ F ) is c ontaine d in the cr oss tr ac e class B 1 ( H ) b ⊗ B 1 ( J ) to o, and the fol lowing r elation holds: ∥ ( A ⊗ B ) C ( E ⊗ F ) ∥ b ⊗ 1 ≤ ∥ A ∥ ∞ ∥ B ∥ ∞ ∥ E ∥ ∞ ∥ F ∥ ∞ ∥ C ∥ b ⊗ 1 . (151) Mor e over, if, in p articular, C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , E = A ∗ and F = B ∗ , then the selfadjoint tr ac e class op er ator ( A ⊗ B ) C ( E ⊗ F ) = ( A ⊗ B ) C ( A ⊗ B ) ∗ is c ontaine d in the selfadjoint cr oss tr ac e class B 1 ( H ) R b ⊗ B 1 ( J ) R to o, and | | | ( A ⊗ B ) C ( A ⊗ B ) ∗ | | | b ⊗ 1 ≤ ( ∥ A ∥ ∞ ∥ B ∥ ∞ ) 2 | | | C | | | b ⊗ 1 . (152) 33 Corollary 3.6. F or any C ∈ B 1 ( H ) b ⊗ B 1 ( J ) and L, M ∈ B ( H ) ˘ ⊗ B ( J ) , L C M ∈ B 1 ( H ) b ⊗ B 1 ( J ) . Among all standard decomp ositions — see Definition 3.6 — of a cross trace class operator, there are some that deserve a sp ecial fo cus. Therefore, coherently with Definition 3.4, and adopting the new notations established in Remark 3.13 and Definition 3.7 for the Banach spaces of cross trace class op erators, we set the following: Definition 3.8. A standard decomp osition P k r k ( X k ⊗ Y k ) of C ∈ B 1 ( H ) b ⊗ B 1 ( J ) ≡ B 1 ( H ⊗ J ) b — with r k ≥ 0 and X k ⊗ Y k ∈ θ ( B 1 ( H ) , B 1 ( J )) ( P k r k < ∞ , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1) — will b e said to be ∥ · ∥ b ⊗ 1 - optimal (or, simply , optimal ) if ∥ C ∥ b ⊗ 1 = P k r k , and a cross trace class op erator admitting an optimal standard decomp osition will b e called ∥ · ∥ b ⊗ 1 - optimal ly de c omp osable (or simply optimal ly de c omp osable ). W e will denote b y ∥ · ∥ b ⊗ 1 - Osd ( C ) the — p ossibly empty — set of all ∥ · ∥ b ⊗ 1 -optimal standard decomp ositions of C , and by Op d  B 1 ( H ) b ⊗ B 1 ( J )  : =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) : ∥ · ∥ b ⊗ 1 - Osd ( C )  = ∅  (153) the set of all ∥ · ∥ b ⊗ 1 -optimally decomp osable op erators in B 1 ( H ) b ⊗ B 1 ( J ). A Hermitian standard decomp osition P k r k ( X k ⊗ Y k ) of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R ≡ B 1 ( H ⊗ J ) b R — with r k ≥ 0 and X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) ( P k r k < ∞ , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1) — will b e said to b e | | | · | | | b ⊗ 1 - optimal if | | | C | | | b ⊗ 1 = P k r k ; moreov er, a selfadjoint cross trace class op erator admitting a | | | · | | | b ⊗ 1 -optimal (Hermitian) standard decomp osition will b e called | | | · | | | b ⊗ 1 -optimal ly de c omp osable . W e will denote b y | | | · | | | b ⊗ 1 - Osd ( C ) the — p ossibly empty — set of all | | | · | | | b ⊗ 1 -optimal (Hermitian) standard decomp ositions of C , and by Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  : =  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : | | | · | | | b ⊗ 1 - Osd ( C )  = ∅  (154) the set of all | | | · | | | b ⊗ 1 -optimally decomp osable op erators in B 1 ( H ) R b ⊗ B 1 ( J ) R . R emark 3.14 . It is clear that Op d  B 1 ( H ) b ⊗ B 1 ( J )  and Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  are line al subsets of B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R , resp ectiv ely; namely , every complex or real multiple of a v ector in Op d  B 1 ( H ) b ⊗ B 1 ( J )  or Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  , resp ectiv ely , b elongs to the same set. R emark 3.15 . A selfadjoin t cross trace class op erator C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R admits both Hermitian standard decompositions and standard decomp ositions that are not Hermitian. In general, a | | | · | | | b ⊗ 1 - optimal Hermitian standard decomp osition (if it exists) will not b e optimal tout c ourt ; in particular, in the case where | | | C | | | b ⊗ 1 > ∥ C ∥ b ⊗ 1 , a | | | · | | | b ⊗ 1 -optimal Hermitian standard decomp osition — if it exists — c annot b e optimal. Therefore, if C is | | | · | | | b ⊗ 1 -optimally decomposable, then it may not b e optimally decomp osable, and vic e versa ; but the picture b ecomes somewhat simpler if w e restrict to Hermitian standard decomp ositions only (see Prop osition 3.11 b elo w). Prop osition 3.11. L et C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R . The fol lowing facts hold true: (O1) If C admits an optimal — i.e., a ∥ · ∥ b ⊗ 1 -optimal — Hermitian standar d de c omp osition, then this de c omp osition is also | | | · | | | b ⊗ 1 -optimal and | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 . (O2) Cle arly, if, c onversely, C admits a | | | · | | | b ⊗ 1 -optimal (Hermitian) standar d de c omp osition and, mor e over, | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 , then this de c omp osition is also optimal. (O3) By the pr evious two p oints, the set ∥ · ∥ b ⊗ 1 - OHsd ( C ) : =  P k r k ( X k ⊗ Y k ) : r k ≥ 0 , X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) , P k r k < ∞ , ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 , P k r k ( X k ⊗ Y k ) = C, ∥ C ∥ b ⊗ 1 = P k r k  , (155) 34 c onsisting of al l optimal Hermitian standar d de c omp ositions of C , is empty if | | | C | | | b ⊗ 1 > ∥ C ∥ b ⊗ 1 ; while, if ∥ · ∥ b ⊗ 1 - OHsd ( C ) is nonempty, then it must c oincide with the set | | | · | | | b ⊗ 1 - Osd ( C ) of al l | | | · | | | b ⊗ 1 -optimal (Hermitian) standar d de c omp ositions of C and | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 . Ther efor e, either ∥ · ∥ b ⊗ 1 - OHsd ( C ) is empty or ∥ · ∥ b ⊗ 1 - OHsd ( C ) = | | | · | | | b ⊗ 1 - Osd ( C )  = ∅ , and, in the latter c ase, | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 . In the ligh t of the previous result, w e set the follo wing: Definition 3.9. A selfadjoint cross trace class op erator C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is said to b e Hermitian-optimal ly de c omp osable if the set ∥ · ∥ b ⊗ 1 - OHsd ( C ) defined by (155) is nonempty , i.e., if it admits an optimal Hermitian standard decomp osition (equiv alen tly , by Proposition 3.11, if C is | | | · | | | b ⊗ 1 -optimally decomposable and | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 ; also, if ∥ · ∥ b ⊗ 1 - OHsd ( C ) = | | | · | | | b ⊗ 1 - Osd ( C )  = ∅ ). W e will denote b y HOp d  B 1 ( H ) R b ⊗ B 1 ( J ) R  : =  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : ∥ · ∥ b ⊗ 1 - OHsd ( C )  = ∅  =  C ∈ Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  : | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1  =  C ∈ Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  : ∥ · ∥ b ⊗ 1 - OHsd ( C ) = | | | · | | | b ⊗ 1 - Osd ( C )  ⊂ Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  (156) the (lineal) set of all Hermitian-optimally decomp osable selfadjoin t cross trace class op erators. The v arious equiv alent c haracterizations of HOp d  B 1 ( H ) R b ⊗ B 1 ( J ) R  in (156) follo w easily from Prop osition 3.11; precisely , the c haracterization on the second line follo ws from (O1) and (O2) , whereas the c haracterization on the third line follo ws from (O3) . Prop osition 3.12. If b oth H and J ar e finite-dimensional Hilb ert sp ac es — N ≡ dim( H ⊗ J ) < ∞ — then every line ar op er ator in B 1 ( H ) b ⊗ B 1 ( J ) is ∥ · ∥ b ⊗ 1 -optimal ly de c omp osable, and every selfadjoint line ar op er ator in B 1 ( H ) R b ⊗ B 1 ( J ) R is | | | · | | | b ⊗ 1 -optimal ly de c omp osable. In the first c ase, ther e is a ∥ · ∥ b ⊗ 1 -optimal standar d de c omp osition of the line ar op er ator c onsisting of, at most, 2 N 2 + 1 simple tensors; in the se c ond c ase, a | | | · | | | b ⊗ 1 -optimal Hermitian standar d de c omp osition c onsisting of, at most, N 2 + 1 simple tensors. Pr o of. By Remark 3.14, it is sufficient to pro ve the result for an y C ∈ B 1 ( H ) b ⊗ B 1 ( J ) suc h that ∥ C ∥ b ⊗ 1 = 1 (resp ectively , for an y C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R suc h that | | | C | | | b ⊗ 1 = 1). Then, let C b elong to the unit spher e S  B 1 ( H ) b ⊗ B 1 ( J )  : =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) : ∥ C ∥ b ⊗ 1 = 1  in the N 2 -dimensional c omplex linear space B ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) (isomorphism of linear spaces), which, regarded as a r e al linear space, b y field restriction, has dimension 2 N 2 . It is known — see, e.g., Prop osition 2.2 of [18] (also see Prop osition 7.D of [20]) — that the closed unit b al l in B 1 ( H ) b ⊗ B 1 ( J ) coincides with the closed con vex h ull of its subset θ ( B ( B 1 ( H )) , B ( B 1 ( J ))), where the closure is tak en wrt an y norm topology , and B ( B 1 ( H )), B ( B 1 ( J )) are the closed unit balls in B 1 ( H ) and B 1 ( J ), resp ectiv ely; i.e., B  B 1 ( H ) b ⊗ B 1 ( J )  = co  θ ( B ( B 1 ( H )) , B ( B 1 ( J )))  = co  θ ( B ( B 1 ( H )) , B ( B 1 ( J )))  . (157) The second equality holds b ecause N ≡ dim( H ⊗ J ) < ∞ , hence the set θ ( B ( B 1 ( H )) , B ( B 1 ( J ))) and its con vex hull are compact (wrt to an y norm on B ( H ⊗ J )). Now — b y Milman’s partial con verse of the Krein-Milman Theorem (see Theorem 2.10.15 of [46]), and since the extreme p oin ts of the unit ball B ( V ) of a Banach space V must b e contained in the unit sphere S ( V ) (F act 2.15) — ext  B  B 1 ( H ) b ⊗ B 1 ( J )  ⊂ θ ( B ( B 1 ( H )) , B ( B 1 ( J ))) ∩ S ( B 1 ( H ) b ⊗ B 1 ( J )) = θ ( S ( B 1 ( H )) , S ( B 1 ( J ))) (one can actually pro ve that the extreme p oin ts of the unit ball of B 1 ( H ) b ⊗ B 1 ( J ) are giv en b y 35 ext  B  B 1 ( H ) b ⊗ B 1 ( J )  = θ ( I 1 ( H ) , I 1 ( J )) ⊂ S  B 1 ( H ) b ⊗ B 1 ( J )  , where I 1 ( H ), I 1 ( J ) are sets of rank-one partial isometries on H and J , resp ectiv ely). Hence, by Carath ´ eo dory’s theorem [50], w e ha ve that C can b e expressed as a conv ex combination of simple tensors of the form C = K X k =1 r k ( X k ⊗ Y k ) , 1 ≤ K ≤ 2 N 2 + 1 , (158) with r k > 0, P k r k = 1 = ∥ C ∥ b ⊗ 1 and X k ⊗ Y k ∈ ext  B  B 1 ( H ) b ⊗ B 1 ( J )  ⊂ θ ( S ( B 1 ( H )) , S ( B 1 ( J ))). Therefore, C admits a ∥ · ∥ b ⊗ 1 -optimal standard decomp osition consisting of, at most, 2 N 2 + 1 sim- ple tensors. The case of the N 2 -dimensional r e al linear space B ( H ⊗ J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R is analogous. Corollary 3.7. If b oth H and J ar e finite-dimensional Hilb ert sp ac es, then HOp d  B 1 ( H ) R b ⊗ B 1 ( J ) R  =  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1  =  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : ∥ · ∥ b ⊗ 1 - OHsd ( C ) = | | | · | | | b ⊗ 1 - Osd ( C )  . (159) 3.7 Natural isometric embeddings and some consequences Giv en closed subspaces V , W of the Hilb ert spaces H and J , resp ectiv ely , it is quite natural to w onder ho w the Banach spaces B 1 ( V ) b ⊗ B 1 ( W ), B 1 ( V ) R b ⊗ B 1 ( W ) R are related to B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R . Before stating the main results, w e start with some useful preliminary facts. First note that, for an y closed subspace V of the Hilb ert space H , and every trace class operator S ∈ B 1 ( V ), there is a natural (isometric, linear) embedding of S in B 1 ( H ) — let us still denote it b y the same symbol S — that is defined by putting S ϕ = S ( ϕ 1 + ϕ 2 ) : = S ϕ 1 , where ϕ = ϕ 1 + ϕ 2 is the decomp osition of a generic vector ϕ ∈ H asso ciated with the orthogonal sum decomp osition H = V ⊕ V ⊥ , where V ⊥ is the orthogonal complemen t of V (whic h is a closed subspace of H to o). Denoting by π 1 , π 2 = I − π 1 the orthogonal pro jections onto V and V ⊥ , resp ectively , we see that S ϕ = S π 1 ϕ = π 1 S ϕ — thus, [ S, π 1 ] = 0 = [ S, I − π 1 ] = [ S, π 2 ] and π 2 S π 2 = S π 2 = 0 — and hence S = π 1 S π 1 + π 1 S π 2 + π 2 S π 1 + π 2 S π 2 = π 1 S π 1 + π 2 S π 2 = π 1 S π 1 . Giv en any closed subspace W of the Hilb ert space J , we then also get a natural (injectiv e, linear) em b edding of B 1 ( V ) ˘ ⊗ B 1 ( W ) into B 1 ( H ) ˘ ⊗ B 1 ( J ), that is isometric wrt the trace norms of the Banach spaces B 1 ( V ⊗ W ) and B 1 ( H ⊗ J ). Analogously , we ha ve a natural (injective, linear) em b edding of B 1 ( V ) R ˘ ⊗ B 1 ( W ) R in to B 1 ( H ) R ˘ ⊗ B 1 ( J ) R , that is isometric wrt the trace norms of the Banac h spaces B 1 ( V ⊗ W ) R and B 1 ( H ⊗ J ) R . Prop osition 3.13. L et V , W b e close d subsp ac es of the Hilb ert sp ac es H and J , r esp e ctively, and let π ∈ B ( H ) , ϖ ∈ B ( J ) b e the ortho gonal pr oje ctions onto V and W . The natur al — inje ctive, line ar — emb e dding map of B 1 ( V ) ˘ ⊗ B 1 ( W ) into B 1 ( H ) ˘ ⊗ B 1 ( J ) is isometric wrt the pr oje ctive norms of these sp ac es. Henc e, it extends to a line ar isometry I of the Banach sp ac e B 1 ( V ) b ⊗ B 1 ( W ) into the Banach sp ac e B 1 ( H ) b ⊗ B 1 ( J ) ; mor e over, for ever C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ( π ⊗ ϖ ) C ( π ⊗ ϖ ) is a cr oss tr ac e class op er ator to o, and ran ( I ) =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) : C = ( π ⊗ ϖ ) C ( π ⊗ ϖ )  =  ( π ⊗ ϖ ) C ( π ⊗ ϖ ) : C ∈ B 1 ( H ) b ⊗ B 1 ( J )  . (160) The line ar map Q : B 1 ( H ) b ⊗ B 1 ( J ) ∋ C 7→ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∈ B 1 ( H ) b ⊗ B 1 ( J ) is a c ontinuous pr oje ction; henc e, ran ( Q ) = ran ( I ) is a c omplemente d subsp ac e of B 1 ( H ) b ⊗ B 1 ( J ) . A n analo gous r esult holds for the Banach sp ac es B 1 ( V ) R b ⊗ B 1 ( W ) R and B 1 ( H ) R b ⊗ B 1 ( J ) R . 36 Pr o of. Let h : B 1 ( V ) → B 1 ( H ), j : B 1 ( W ) → B 1 ( J ) b e the natural linear isometric em b eddings of B 1 ( V ), B 1 ( W ) in to B 1 ( H ) and B 1 ( J ), resp ectiv ely . Let us denote by π ∈ B ( H ), ϖ ∈ B ( J ) the orthogonal pro jections onto the closed subspaces V , W of H and J , resp ectively . Then, we ha ve: ran ( h ) = { S ∈ B 1 ( H ) : S = π S π } = { π S π : S ∈ B 1 ( H ) } , ran ( j ) = { T ∈ B 1 ( J ) : T = ϖ T ϖ } = { ϖ T ϖ : T ∈ B 1 ( J ) } . (161) Let us define the linear map h ⊗ j : B 1 ( V ) ˘ ⊗ B 1 ( W ) → B 1 ( H ) b ⊗ B 1 ( J ) as the standard tensor pro duct of the linear maps h and j (see, e.g., Section 3.3 of [20]), i.e.,  h ⊗ j  ( S ⊗ T ) : = h ( S ) ⊗ j ( T ), for all S ∈ B 1 ( V ) and T ∈ B 1 ( W ). Here, with a sligh t abuse, we hav e assumed the codomain of the map h ⊗ j to b e B 1 ( H ) b ⊗ B 1 ( J ) rather than B 1 ( H ) ˘ ⊗ B 1 ( J ). Since the maps h and j are injectiv e, then h ⊗ j is injectiv e to o, and the linear subspace ran ( h ⊗ j ) of B 1 ( H ) ˘ ⊗ B 1 ( J ) can b e iden tified with — i.e., is linearly isomorphic to — ran ( h ) ˘ ⊗ ran ( j ) = B 1 ( V ) ˘ ⊗ B 1 ( W ). Th us, by natural em b edding, B 1 ( V ) ˘ ⊗ B 1 ( W ) can b e iden tified with the linear subspace of B 1 ( H ) ˘ ⊗ B 1 ( J ) consisting of all op erators of the latter space living on the closed subspace V ⊗ W of H ⊗ J ; i.e., B 1 ( V ) ˘ ⊗ B 1 ( W ) = ran ( h ) ˘ ⊗ ran ( j ) ≡ ran ( h ⊗ j ) =  C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) : C = ( π ⊗ ϖ ) C ( π ⊗ ϖ )  . Note that, b y this remark, denoting b y W V ∥ · ∥ b ⊗ 1 the pro jective norm of the algebraic tensor pro duct B 1 ( V ) ˘ ⊗ B 1 ( W ), it is clear that ∥ C ∥ b ⊗ 1 ≤ W V ∥ C ∥ b ⊗ 1 , for all C ∈ B 1 ( V ) ˘ ⊗ B 1 ( W ), b ecause the ev aluation of the pro jective norm W V ∥ C ∥ b ⊗ 1 in volv es decomp ositions of C into elemen tary tensors living on the subsp ac e V ⊗ W only . Let us prov e that the rev erse inequalit y holds to o. In fact, for ev ery C ∈ B 1 ( V ) ˘ ⊗ B 1 ( W ), if C = P k S k ⊗ T k is any finite decomp osition, then w e also hav e that C = P k ( π ⊗ ϖ ) ( S k ⊗ T k )( π ⊗ ϖ ); hence: ∥ C ∥ b ⊗ 1 : = inf  P k ∥ S k ∥ 1 ∥ T k ∥ 1 : P k S k ⊗ T k = C = ( π ⊗ ϖ ) C ( π ⊗ ϖ )  ≥ inf  P k ∥ π S k π ∥ 1 ∥ ϖ T k ϖ ∥ 1 : P k S k ⊗ T k = C = ( π ⊗ ϖ ) C ( π ⊗ ϖ )  = inf  P k ∥ S k ∥ 1 ∥ T k ∥ 1 : P k S k ⊗ T k = C, S k = π S k π , T k = ϖ T k ϖ  = W V ∥ C ∥ b ⊗ 1 , (162) where the infim um on the first line is extended o ver al l (finite) decomp ositions of C in to elemen tary tensors, and, for obtaining the second line, we ha ve used the fact that, for any S ∈ B 1 ( H ) and T ∈ B 1 ( J ), ∥ π S π ∥ 1 ≤ ( ∥ π ∥ ∞ ) 2 ∥ S ∥ 1 = ∥ S ∥ 1 and ∥ ϖ T ϖ ∥ 1 ≤ ∥ T ∥ 1 . Therefore, actually , for ev ery C ∈ ran ( h ⊗ j ) ≡ B 1 ( V ) ˘ ⊗ B 1 ( W ), we hav e that W V ∥ C ∥ b ⊗ 1 = ∥ C ∥ b ⊗ 1 ; i.e., the linear map h ⊗ j : B 1 ( V ) ˘ ⊗ B 1 ( W ) → B 1 ( H ) b ⊗ B 1 ( J ) is isometric wrt the norms W V ∥ · ∥ b ⊗ 1 and ∥ · ∥ b ⊗ 1 . Hence, by con tinuous extension w e obtain an isometric em b edding — a linear isometry — of B 1 ( V ) b ⊗ B 1 ( W ) in to B 1 ( H ) b ⊗ B 1 ( J ). Summarizing, we ha ve prov ed that there is a linear isometry I : B 1 ( V ) b ⊗ B 1 ( W ) → B 1 ( H ) b ⊗ B 1 ( J ) such that I ( S ⊗ T ) = h ( S ) ⊗ j ( T ), for all S ∈ B 1 ( V ) and T ∈ B 1 ( W ). Hence, for every C = P k S k ⊗ T k ∈ B 1 ( V ) b ⊗ B 1 ( W ) — con vergence wrt the norm W V ∥ · ∥ b ⊗ 1 b eing understo od — we ha ve that I ( C ) = ∥ · ∥ b ⊗ 1 – X k h ( S k ) ⊗ j ( T k ) = ∥ · ∥ b ⊗ 1 – X k  π h ( S k ) π  ⊗  ϖ j ( T k ) ϖ  = ( π ⊗ ϖ )  ∥ · ∥ b ⊗ 1 – P k h ( S k ) ⊗ j ( T k )  ( π ⊗ ϖ ) = ( π ⊗ ϖ ) I ( C ) ( π ⊗ ϖ ) . (163) Here, the first equality holds b y the contin uit y of I , and the second equalit y holds b y relations (161), whereas, for the third equalit y , observe that, b y Prop osition 3.10, the linear map Q : B 1 ( H ) b ⊗ B 1 ( J ) ∋ C 7→ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∈ B 1 ( H ) b ⊗ B 1 ( J ) (164) is well defined — i.e., ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∈ B 1 ( H ) b ⊗ B 1 ( J ) — and b ounded. Therefore, ran ( I ) ⊂  ( π ⊗ ϖ ) C ( π ⊗ ϖ ) : C ∈ B 1 ( H ) b ⊗ B 1 ( J )  =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) : C = ( π ⊗ ϖ ) C ( π ⊗ ϖ )  . Let 37 us show that the reverse inclusion relation holds to o. Indeed, for every C = ∥ · ∥ b ⊗ 1 – P k S k ⊗ T k ∈ B 1 ( H ) b ⊗ B 1 ( J ) suc h that C = ( π ⊗ ϖ ) C ( π ⊗ ϖ ), we ha ve: C = ∥ · ∥ b ⊗ 1 – X k S k ⊗ T k = ( π ⊗ ϖ )  ∥ · ∥ b ⊗ 1 – P k S k ⊗ T k  ( π ⊗ ϖ ) = ∥ · ∥ b ⊗ 1 – X k  π S k π  ⊗  ϖ T k ϖ  = ∥ · ∥ b ⊗ 1 – X k h ( S k ) ⊗ j ( T k ) = I  P k S k ⊗ T k  , (165) for some sets { S k } ⊂ B 1 ( V ), { T k } ⊂ B 1 ( W ) — note: π S k π ∈ ran ( h ), ϖ T k ϖ ∈ ran ( j ) — satisfying ∥ S k ∥ 1 = ∥ h ( S k ) ∥ 1 = ∥ π S π ∥ 1 ≤ ∥ S ∥ 1 and ∥ T k ∥ 1 = ∥ j ( T k ) ∥ 1 = ∥ ϖ T ϖ ∥ 1 ≤ ∥ T ∥ 1 , so that P k S k ⊗ T k is absolutely con vergen t wrt the norm W V ∥ · ∥ b ⊗ 1 of B 1 ( V ) b ⊗ B 1 ( W ). No w, let us consider the b ounded linear map Q : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ) b ⊗ B 1 ( J ) defined b y (164). Clearly , Q 2 = Q ; hence, Q is a con tin uous pro jection. Therefore, b y a w ell-known result, the closed subspace ran ( Q ) = ran ( I ) of B 1 ( H ) b ⊗ B 1 ( J ) is complemented (see Corollary 3.2.15 of [46], or Remark 5.3 of [20]). The case of the Banac h spaces B 1 ( V ) R b ⊗ B 1 ( W ) R and B 1 ( H ) R b ⊗ B 1 ( J ) R is analogous. Prop osition 3.14. L et π ∈ B ( H ) , ϖ ∈ B ( J ) b e the ortho gonal pr oje ctions onto finite-dimensional subsp ac es V , W of H and J , r esp e ctively. Then, every cr oss tr ac e class op er ator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) such that C = ( π ⊗ ϖ ) C ( π ⊗ ϖ ) is optimal ly de c omp osable. Analo gously, every selfadjoint cr oss tr ac e class op er ator C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R such that C = ( π ⊗ ϖ ) C ( π ⊗ ϖ ) is | | | · | | | b ⊗ 1 -optimal ly de c omp osable. Pr o of. By Prop osition 3.13 — given an y pair π ∈ B ( H ), ϖ ∈ B ( J ) of orthogonal pro jections — if C ∈ B 1 ( H ) b ⊗ B 1 ( J ) is such that C = ( π ⊗ ϖ ) C ( π ⊗ ϖ ), then — putting V = ran ( π ) and W = ran ( ϖ ), and considering linear isometry I : B 1 ( V ) b ⊗ B 1 ( W ) → B 1 ( H ) b ⊗ B 1 ( J ) — we can iden tify C with the unique element C of B 1 ( V ) b ⊗ B 1 ( W ) such that I ( C ) = C . Therefore, we hav e: ∥ C ∥ b ⊗ 1 = ∥ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∥ b ⊗ 1 = W V ∥ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∥ b ⊗ 1 . (166) Here, W V ∥ · ∥ b ⊗ 1 denotes the norm of the pro jectiv e tensor pro duct B 1 ( V ) b ⊗ B 1 ( W ), and, with a slight abuse of notation, C = ( π ⊗ ϖ ) C ( π ⊗ ϖ ) is regarded as an element of b oth B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( V ) b ⊗ B 1 ( W ). If the subspaces V , W are finite-dimensional, then, b y Proposition 3.12, there is an optimal standard decomp osition P k r k ( X k ⊗ Y k ) of the op erator ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∈ B 1 ( V ) b ⊗ B 1 ( W ); i.e., P k r k = W V ∥ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∥ b ⊗ 1 = ∥ C ∥ b ⊗ 1 . Then, b y the isometric em b edding of B 1 ( V ) b ⊗ B 1 ( W ) in B 1 ( H ) b ⊗ B 1 ( J ), w e obtain an optimal decomp osition of C . W e hav e argued that, if dim( H ⊗ J ) < ∞ , then b oth Op d  B 1 ( H ) b ⊗ B 1 ( J )  = B 1 ( H ) b ⊗ B 1 ( J ) and Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  = B 1 ( H ) R b ⊗ B 1 ( J ) R (see Prop osition 3.12); moreov er, b y Corollary 3.7, the set HOp d  B 1 ( H ) R b ⊗ B 1 ( J ) R  coincides with  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1  . Now, as a consequence of Prop osition 3.14, w e obtain the follo wing natural generalization of this fact: Corollary 3.8. The sets Op d  B 1 ( H ) b ⊗ B 1 ( J )  and Op d  B 1 ( H ) R b ⊗ B 1 ( J ) R  ar e norm-dense in B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ) R b ⊗ B 1 ( J ) R , r esp e ctively. HOp d  B 1 ( H ) R b ⊗ B 1 ( J ) R  is a norm-dense sub- set of  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1  . Prop osition 3.15. F or every cr oss tr ac e class op er ator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , and for every p air π ∈ B ( H ) , ϖ ∈ B ( J ) of ortho gonal pr oje ction op er ators, ( π ⊗ ϖ ) C ( π ⊗ ϖ ) is a cr oss tr ac e class op er ator to o, and the fol lowing ine quality holds: ∥ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∥ b ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 . (167) 38 L et H = ⊕ j H j , J = ⊕ l J l b e ortho gonal sum de c omp ositions of the Hilb ert sp ac es H and J , and let π j ∈ B ( H ) , ϖ l ∈ B ( J ) b e the ortho gonal pr oje ction op er ators onto the close d subsp ac es H j ⊂ H and J l ⊂ J , r esp e ctively. Then, for every cr oss tr ac e class op er ator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ( π j ⊗ ϖ l ) C ( π m ⊗ ϖ n ) is a cr oss tr ac e class op er ator to o, and we have: ∥ C ∥ b ⊗ 1 ≤ X j lmn   C mn j l   b ⊗ 1 , wher e C mn j l : = ( π j ⊗ ϖ l ) C ( π m ⊗ ϖ n ) ∈ B 1 ( H ) b ⊗ B 1 ( J ) . (168) If, in p articular, the pr evious ortho gonal sum de c omp ositions of the Hilb ert sp ac es H and J ar e finite — i.e., if H = ⊕ n 1 j =1 H j and J = ⊕ n 2 l =1 J l , with n 1 , n 2 ∈ N — then we have that 1 n X j lmn   C mn j l   b ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ X j lmn   C mn j l   b ⊗ 1 = n 1 X j, m =1 n 2 X l, n =1   C mn j l   b ⊗ 1 , (169) wher e n : = card  C mn j l  = 0 : j, m = 1 , . . . , n 1 , l , n = 1 , . . . , n 2  ≤ ( n 1 n 2 ) 2 ; (170) henc e, in the c ase wher e [ C , π j ⊗ ϖ l ] = 0 , for al l j ∈ { 1 , . . . , n 1 } and l ∈ { 1 , . . . , n 2 } — e quivalently, for C = P j l C j l j l , with C j l j l : = ( π j ⊗ ϖ l ) C ( π j ⊗ ϖ l ) — n ≤ n 1 n 2 and we have: 1 n 1 n 2 X j l    C j l j l    b ⊗ 1 ≤ 1 n X j l    C j l j l    b ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ X j l    C j l j l    b ⊗ 1 . (171) Pr o of. The first assertion follo ws immediately from Prop osition 3.10 — see relation (151) — taking in to account the fact that ∥ π ∥ ∞ = ∥ ϖ ∥ ∞ = 1. Regarding the second assertion (relation (168)), note that — since P j l π j ⊗ ϖ l = I , where the double sum, whenever infinite, con verges (unconditionally , i.e., for ev ery ordering of the index set) wrt the strong op erator topology of B ( H ⊗ J ) — for every a ∈ H ⊗ J we ha ve: C a = C X mn  ( π m ⊗ ϖ n ) a  = X mn  C ( π m ⊗ ϖ n ) a  = X j l X mn  ( π j ⊗ ϖ l ) C ( π m ⊗ ϖ n ) a  = X j l X mn  C mn j l a  . (172) Here, the second equalit y holds because C is a bounded op erator, and C mn j l : = ( π j ⊗ ϖ l ) C ( π m ⊗ ϖ n ) b elongs to B 1 ( H ) b ⊗ B 1 ( J ). Thus, b y relation (172), the iterated sums P j l P mn C mn j l — whenever infinite — con verge (unconditionally) to C wrt the strong op erator top ology . If P j lmn ∥ C mn j l ∥ b ⊗ 1 = ∞ , relation (168) is obvious. Let us supp ose that P j lmn ∥ C mn j l ∥ b ⊗ 1 < ∞ , instead, so that the sum P j lmn C mn j l — whenev er infinite — is absolutely (hence, unconditionally) con vergen t wrt the pro jective norm ∥ · ∥ b ⊗ 1 . Hence, it will con verge unconditionally , wrt the (subspace top ology induced by the) strong op erator top ology — that is weak er than the pro jectiv e norm top ology — to the same limit, which, b y relation (172), must b e precisely C ; i.e., ∥ · ∥ b ⊗ 1 – X j lmn C mn j l = ∥ · ∥ b ⊗ 1 – X j l ∥ · ∥ b ⊗ 1 – X mn C mn j l = X j l X mn C mn j l = C, (173) where the first series con verges absolutely , whence w e immediately get relation (168). Assuming, now, that b oth the orthogonal sum decomp ositions H = ⊕ j H j and J = ⊕ l J l are finite — sp ecifically , that they in volv e n 1 and n 2 subspaces, resp ectiv ely — the inequality ∥ C mn j l ∥ b ⊗ 1 = ∥ ( π j ⊗ ϖ l ) C ( π m ⊗ ϖ n ) ∥ b ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 also entails that P j lmn ∥ C mn j l ∥ b ⊗ 1 ≤ n ∥ C ∥ b ⊗ 1 , where the n umber n ≤ ( n 1 n 2 ) 2 is defined b y (170). It follows that relation (169) and — in the case where [ C, π j ⊗ ϖ l ] = 0, with j ∈ { 1 , . . . , n 1 } , l ∈ { 1 , . . . , n 2 } — (171) hold to o. 39 R emark 3.16 . By relation (167) in Prop osition 3.15, and taking into accoun t Prop osition 3.13, for ev ery pair of orthogonal pro jections π ∈ B ( H ) and ϖ ∈ B ( J ), setting V = ran ( π ) and W = ran ( ϖ ), W V ∥ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∥ b ⊗ 1 = ∥ ( π ⊗ ϖ ) C ( π ⊗ ϖ ) ∥ b ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , (174) where W V ∥ · ∥ b ⊗ 1 is the norm of the pro jectiv e tensor pro duct B 1 ( V ) b ⊗ B 1 ( W ). 3.8 The bipartite trace class vs the cross trace class Exploiting the to ols developed so far, we are no w able to obtain further information ab out the Banac h space B 1 ( H ) b ⊗ B 1 ( J ) of all cross trace class op erators, with a sp ecial fo cus on its relation with the bip artite tr ac e class B 1 ( H ⊗ J ), that coincides, as we hav e sho wn in Subsection 3.2 (see Theorem 3.1), with the natural (or spatial) tensor pro duct B 1 ( H ) ⊗ B 1 ( J ) of the trace classes B 1 ( H ) and B 1 ( J ), defined by (31). Ev entually , it will turn out that there is a sharp distinction b etw een the case where at least one of the Hilb ert spaces of the bipartition H ⊗ J is finite-dimensional and the genuinely infinite-dimensional setting, i.e., the case where dim( H ) = dim( J ) = ∞ . Let us fo cus, at first, on the case where at most one of the Hilbert spaces H and J is infinite-dimensional. Prop osition 3.16. The norme d ve ctor sp ac e  B 1 ( H ) ˘ ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1  is c omplete — otherwise state d, B 1 ( H ) ˘ ⊗ B 1 ( J ) = B 1 ( H ) b ⊗ B 1 ( J ) — iff H and/or J ar e finite-dimensional. In the c ase wher e M = min { dim( H ) , dim( J ) } < ∞ , every element C of B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) ˘ ⊗ B 1 ( J ) admits a (finite) standar d de c omp osition of the form C = K X k =1 r k ( X k ⊗ Y k ) , (175) with K = M 2 , r k ≥ 0 , X k ⊗ Y k ∈ θ ( B 1 ( H ) , B 1 ( J )) and ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 ; mor e over, { X k } K k =1 , if dim( H ) = min { dim( H ) , dim( J ) } < ∞ — or { Y k } K k =1 , if dim( J ) = min { dim( H ) , dim( J ) } < ∞ — c an b e assume d to b e any fixe d (maximal) line arly indep endent set and, in p articular, any fixe d orthonormal b asis wrt the Hilb ert-Schmidt sc alar pr o duct. A nalo gously, B 1 ( H ) R ˘ ⊗ B 1 ( J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R — iff H and/or J ar e finite-dimensional, and, in the c ase wher e M = min { dim( H ) , dim( J ) } < ∞ , every element C of the r e al Banach sp ac e B 1 ( H ) R b ⊗ B 1 ( J ) R = B 1 ( H ) R ˘ ⊗ B 1 ( J ) R admits a (finite) standar d de c omp osition of the form (175) , with K = M 2 , r k ≥ 0 , X k ⊗ Y k ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) and ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 ; the set { X k } K k =1 , if dim( H ) = min { dim( H ) , dim( J ) } < ∞ — or { Y k } K k =1 , if dim( J ) = min { dim( H ) , dim( J ) } < ∞ — c an b e assume d to b e any fixe d orthonormal b asis wrt the r e al Hilb ert-Schmidt sc alar pr o duct. Our next step is to argue that, in the case where min { dim( H ) , dim( J ) } < ∞ , the cross trace class on H ⊗ J is the trace class tout c ourt — i.e., B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ⊗ J ) (where a set equalit y , or an isomorphism of linear spaces, is understoo d) — and, moreo ver, the pro jective norm and the trace norm are equiv alent on this linear space; otherwise stated, that the Banach space  B 1 ( H ) b ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1  can b e regarded as a r enorming [64] of the trace class  B 1 ( H ⊗ J ) , ∥ · ∥ ⊗ 1  . Theorem 3.3. Assume that M = min { dim( H ) , dim( J ) } < ∞ . Then, we have that B 1 ( H ⊗ J ) = B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) ˘ ⊗ B 1 ( J ) (176) and, henc e, D ( H ⊗ J ) b : = D ( H ⊗ J ) ∩ B 1 ( H ) b ⊗ B 1 ( J ) = D ( H ⊗ J ) . (177) 40 The norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 ar e e quivalent on the line ar sp ac e B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) . Inde e d, for every A ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) , the fol lowing ine qualities hold: ∥ A ∥ ⊗ 1 ≤ ∥ A ∥ b ⊗ 1 ≤ N ∥ A ∥ ⊗ 1 , wher e N = min  4 M , M 2  . (178) F or the selfadjoint tr ac e class op er ators and for the density op er ators on H ⊗ J , in p articular, str onger estimates hold, i.e., ∥ A ∥ ⊗ 1 ≤ ∥ A ∥ b ⊗ 1 ≤ 2 M ∥ A ∥ ⊗ 1 , ∀ A ∈ B 1 ( H ⊗ J ) R , (179) 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ M , ∀ D ∈ D ( H ⊗ J ) = D ( H ⊗ J ) b . (180) A ctual ly, in r elation (180) , b oth ine qualities c an b e satur ate d; in fact, ∥D ( H ⊗ J ) ∥ b ⊗ 1 = [1 , M ] . A nalo gously, we have that B 1 ( H ⊗ J ) R = B 1 ( H ) R ⊗ B 1 ( J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R = B 1 ( H ) R ˘ ⊗ B 1 ( J ) R . (181) The norms ∥ · ∥ ⊗ 1 , ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 ar e mutual ly e quivalent on B 1 ( H ⊗ J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R , b e c ause, for every A ∈ B 1 ( H ⊗ J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R , we have that ∥ A ∥ ⊗ 1 ≤ ∥ A ∥ b ⊗ 1 ≤ | | | A | | | b ⊗ 1 ≤ 2 ∥ A ∥ b ⊗ 1 ≤ 4 M ∥ A ∥ ⊗ 1 ; (182) in p articular, for every density op er ator D ∈ D ( H ⊗ J ) = D ( H ⊗ J ) b , 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ | | | D | | | b ⊗ 1 ≤ 2 ∥ D ∥ b ⊗ 1 ≤ 2 M . (183) Pr o of. Assume that M ≡ dim( H ) < ∞ and dim( H ) ≤ dim( J ) ≤ ∞ (obviously , the case where dim( J ) ≤ dim( H ) ≤ ∞ and dim( J ) = M < ∞ is analogous). Let us first pro ve relations (176) and (181). T o this end, it is sufficien t to sho w that — with our initial assumption — B 1 ( H ⊗ J ) ⊂ B 1 ( H ) b ⊗ B 1 ( J ). In fact, giv en an y trace class op erator A ∈ B 1 ( H ⊗ J ), A  = 0, w e can write the singular v alue decomp osition (recall Remark 2.1) A = X k s k [ a k c k , s k > 0 , [ a k c k ≡ | a k ⟩ ⟨ c k | (184) — that con verges wrt the (bipartite) trace norm ∥ · ∥ ⊗ 1 — where { a k } , { c k } are orthonormal systems in H ⊗ J . Now, let us consider extende d Schmidt decomp ositions of the vectors a k , c k ∈ H ⊗ J (recall Remark 3.2), i.e., a k = P M m =1 t k m  η k m ⊗ χ k m  and c k = P M n =1 u k n  ϕ k n ⊗ ψ k n  , where { η k m } M m =1 , { ϕ k n } M n =1 are orthonormal bases in H , { χ k m } M m =1 , { ψ k n } M n =1 are orthonormal systems in J , and t k m , u k n are non-negative real num b ers such that P M m =1  t k m  2 = 1 = P M n =1  u k n  2 = ⇒ 0 ≤ t k m , u k n ≤ 1. A t this p oin t — putting, as usual, c η ϕ ≡ | η ⟩ ⟨ ϕ | — we can write A = ∥ · ∥ ⊗ 1 – X k s k  P M m, n =1 t k m u k n  \ η k m ϕ k n ⊗ \ χ k m ψ k n  = M X m, n =1 ∥ · ∥ ⊗ 1 – X k s k t k m u k n  \ η k m ϕ k n ⊗ \ χ k m ψ k n  = M X m, n =1 A mn , (185) where the trace class op erator A mn ∈ B 1 ( H ⊗ J ), A mn : = ∥ · ∥ ⊗ 1 – X k s k t k m u k n  \ η k m ϕ k n ⊗ \ χ k m ψ k n  , (186) 41 is contained in B 1 ( H ) b ⊗ B 1 ( J ), as well, b ecause 0 < X k s k t k m u k n ≤ X k s k = ∥ A ∥ ⊗ 1 ; (187) hence, the series in (186) conv erges (absolutely) wrt the pro jective norm ∥ · ∥ b ⊗ 1 to o. Therefore, a generic element of B 1 ( H ⊗ J ) can alwa ys b e expressed as the sum of a finite n umber of elemen ts of B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) ˘ ⊗ B 1 ( J ), so that B 1 ( H ⊗ J ) ⊂ B 1 ( H ) b ⊗ B 1 ( J ); hence, actually , B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ). It follows that B 1 ( H ⊗ J ) R ⊂ B 1 ( H ) b ⊗ B 1 ( J ), and hence — recalling relation (120) and Remark 3.13 — B 1 ( H ⊗ J ) R = B 1 ( H ⊗ J ) R ∩ B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) R b ⊗ B 1 ( J ) R , as w ell. Note that for every A ∈ B 1 ( H ⊗ J ) — since, from (186) and (187), one e asily obtains the inequalit y ∥ A mn ∥ b ⊗ 1 ≤ P k s k t k m u k n ≤ ∥ A ∥ ⊗ 1 — the follo wing simple estimate holds: ∥ A ∥ b ⊗ 1 =   P M m, n =1 A mn   b ⊗ 1 ≤ P M m, n =1 ∥ A mn ∥ b ⊗ 1 ≤ M 2 ∥ A ∥ ⊗ 1 . (188) Ho wev er, in the case where M ≥ 5 or A = A ∗ — and, in particular, for a density op erator — stronger estimates hold. In fact, b y Theorem 14.1 of [25], relation (180) holds true and, actually , ∥D ( H ⊗ J ) ∥ b ⊗ 1 = [1 , M ]. F rom this result, we easily conclude that relation (179) holds to o. In fact, ev ery A ∈ B 1 ( H ⊗ J ) R can be expressed as a linear com bination of tw o positive op erators (F act 2.3), i.e., A = A + − A − = a + D + − a − D − , where A + + A − = | A | , A + A − = 0 = A − A + , a + , a − ∈ R + , D + , D − ∈ D ( H ⊗ J ), A + = a + D + and A − = a − D − ; hence: a + , a − ≤ ∥ A ∥ ⊗ 1 : = tr( | A | ) = tr( A + + A − ) = a + + a − . It follows that ∥ A ∥ b ⊗ 1 ≤ a + ∥ D + ∥ b ⊗ 1 + a − ∥ D − ∥ b ⊗ 1 ≤ ∥ A ∥ ⊗ 1  ∥ D + ∥ b ⊗ 1 + ∥ D − ∥ b ⊗ 1  ≤ 2 M ∥ A ∥ ⊗ 1 . Therefore, b oth relations (179) and (180) hold true, and — since, for ev ery A ∈ B 1 ( H ⊗ J ) R , | | | A | | | b ⊗ 1 ≤ 2 ∥ A ∥ b ⊗ 1 — the estimates (182) and (183) immediately follow. Finally from the estimate (179) — since ev ery A ∈ B 1 ( H ⊗ J ) can b e written as A = A 1 + i A 2 , where A 1 , A 2 are selfadjoint trace class op erators in B 1 ( H ⊗ J ) suc h that ∥ A 1 ∥ ⊗ 1 , ∥ A 2 ∥ ⊗ 1 ≤ ∥ A ∥ ⊗ 1 (e.g., ∥ A 1 ∥ ⊗ 1 ≤ 1 2 ( ∥ A ∥ ⊗ 1 + ∥ A ∗ ∥ ⊗ 1 ) = ∥ A ∥ ⊗ 1 ) — w e see that ∥ A ∥ b ⊗ 1 ≤ ∥ A 1 ∥ b ⊗ 1 + ∥ A 2 ∥ b ⊗ 1 ≤ 2 M ( ∥ A 1 ∥ ⊗ 1 + ∥ A 2 ∥ ⊗ 1 ) ≤ 4 M ∥ A ∥ ⊗ 1 . (189) F rom (188) and (189), we no w conclude that relation (178) holds true. Corollary 3.9. Assume that M = min { dim( H ) , dim( J ) } < ∞ . In this c ase, the Banach sp ac e dual  B 1 ( H ) b ⊗ B 1 ( J )  ∗ of B 1 ( H ) b ⊗ B 1 ( J ) c an b e identifie d — via the tr ac e functional — with the algebr aic tensor pr o duct B ( H ) ˘ ⊗ B ( J ) , endowe d with a norm that is e quivalent to the standar d op er ator norm ∥ · ∥ ⊗ ∞ . In fact, with our initial assumption, ( B ( H ) ˘ ⊗ B ( J ) , ∥ · ∥ ⊗ ∞ ) is a Banach sp ac e, that c oincides with B ( H ⊗ J ) , and c an b e identifie d — via the tr ac e functional — with a r enorming of  B 1 ( H ) b ⊗ B 1 ( J )  ∗ . Ther efor e, in p articular, every element of  B 1 ( H ) b ⊗ B 1 ( J )  ∗ c an b e expr esse d in the form J X j =1 tr  ( · )( A j ⊗ B j )  : B 1 ( H ) b ⊗ B 1 ( J ) → C , A j ∈ B ( H ) , B j ∈ B ( J ) , (190) with J ≤ K = M 2 . Pr o of. Suppose that M = min { dim( H ) , dim( J ) } < ∞ . Then, b y Theorem 3.3, the linear spaces B 1 ( H ) ˘ ⊗ B 1 ( J ), B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ⊗ J ) coincide, and, moreov er, the norms ∥ · ∥ b ⊗ 1 and ∥ · ∥ ⊗ 1 are equiv alen t (hence, they induce the same top ology). Therefore,  B 1 ( H ) b ⊗ B 1 ( J )  ∗ can b e identified (by means of the trace functional) — as a linear space — with B ( H ⊗ J ), and, as 42 a Banach space, with a r enorming of the dual B ( H ⊗ J ) = B 1 ( H ⊗ J ) ∗ of B 1 ( H ⊗ J ) (this is a standard fact of the theory of renormings of Banac h spaces; see, e.g., Chapter 3 of [64]). Let us no w focus on B ( H ⊗ J ). It is a standard fact of the theory of operator algebras (see [44], Section 11.1.5 and Section 11.1.6) that — in the case where at least one of the Hilb ert spaces H , J is finite-dimensional — denoting by B ( H ) ⊗ B ( J ) the C ∗ -algebraic tensor pro duct of B ( H ) and B ( J ), we hav e that B ( H ) ˘ ⊗ B ( J ) = B ( H ) ⊗ B ( J ) = B ( H ⊗ J ); moreo ver, every element of B ( H ⊗ J ) is of the form P J j =1 A j ⊗ B j , where A j ∈ B ( H ), B j ∈ B ( J ) and J ≤ K = M 2 , with M = min { dim( H ) , dim( J ) } < ∞ . (F or further details, see our previous Remark 3.5.) Ev entually , com bining all the preceding facts, we conclude that, with our previous assumptions, ev ery elemen t of the Banac h space  B 1 ( H ) b ⊗ B 1 ( J )  ∗ can b e expressed in the form (190). R emark 3.17 . In the case where M = min { dim( H ) , dim( J ) } < ∞ , we hav e an isomorphism of Banac h spaces B ( H ) ˇ ⊗ B ( J ) ∼ =  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , (191) where B ( H ) ˇ ⊗ B ( J ) is the inje ctive tensor pr o duct [17–20] of the Banach spaces B ( H ) and B ( J ); i.e., the Banach space completion of the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) wrt the inje ctive norm ∥ · ∥ ˇ ⊗ ∞ . F or every P j A j ⊗ B j ∈ B ( H ) ˘ ⊗ B ( J ), we ha ve the following expression:   P j A j ⊗ B j   ˇ ⊗ ∞ : = sup    P j s ( A j ) t ( B j )   : s ∈ B ( H ) ∗ , t ∈ B ( J ) ∗ , ∥ s ∥ B = ∥ t ∥ B = 1  = sup    P j tr( S A j ) tr( T B j )   : S ∈ B 1 ( H ) , T ∈ B 1 ( J ) , ∥ S ∥ 1 = ∥ T ∥ 1 = 1  . (192) Here, w e hav e used the fact that the unit balls B ( B 1 ( H )) and B ( B 1 ( J )) — regarded as subsets of the biduals B 1 ( H ) ∗∗ and B 1 ( J ) ∗∗ — are norming sets for B 1 ( H ) ∗ = B ( H ) and B 1 ( J ) ∗ = B ( J ), resp ectiv ely; see Subsection 4.1 in Chapter I of [17], or form ula (3.3) in Section 3.1 of [18]. Note that the previous expression (192) of the injective norm, as w ell as all our further argumen ts b elo w, do not dep end on the specific (finite) representation of an element of the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ). It turns out that, with our initial assumption (min { dim( H ) , dim( J ) } < ∞ ), B ( H ) ˘ ⊗ B ( J ) — which, as a linear space, coincides with B ( H ⊗ J ) (Corollary 3.9) — is already complete wrt the injectiv e norm ∥ · ∥ ˇ ⊗ ∞ . In fact, b y p oin ts (b1) and (c1) in Theorem 7.17 of [20] (also see the standard isometric embeddings on the third line of (3.4) in Section 3.1 of [18]), if B ( H ) ˘ ⊗ B ( J ) is endo wed with the norm ∥ · ∥ ˇ ⊗ ∞ , then the mapping B ( H ) ˘ ⊗ B ( J ) ∋ X j A j ⊗ B j 7→ X j tr  ( · ) A j  B j ∈ B ( B 1 ( H ); B ( J )) (193) is an isometric em b edding, which extends to an isometric em b edding of the injectiv e tensor product B ( H ) ˇ ⊗ B ( J ) in to B ( B 1 ( H ); B ( J )). Note that in (193), and in (194) below, the sums are supp osed to b e finite . Now — assuming, e.g., that dim( J ) < ∞ (the case where dim( H ) < ∞ is analogous) — it is easy to see that the map (193) is surjective; hence, actually , ( B ( H ) ˘ ⊗ B ( J ) , ∥ · ∥ ˇ ⊗ ∞ ) is a Banac h space isomorphic to B ( B 1 ( H ); B ( J )). At this p oin t, it is sufficient to recall that the map B ( B 1 ( H ); B ( J )) ∋ X j tr  ( · ) A j  B j 7→ X j tr  ( · )( A j ⊗ B j )  ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ (194) is an isomorphism of Banac h spaces (see Section 2.2 of [18]; also, p oin t (e) in Remark 7.13 of [20])); hence, by composing the maps (193) and (194), the isomorphism of Banac h spaces (191) holds. It is worth observing that, by this isomorphism and by relation (192), w e ha ve:   P j A j ⊗ B j   ˇ ⊗ ∞ = sup    P j tr( C ( A j ⊗ B j ))   : C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ∥ C ∥ b ⊗ 1 = 1  . (195) 43 By this expression of the injective norm ∥ · ∥ ˇ ⊗ ∞ and by the estimate (178), we see that N − 1   P j A j ⊗ B j   ⊗ ∞ ≤   P j A j ⊗ B j   ˇ ⊗ ∞ ≤   P j A j ⊗ B j   ⊗ ∞ , where N = min  4 M , M 2  . (196) Th us, as stated in Corollary 3.9, the operator norm ∥ · ∥ ⊗ ∞ and the injectiv e norm ∥ · ∥ ˇ ⊗ ∞ — whic h can b e iden tified with the norm of  B 1 ( H ) b ⊗ B 1 ( J )  ∗ — are equiv alent on the linear space B ( H ) ˘ ⊗ B ( J ); i.e., the injective tensor pro duct B ( H ) ˇ ⊗ B ( J ) ∼ =  B 1 ( H ) b ⊗ B 1 ( J )  ∗ is a renorming of the Banach space ( B ( H ) ˘ ⊗ B ( J ) , ∥ · ∥ ⊗ ∞ ) = B ( H ⊗ J ). By Theorem 3.3, we are now able to derive tw o further results characterizing the trace class v ersus the cr oss trace class of the Hilb ert space H ⊗ J , in the genuinely infinite-dimensional setting. Corollary 3.10. In the c ase wher e dim( H ) = dim( J ) = ∞ , the pr oje ctive norm ∥ · ∥ b ⊗ 1 — and, henc e, the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 — is unb ounde d on the set D ( H ⊗ J ) b of cr oss states. Pr o of. Let { ψ m } ∞ m =1 b e an orthonormal basis in the Hilb ert space J , and, given any M ∈ N , with M ≥ 2, put J ( M ) : = span { ψ m } M m =1 . By Prop osition 3.13, the space B 1 ( H ) b ⊗ B 1 ( J ( M ) ) = B 1 ( H ⊗ J ( M ) ) can b e (isometrically) identified with a linear subspace of B 1 ( H ) b ⊗ B 1 ( J ). Denoting b y M ∥ · ∥ b ⊗ 1 the pro jective norm of B 1 ( H ) b ⊗ B 1 ( J ( M ) ), by the assertion of Theorem 3.3 that follo ws relation (180), there is some density op erator D ∈ B 1 ( H ) b ⊗ B 1 ( J ( M ) ) ⊂ B 1 ( H ) b ⊗ B 1 ( J ) such that M ∥ D ∥ b ⊗ 1 = M . Since ∥ D ∥ b ⊗ 1 = M ∥ D ∥ b ⊗ 1 = M , then, by the arbitrariness of M ≥ 2, we conclude that the pro jectiv e norm ∥ · ∥ b ⊗ 1 — hence, the Hermitian pro jective norm | | | · | | | b ⊗ 1 — is unbounded on D ( H ⊗ J ) b . T o prov e the second announced consequence of Theorem 3.3 — i.e., to sho w that, in the gen- uinely infinite-dimensional setting , there are states that are not cross states — we first note a tec hnical p oin t that follo ws immediately from Prop osition 3.15. Lemma 3.2. L et {J l } N l =1 , N ∈ N , b e a set of mutual ly ortho gonal close d subsp ac es of the the Hilb ert sp ac e J , and let ϖ l ∈ B ( J ) b e the ortho gonal pr oje ction op er ator onto the close d subsp ac e J l . F or every cr oss tr ac e class op er ator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ( I ⊗ ϖ l ) C ( I ⊗ ϖ l ) is a cr oss tr ac e class op er ator to o, and, if C = P N l =1 ( I ⊗ ϖ l ) C ( I ⊗ ϖ l ) , then we have that N − 1 N X l =1 ∥ C l ∥ b ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ N X l =1 ∥ C l ∥ b ⊗ 1 , wher e C l : = ( I ⊗ ϖ l ) C ( I ⊗ ϖ l ) ∈ B 1 ( H ) b ⊗ B 1 ( J ) . (197) Corollary 3.11. In the c ase wher e dim( H ) = dim( J ) = ∞ , the fol lowing strict set inclusions hold: B 1 ( H ) b ⊗ B 1 ( J ) ⊊ B 1 ( H ⊗ J ) and, in p articular, D ( H ⊗ J ) b ⊊ D ( H ⊗ J ) . Pr o of. Let us explicitly construct a suitable densit y operator D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . T o this end, given an y orthonormal basis { ψ n } ∞ n =1 in J , we put J l : = span { ψ j ( l ) , . . . , ψ k ( l ) } , l ∈ N , with j ( l ) : = l − 1 X n =0 2 2 n = 2 2 l − 1 3 ∈ { 1 , 5 , . . . } , k ( l ) : = l X n =1 2 2 n = 4 j ( l ) = 4  2 2 l − 1  3 ∈ { 4 , 20 , . . . } , (198) so that dim( J l ) = k ( l ) − j ( l ) + 1 = 2 2 l and J = ⊕ ∞ l =1 J l . By the assertion of Theorem 3.3 that follows relation (180), w e can choose some densit y operator ∆ l ∈ D ( H ⊗ J l ) ⊂ B 1 ( H ⊗ J l ) = B 1 ( H ) b ⊗ B 1 ( J l ) — which, by Prop osition 3.13, can b e isometrically embedded in B 1 ( H ) b ⊗ B 1 ( J ) — such that ∥ ∆ l ∥ b ⊗ 1 = dim( J l ) = 2 2 l . Let us then select a densit y op erator D in the ∥ · ∥ ⊗ 1 -closed con v ex hull co ( { ∆ l } ) ⊂ D ( H ⊗ J ); precisely , we set D : = ∥ · ∥ ⊗ 1 – ∞ X l =1 2 − l ∆ l ∈ D ( H ⊗ J ) . (199) 44 W e will no w argue that D cannot belong to the set D ( H ⊗ J ) b of all cross states wrt the bipartition H ⊗ J . In fact, putting D ( N ) : = P N l =1 2 − l ∆ l ∈ B 1 ( H ) b ⊗ B 1 ( J ), for any N ∈ N , we ha ve that 2 − l ∥ ∆ l ∥ b ⊗ 1 = 2 l and ∥ D ( N ) ∥ b ⊗ 1 ≥ N − 1 P N l =1 2 − l ∥ ∆ l ∥ b ⊗ 1 = N − 1 P N l =1 2 l =  2 N +1 − 2  / N . (200) Here, the inequality ∥ D ( N ) ∥ b ⊗ 1 ≥ N − 1 P N l =1 2 − l ∥ ∆ l ∥ b ⊗ 1 follo ws directly from Lemma 3.2 (with C = D ( N ) ), b ecause, by construction, 2 − l ∆ l = D l = ( I ⊗ ϖ l ) D ( I ⊗ ϖ l ) = ( I ⊗ ϖ l ) D ( N ) ( I ⊗ ϖ l ), where ϖ l ∈ B ( J ) is the orthogonal pro jection op erator on to the closed subspace J l of J . Hence, w e ha ve: ∥ D ( N ) ∥ b ⊗ 1 ≥  2 N +1 − 2  / N ≥ 2 N / N , ∀ N ∈ N = ⇒ lim N ∥ D ( N ) ∥ b ⊗ 1 = ∞ . (201) Therefore, w e must conclude that the series (199) cannot con verge wrt the pro jective norm, but this fact alone do es not entail that D ∈ D ( H ⊗ J ) b . T o prov e that the density op erator D c annot b e a cross state, we will argue b y contradiction. Indeed, first note that D ( N ) = ( I ⊗ ϖ ( N ) ) D ( I ⊗ ϖ ( N ) ), where ϖ ( N ) is the pro jection op erator defined b y ϖ ( N ) : = P N l =1 ϖ l ; i.e., the orthogonal pro jection on to ⊕ N l =1 J l . Next, supp ose that D ∈ D ( H ⊗ J ) b . Then, by relation (167) — with the ob vious iden tifications π ≡ I and ϖ ≡ ϖ ( N ) — we should hav e that ∥ D ∥ b ⊗ 1 ≥ ∥ ( I ⊗ ϖ ( N ) ) D ( I ⊗ ϖ ( N ) ) ∥ b ⊗ 1 = ∥ D ( N ) ∥ b ⊗ 1 ≥ 2 N / N , ∀ N ∈ N . (202) Th us, w e find a con tradiction, and hence D = ∥ · ∥ ⊗ 1 – lim N D ( N ) ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . R emark 3.18 . The state D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b constructed in the pro of of Corollary 3.11 — see (199) — is not pur e . In Section 6 — see, in particular, Example 6.1 — we will show that, in the genuinely infinite-dimensional setting, there also exist pur e states whic h are not cross states. 3.9 The signed decomp osition of a selfadjoint cross trace class op erator W e will now introduce a suitable simple-tensor decomp osition, of the general form (95), of an elemen t of the selfadjoint cross trace class B 1 ( H ) R b ⊗ B 1 ( J ) R , where we relax the assumption that the scalar co efficien ts are non-negative — lik e in the standard decomp osition (118) — but , instead, w e assume the p ositivity of the op erators inv olv ed in the elementary tensor pro duct. Sp ecifically , since these op erators are also assumed to b e normalized, we require them to b e density op erators, i.e., to b elong to the set θ ( D ( H ) , D ( J )). Lemma 3.3. F or every S ∈ B 1 ( H ) , | tr( S ) | ≤ ∥ S ∥ 1 , and | tr( S ) | = ∥ S ∥ 1 iff, for some c ∈ T , c S ≥ 0 ; i.e., tr( S ) = ∥ S ∥ 1 iff S ∈ B 1 ( H ) + . Lemma 3.4. If X ∈ B 1 ( H ) and Y ∈ B 1 ( J ) ar e such that X ⊗ Y ≥ 0 and ∥ X ⊗ Y ∥ ⊗ 1 = 1 , then X ⊗ Y = ρ ⊗ σ , for some (uniquely determine d) density op er ators ρ ∈ D ( H ) and σ ∈ D ( J ) . Theorem 3.4 (The signed decomposition) . The r e al Banach sp ac e B 1 ( H ) R b ⊗ B 1 ( J ) R of selfadjoint cr oss tr ac e class op er ators is char acterize d as the set of al l tr ac e class op er ators on H ⊗ J of the form C = X k t k ( ρ k ⊗ σ k ) , t k ∈ R ( P k | t k | < ∞ ) , ρ k ⊗ σ k ∈ θ ( D ( H ) , D ( J )) , (203) wher e the (p ossibly c ountably infinite) sum is absolutely c onver gent wrt to the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 — e quivalently, wrt the pr oje ctive norm ∥ · ∥ b ⊗ 1 or the tr ac e norm ∥ · ∥ ⊗ 1 — and tr( C ) = X k t k , ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ | | | C | | | b ⊗ 1 ≤ X k | t k | . (204) 45 In fact, if C = P j r j ( X j ⊗ Y j ) is any Hermitian standar d de c omp osition of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , then C admits a de c omp osition of the form (203) such that P k | t k | = P j r j (and c onversely). Ther efor e, de c omp osition (203) is | | | · | | | b ⊗ 1 -norming and, if C is | | | · | | | b ⊗ 1 -optimal ly de c omp osable (in p articular, Hermitian-optimal ly de c omp osable), then it admits a de c omp osition of the form (203) satisfying | | | C | | | b ⊗ 1 = P k | t k | (in p articular, | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 = P k | t k | ). Mor e over, one c an always assume — stil l ke eping the pr evious pr op erties — that in de c omp osition (203) the density op er ators { ρ k } , { σ k } ar e, sp e cific al ly, pur e states; i.e., ρ k = π k ∈ P ( H ) and σ k = ϖ k ∈ P ( J ) , for every k . If C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is not | | | · | | | b ⊗ 1 -optimal ly de c omp osable, then e ach exp ansion of C of the form (203) must c ontain b oth (strictly) p ositive and (strictly) ne gative sc alar c o efficients. In the c ase wher e M = min { dim( H ) , dim( J ) } < ∞ , for every C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , ther e is a finite de c omp osition of the form (203) c onsisting of, at most, 4 M 2 summands. L et R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R . The fol lowing c onditions ar e e quivalent: (P1) R admits a de c omp osition of the form (203) , that is, simultane ously, a Hermitian standard de c omp osition; i.e., if R is of the form R = X k r k ( ρ k ⊗ σ k ) , r k ≥ 0 ( P k r k < ∞ ) , ρ k ⊗ σ k ∈ θ ( D ( H ) , D ( J )) . (205) (P2) R is p ositive, Hermitian-optimal ly de c omp osable and such that ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 . (P3) R is p ositive, | | | · | | | b ⊗ 1 -optimal ly de c omp osable and such that | | | R | | | b ⊗ 1 = ∥ R ∥ ⊗ 1 . (P4) R is p ositive, optimal ly de c omp osable and such that ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 . (P5) R is p ositive, optimal ly de c omp osable and such that | | | R | | | b ⊗ 1 = ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 . If any of the e quivalent c onditions (P1) – (P5) holds, then any exp ansion of R of the form (205) is an optimal Hermitian standar d de c omp osition; thus, it is, at the same time, an optimal standar d de c omp osition and a | | | · | | | b ⊗ 1 -optimal (Hermitian) standar d de c omp osition, and tr( R ) = ∥ R ∥ ⊗ 1 = ∥ R ∥ b ⊗ 1 = | | | R | | | b ⊗ 1 = X k r k . (206) Pr o of. W e no w pro v e the first part of the theorem, concerning the c haracterization of the selfadjoin t cross trace class B 1 ( H ) R b ⊗ B 1 ( J ) R and the related facts. Let C = P j r j ( X j ⊗ Y j ) b e a Hermitian standard decomposition of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R . Let us express eac h of the selfadjoint trace class op erators X j ∈ B 1 ( H ) R , Y j ∈ B 1 ( J ) R as a (real) linear com bination of t wo densit y op erators — by F act 2.3 — i.e., X j = x j ; + ρ j ; + + x j ; − ρ j ; − , Y j = y j ; + σ j ; + + y j ; − σ j ; − , (207) where x j ; + = 1 2 tr( X j + | X j | ) ≥ 0 , . . . , y j ; − = 1 2 tr( Y j − | Y j | ) ≤ 0, ρ j ; + , ρ j ; − ∈ D ( H ), σ j ; + , σ j ; − ∈ D ( J ) and x j ; + ρ j ; + = 1 2 ( X j + | X j | ) ≥ 0 , x j ; − ρ j ; − = 1 2 ( X j − | X j | ) ≤ 0 , y j ; + σ j ; + = 1 2 ( Y j + | Y j | ) ≥ 0 , y j ; − σ j ; − = 1 2 ( Y j − | Y j | ) ≤ 0 ; (208) hence: x j ; + = 0 for X j + | X j | = 0 , . . . , y j ; − = 0 for Y j − | Y j | = 0, and ( x j ; + ρ j ; + )( x j ; − ρ j ; − ) = 0 = ( y j ; + σ j ; + )( y j ; − σ j ; − ), | X j | = x j ; + ρ j ; + − x j ; − ρ j ; − , | Y j | = y j ; + σ j ; + − y j ; − σ j ; − . (209) 46 (Clearly , if, say , x j ; + = 0, then ρ j ; + ∈ D ( H ) can b e chosen arbitrarily , etc.) Next, let us put s j ; ++ : = x j ; + y j ; + ≥ 0 , s j ; + − : = x j ; + y j ; − ≤ 0 , s j ; − + : = x j ; − y j ; + ≤ 0 , s j ; −− : = x j ; − y j ; − ≥ 0 , (210) so that X j ⊗ Y j = s j ; ++ ( ρ j ; + ⊗ σ j ; + ) + s j ; + − ( ρ j ; + ⊗ σ j ; − ) + s j ; − + ( ρ j ; − ⊗ σ j ; − ) + s j ; −− ( ρ j ; − ⊗ σ j ; − ) , (211) and — since ∥ X j ∥ 1 = tr( | X j | ) = tr( x j ; + ρ j ; + ) − tr( x j ; − ρ j ; − ) = x j ; + − x j ; − , ∥ Y j ∥ 1 = y j ; + − y j ; − (b y (209)) — we ha ve: 1 = ∥ X j ⊗ Y j ∥ ⊗ 1 = ∥ X j ∥ 1 ∥ Y j ∥ 1 = ( x j ; + − x j ; − )( y j ; + − y j ; − ) = | s j ; ++ | + | s j ; + − | + | s j ; − + | + | s j ; −− | . (212) By the last relation, w e see that 0 ≤ X j r j = X j r j ( | s j ; ++ | + | s j ; + − | + | s j ; − + | + | s j ; −− | ) . (213) Ev entually , b y applying the expressions (211) to the Hermitian standard decomp osition of C , and b y suitably renaming the co efficien ts and their indices — i.e., in tro ducing a new set of co efficien ts { t k } ⊂ R (e.g., if j = 1 , 2 , . . . , w e can put t 4 j − 3 = r 1 s j ; ++ , t 4 j − 2 = r 1 s j ; + − , t 4 j − 1 = r 1 s j ; − + and t 4 j = r 1 s j ; −− ) — w e obtain a relation of the form C = X j r j ( X j ⊗ Y j ) = X k t k ( ρ k ⊗ σ k ) , t k ∈ R , ρ k ∈ D ( H ) , σ k ∈ D ( J ) , (214) where 0 ≤ P k | t k | = P j r j ( | s j ; ++ | + | s j ; + − | + | s j ; − + | + | s j ; −− | ) = P j r j < ∞ . Therefore, it is prov en that decomp osition (203), with the sp ecified prop erties, holds, and con verges absolutely wrt the Hermitian pro jective norm; it also conv erges (to the same limit) wrt the pro jective norm, and wrt the trace norm, so that tr( C ) = P k t k tr( ρ k ⊗ σ k ) = P k t k . (Conv ersely , from ev ery decomposition of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R of the form (203) — setting, b y conv ention, sgn (0) ≡ 1 — we immediately get the Hermitian standard decomp osition C = P k | t k |  (sgn ( t k ) ρ k ) ⊗ σ k  .) No w, take an y expansion of C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R of the form (203) and, exploiting the spectral decomp osition of a density op erator, express each pro duct state ρ k ⊗ σ k as a conv ex combination of pur e states, i.e., ρ k ⊗ σ k = X m,n p k ; m q k ; n ( π k ; m ⊗ ϖ k ; n ) , (215) where π k ; m ∈ P ( H ), ϖ k ; n ∈ P ( J ), p k ; m , q k ; n > 0 and, for ev ery k , P m p k ; m = 1 = P n q k ; n ; hence, the — p ossibly countably infinite — sum is absolutely con vergen t wrt the Hermitian pro jectiv e norm (equiv alently , wrt the pro jective norm or the trace norm). Then, by suitably renaming the co efficien ts and their indices, w e get a decomp osition of the form C = X l s l ( π l ⊗ ϖ l ) , s l ∈ R , π l ∈ P ( H ) , ϖ l ∈ P ( J ) . (216) Since decomposition (203) is | | | · | | | b ⊗ 1 -norming and ∥ ρ k ⊗ σ k ∥ ⊗ 1 = 1 = P m,n p k ; m q k ; n ∥ π k ; m ⊗ ϖ k ; n ) ∥ ⊗ 1 , then decomp osition (216) is | | | · | | | b ⊗ 1 -norming to o; moreo ver, if D is | | | · | | | b ⊗ 1 -optimally decomp osable (in particular, Hermitian-optimally decomp osable), then it admits a decomp osition of the form (216) satisfying | | | C | | | b ⊗ 1 = P l | s l | (in particular, | | | C | | | b ⊗ 1 = ∥ C ∥ b ⊗ 1 = P l | s l | ). Let C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R b e not | | | · | | | b ⊗ 1 -optimally decomp osable (hence, in particular, C  = 0). F or ev ery decomp osition of the form C = P k t k ( ρ k ⊗ σ k ), we hav e that 0 < | | | C | | | b ⊗ 1 < P k | t k | 47 (otherwise, b y the first part of the pro of, the second strict inequality would b e violated b y some decomp osition of C of this form). By this fact, and by Lemma 3.3, we see that | P k t k | = | tr( C ) | ≤ ∥ C ∥ ⊗ 1 ≤ | | | C | | | b ⊗ 1 < P k | t k | = ⇒ { t k } ∩ R + ∗  = ∅  = { t k } ∩ R − ∗ . (217) By Prop osition 3.16, if M = min { dim( H ) , dim( J ) } < ∞ , then every C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R admits a finite Hermitian standard decomp osition consisting of, at most, M 2 summands. Hence, arguing as ab o ve, one concludes that C admits a finite decomp osition of the form (203) consisting of, at most, 4 M 2 summands (because X ⊗ Y ∈ θ ( B 1 ( H ) R , B 1 ( J ) R ) can b e expressed as a real linear com bination of, at most, four densit y op erators). W e next pro ve the second part of the theorem, i.e., the equiv alence of prop erties (P1) – (P5) . If R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is of the form (205), then it is p ositiv e — in fact, for every D ∈ D ( H ⊗ J ) (in particular, for ev ery pure state in P ( H ⊗ J )), since decomp osition (205) con verges wrt the trace norm, we ha ve that tr( D R ) = P k r k tr( D ( ρ k ⊗ σ k )) ≥ 0 — and, by relations (141), X k r k = tr( R ) = ∥ R ∥ ⊗ 1 ≤ ∥ R ∥ b ⊗ 1 ≤ | | | R | | | b ⊗ 1 ≤ X k r k ∥ ρ k ⊗ σ k ∥ ⊗ 1 = X k r k , (218) so that, actually , | | | R | | | b ⊗ 1 = ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 = tr( R ) = P k r k ; i.e., the decomp osition (205) is b oth an optimal and a | | | · | | | b ⊗ 1 -optimal Hermitian standard decomp osition of R and | | | R | | | b ⊗ 1 = ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 . Th us, condition (P1) implies all prop erties (P2) – (P5) . Moreo ver, condition (P2) implies (P3) , b ecause, if R is Hermitian-optimally decomp osable and ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 , then, b y p oint (O1) of Prop osition 3.11, ev ery optimal Hermitian standard decomp osition of R is also | | | · | | | b ⊗ 1 -optimal and | | | R | | | b ⊗ 1 = ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 . Next, if R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is | | | · | | | b ⊗ 1 -optimally decomp osable and | | | R | | | b ⊗ 1 = ∥ R ∥ ⊗ 1 , then — by (141) ( | | | R | | | b ⊗ 1 ≥ ∥ R ∥ b ⊗ 1 ≥ ∥ R ∥ ⊗ 1 ) — | | | R | | | b ⊗ 1 = ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 , so that ev ery | | | · | | | b ⊗ 1 -optimal Hermitian standard decomp osition of R is also optimal tout c ourt . Therefore, condition (P3) implies (P4) . Let us finally pro ve that condition (P4) implies (P1) , and hence conditions (P1) – (P5) are m utually equiv alent (b ecause (P1) = ⇒ (P2) = ⇒ (P3) = ⇒ (P4) = ⇒ (P1) = ⇒ (P5) and (P5) trivially implies (P4) ). In fact, if R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is positive and optimally decomp osable, with R  = 0 (otherwise, there is nothing to prov e), then, pic king any optimal standard decomposition R = P k r k ( X k ⊗ Y k ), with r k > 0 (for ev ery k ) — i.e, a she er decomp osition — we ha v e: 0 < ∥ R ∥ ⊗ 1 = tr( R ) = X k r k tr( X k ) tr( Y k ) =   P k r k tr( X k ) tr( Y k )   ≤ X k r k | tr( X k ) | | tr( Y k ) | ≤ X k r k ∥ X k ∥ 1 ∥ Y k ∥ 1 = X k r k = ∥ R ∥ b ⊗ 1 . (219) Here, the second inequality follows from the fact that | tr( X k ) | ≤ ∥ X k ∥ 1 and | tr( Y k ) | ≤ ∥ Y k ∥ 1 ; see the first assertion of Lemma 3.3. No w, if, moreov er, w e assume that ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 , then the inequalities on the second and third lines of (219) must be saturated; i.e., w e must ha ve that P k r k tr( X k ) tr( Y k ) = P k r k , or, equiv alently , X k r k tr( X k ⊗ Y k ) = X k r k . (220) T aking in to account that r k > 0 (for ev ery k ), this relation holds iff P k p k tr( X k ⊗ Y k ) = 1, where | tr( X k ⊗ Y k ) | = | tr( X k ) tr( Y k ) | ≤ ∥ X k ∥ 1 ∥ Y k ∥ 1 = 1 and w e hav e put p k : = r k / ( P k r k ) > 0 48 ( P k p k = 1). Since the extreme p oin ts of the closed unit disc { z ∈ C : | z | ≤ 1 } form the unit circle T , this last condition is equiv alent to requiring that, for ev ery k , tr( X k ⊗ Y k ) = 1 = ∥ X k ⊗ Y k ∥ ⊗ 1 ; namely , by the second assertion of Lemma 3.3, that, for ev ery k , X k ⊗ Y k is a p ositiv e selfadjoin t trace class op erator. Summarizing, by the previous argumen ts, if R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , with R  = 0 (otherwise, there is nothing to prov e), is p ositiv e, optimally decomp osable and, moreov er, ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 , then, giv en any sheer optimal standard decomp osition R = P k r k ( X k ⊗ Y k ), the simple tensor X k ⊗ Y k ∈ θ ( B 1 ( H ) , B 1 ( J )), for every k , must b e a p ositiv e selfadjoin t trace class op erator; i.e., since ∥ X k ⊗ Y k ∥ ⊗ 1 = 1, by Lemma 3.4 we m ust hav e that X k ⊗ Y k = ρ k ⊗ σ k , for some ρ k ∈ D ( H ), σ k ∈ D ( J ) . (221) Therefore, condition (P4) implies (P1) , as we wan ted to prov e, and hence conditions (P1) – (P5) are mutually equiv alen t. Note that, if condition (P1) , or any of the other equiv alent conditions (P2) – (P5) , is satisfied, with R  = 0, then every sheer optimal standard decomp osition of R is of the form (203), provided that w e do not distinguish b etw een the v arious representations of the same elementary tensor ρ k ⊗ σ k — sp ecifically , a decomp osition of the form (205), with r k = t k > 0 (i.e., a decomp osition of the form (203), with strictly p ositiv e co efficien ts) — and, moreo ver, b y our previous argumen ts, ev ery expansion of R of the form (205) is an optimal Hermitian standard decomp osition, so that relation (206) holds. Also, using essentially the same arguments as ab ov e, based on an estimate for the norm | | | · | | | b ⊗ 1 analogous to (219), one pro ves that, if one of the equiv alent conditions (P1) – (P5) is v erified, then ev ery sheer | | | · | | | b ⊗ 1 -optimal Hermitian standard decomp osition of R  = 0 is of the form (205) (with strictly p ositiv e co efficien ts { r k } ). Definition 3.10. An expansion of a selfadjoint cross trace class operator C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R of the form (203) will be called a signe d de c omp osition of C . Sp ecifically , if in decomposition (203) the densit y op erators { ρ k } , { σ k } are pure states — i.e., for every k , ρ k = π k ∈ P ( H ), σ k = ϖ k ∈ P ( J ) — then it will b e called a pur e-state signe d de c omp osition . If a (p ositiv e) selfadjoin t op erator R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R admits a decomp osition of the form (205), then it will b e called p ositively de c omp osable , and an expansion of the form (205) will b e called a p ositive de c omp osition of R . In the case where C  = 0 (or R  = 0), a signed decomposition (alternativ ely , a p ositiv e decomp osition) of C (resp ectively , of R ) is said to b e she er if it do es not contain zero summands. The set of all p ositiv ely decomp osable op erators in B 1 ( H ) R b ⊗ B 1 ( J ) R w e will denoted b y P o d  B 1 ( H ) R b ⊗ B 1 ( J ) R  . R emark 3.19 . By Theorem 3.4, a p ositiv e cross trace class op erator R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R is p osi- tiv ely decomp osable iff it is optimally decomp osable and such that ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 . F rom the pro of of the theorem, it is also clear that, if R ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , with R  = 0, is p ositiv ely decom- p osable, then every sheer optimal standard decomp osition — as w ell as every sheer | | | · | | | b ⊗ 1 -optimal Hermitian standard decomp osition — of R is actually a signed decomp osition of the form (203) (pro vided that we do not distinguish b et ween different represen tations of the same elemen tary ten- sor pro duct op erator), with strictly p ositive c o efficients , i.e., a (sheer) p ositiv e decomp osition of the form (205); also, ev ery signed decomp osition R = P k t k ( ρ k ⊗ σ k ) such that P k | t k | = ∥ R ∥ b ⊗ 1 , m ust b e a p ositiv e decomp osition (indeed, if R is p ositively decomp osable, then w e hav e that ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 = ⇒ ∥ R ∥ ⊗ 1 = tr( R ) = P k t k ≤ P k | t k | = ∥ R ∥ b ⊗ 1 = ∥ R ∥ ⊗ 1 = ⇒ { t k } ∩ R − ∗ = ∅ ). Prop osition 3.17 (Characterization of the cross trace class) . Every selfadjoint cr oss tr ac e class op er ator — wrt the bip artition H ⊗ J — c an b e expr esse d as a r e al line ar c ombination of (at most) two p ositively de c omp osable cr oss tr ac e class op er ators; henc e, every cr oss tr ac e class op er ator c an 49 b e expr esse d as a c omplex line ar c ombination of (at most) four p ositively de c omp osable cr oss tr ac e class op er ators. Pr e cisely, the fol lowing r elations hold: B 1 ( H ) R b ⊗ B 1 ( J ) R = P o d  B 1 ( H ) R b ⊗ B 1 ( J ) R  − P o d  B 1 ( H ) R b ⊗ B 1 ( J ) R  =  R 1 − R 2 : R 1 , R 2 ∈ P o d  B 1 ( H ) R b ⊗ B 1 ( J ) R  (222) and B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) R b ⊗ B 1 ( J ) R + i  B 1 ( H ) R b ⊗ B 1 ( J ) R  =  ( R 1 − R 2 ) + i ( R 3 − R 4 ) : R 1 , R 2 , R 3 , R 4 ∈ P o d  B 1 ( H ) R b ⊗ B 1 ( J ) R  . (223) Pr o of. F or every C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , with C  = 0 (the case where C = 0 b eing trivial), we can re-express any signed decomp osition of C as follows: C = X k t k ( ρ k ⊗ σ k ) = X t k > 0 t k ( ρ k ⊗ σ k ) − X t k < 0 | t k | ( ρ k ⊗ σ k ) = R 1 − R 2 , R 1 , R 2 ∈ P o d  B 1 ( H ) R b ⊗ B 1 ( J ) R  , (224) whic h prov es the c haracterization (222) of the selfadjoint cross trace class. Then, b y Prop osition 3.7, relation (223) follo ws immediately . Prop osition 3.18 (Characterization of cross states) . The set D ( H ⊗ J ) b of cr oss states wrt to the bip artition H ⊗ J is a c onvex subset of B 1 ( H ⊗ J ) , that is char acterize d as the class of al l p ositive tr ac e class op er ators on H ⊗ J that admit a signe d de c omp osition of the form D = X k t k ( ρ k ⊗ σ k ) , t k ∈ R ∗ ( P k | t k | < ∞ ) , ρ k ∈ D ( H ) , σ k ∈ D ( J ) , (225) wher e the (p ossibly c ountably infinite) sum is absolutely c onver gent wrt to the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 — e quivalently, wrt the pr oje ctive norm ∥ · ∥ b ⊗ 1 or the tr ac e norm ∥ · ∥ ⊗ 1 — and 1 = tr( D ) = X k t k , 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ | | | D | | | b ⊗ 1 ≤ X k | t k | . (226) De c omp osition (225) is | | | · | | | b ⊗ 1 -norming and, if D is | | | · | | | b ⊗ 1 -optimal ly de c omp osable (in p articular, Hermitian-optimal ly de c omp osable), then it admits a signe d de c omp osition of the form (225) such that | | | D | | | b ⊗ 1 = P k | t k | (in p articular, | | | D | | | b ⊗ 1 = ∥ D ∥ b ⊗ 1 = P k | t k | ). If D ∈ D ( H ⊗ J ) b is not | | | · | | | b ⊗ 1 -optimal ly de c omp osable, then e ach exp ansion of D of the form (225) must c ontain b oth (strictly) p ositive and (strictly) ne gative sc alar c o efficients. In the c ase wher e M = min { dim( H ) , dim( J ) } < ∞ , every D ∈ D ( H ⊗ J ) b admits a finite de c omp osition of the form (225) c onsisting of, at most, 4 M 2 summands. F or every cr oss state D ∈ D ( H ⊗ J ) b , the fol lowing c onditions ar e e quivalent: (D1) D admits a (she er) p ositive de c omp osition of the form D = X k p k ( ρ k ⊗ σ k ) , p k > 0  P k p k = 1  , ρ k ∈ D ( H ) , σ k ∈ D ( J ) . (227) (D2) D is Hermitian-optimal ly de c omp osable and ∥ D ∥ b ⊗ 1 = 1 . (D3) D is | | | · | | | b ⊗ 1 -optimal ly de c omp osable and | | | D | | | b ⊗ 1 = 1 . 50 (D4) D is optimal ly de c omp osable and ∥ D ∥ b ⊗ 1 = 1 . (D5) D is optimal ly de c omp osable and | | | D | | | b ⊗ 1 = ∥ D ∥ b ⊗ 1 = 1 . If any of the e quivalent c onditions (D1) – (D5) is satisfie d, then the exp ansion (227) is a (she er) optimal Hermitian standar d de c omp osition and, thus, it is b oth an optimal standar d de c omp osition and a | | | · | | | b ⊗ 1 -optimal Hermitian standar d de c omp osition; mor e over, every she er optimal standar d de c omp osition, and every she er | | | · | | | b ⊗ 1 -optimal Hermitian standar d de c omp osition, of the cr oss state D — as wel l as every she er signe d de c omp osition D = P k t k ( ρ k ⊗ σ k ) such that P k | t k | = 1 — is actual ly a p ositive de c omp osition of the form (227) , pr ovide d that one do es not distinguish b etwe en differ ent r epr esentations of the same elementary tensor pr o duct op er ator. Pr o of. All claims follow immediately from Theorem 3.4 and Remark 3.19. 3.10 The univ ersal bilinear map on trace classes The application b θ : B 1 ( H ) × B 1 ( J ) ∋ ( S, T ) 7→ S ⊗ T ∈ B 1 ( H ) b ⊗ B 1 ( J ) (228) is a bounded bilinear map, that we will call henceforth the universal biline ar map on B 1 ( H ) × B 1 ( J ). Note that this map shares b oth the same domain (the Cartesian pro duct B 1 ( H ) × B 1 ( J )) and r ange (the elementary tensor pro ducts of trace class operators) with the natural bilinear map θ : B 1 ( H ) × B 1 ( J ) → B 1 ( H ⊗ J ) in tro duced in Subsection 3.2 — and, moreov er, θ ( S, T ) = b θ ( S, T ) — but it has a different c o domain , i.e., the Banac h space ( B 1 ( H ) b ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1 ) versus the trace class ( B 1 ( H ⊗ J ) , ∥ · ∥ ⊗ 1 ). This fact has remark able consequences; see the next subsection. Clearly , b θ ∈ B  B 1 ( H ) , B 1 ( J ); B 1 ( H ) b ⊗ B 1 ( J )  — like the natural bilinear map θ — is a bilinear isometry since ∥ b θ ( S, T ) ∥ b ⊗ 1 = ∥ S ⊗ T ∥ ⊗ 1 = ∥ S ∥ 1 ∥ T ∥ 1 , ∀ ( S, T ) ∈ B 1 ( H ) × B 1 ( J ) , (229) so that ∥ b θ ∥ B = 1. Notation 3.3. In analogy with our previously in tro duced Notation 3.2, giv en (nonempt y) subsets X , Y of B 1 ( H ) and B 1 ( J ), resp ectiv ely , w e put X b ⊗ Y : = co b ⊗  b θ ( X , Y )  = cl ∥ · ∥ b ⊗ 1  X ˘ ⊗ Y  ⊂ B 1 ( H ) b ⊗ B 1 ( J ) ⊂ B 1 ( H ⊗ J ) ; (230) i.e., X b ⊗ Y is the ∥ · ∥ b ⊗ 1 -closed con vex hull co b ⊗ ( b θ ( X , Y )) of b θ ( X , Y ). Since b θ ( X , Y ) = θ ( X , Y ) is a subset of b oth the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) and the bipartite trace class B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ⊗ J ), for the sake of simplicity in the following we will use the notation θ ( X , Y ) for this set ev en in the case where it should b e regarded as a subset of B 1 ( H ) b ⊗ B 1 ( J ). E.g., w e will write X b ⊗ Y : = co b ⊗ ( θ ( X , Y )) ≡ co b ⊗  b θ ( X , Y )  . In the case where X = B 1 ( H ) and Y = B 1 ( J ), Notation 3.3 is consisten t with our definition of the Banach space B 1 ( H ) b ⊗ B 1 ( J ); see relation (231) b elo w. Once again, privileging the con vex structures wrt to the linear ones in Notation 3.3, as previously done for Notation 3.2, is motiv ated b y the applications w e ha ve in mind. Prop osition 3.19. L et X , Y b e line ar subsp ac es of B 1 ( H ) and B 1 ( J ) , r esp e ctively. Then, we have: X b ⊗ Y : = co b ⊗ ( θ ( X , Y )) = span b ⊗ ( θ ( X , Y )) ≡ span ∥ · ∥ b ⊗ 1 ( θ ( X , Y )) : = cl ∥ · ∥ b ⊗ 1  span ( θ ( X , Y ))  . (231) 51 Mor e over, the fol lowing r elations hold: D ( H ) b ⊗ D ( J ) : = co b ⊗  θ ( D ( H ) , D ( J ))  = co b ⊗  θ ( P ( H ) , P ( J ))  = : P ( H ) b ⊗ P ( J ) (232) and B 1 ( H ) b ⊗ B 1 ( J ) = span b ⊗  θ ( D ( H ) , D ( J ))  = span b ⊗  θ ( P ( H ) , P ( J ))  , (233) B 1 ( H ) b ⊗ B 1 ( J ) = span b ⊗  D ( H ) b ⊗ D ( J )  = span b ⊗  P ( H ) b ⊗ P ( J )  . (234) 3.11 The canonical linearization of bilinear maps on trace classes The term “univ ersal bilinear map” introduced for the application (228) is justified b y the following: Theorem 3.5 (Univ ersal mapping prop ert y of the cross trace class) . L et S b e a c omplex Banach sp ac e, and let Λ b ⊗ : B 1 ( H ) b ⊗ B 1 ( J ) → S b e a b ounde d line ar map. Then, λ : = Λ b ⊗ ◦ b θ : B 1 ( H ) × B 1 ( J ) → S (235) is a b ounde d biline ar map and, mor e over, ∥ λ ∥ B = ∥ Λ b ⊗ ∥ B ; i.e., ∥ λ ∥ B : = sup 0  = S ∈B 1 ( H ) 0  = T ∈B 1 ( J ) ∥ λ ( S, T ) ∥ ∥ S ∥ 1 ∥ T ∥ 1 = sup 0  = S ∈B 1 ( H ) 0  = T ∈B 1 ( J ) ∥ Λ b ⊗ ( S ⊗ T ) ∥ ∥ S ∥ 1 ∥ T ∥ 1 = sup 0  = C ∈B 1 ( H ) b ⊗ B 1 ( J ) ∥ Λ b ⊗ ( C ) ∥ ∥ C ∥ b ⊗ 1 = : ∥ Λ b ⊗ ∥ B . (236) Conversely, let S b e a c omplex Banach sp ac e and λ : B 1 ( H ) × B 1 ( J ) → S a b ounde d biline ar map. Then, ther e exists a unique b ounde d line ar map Λ b ⊗ : B 1 ( H ) b ⊗ B 1 ( J ) → S such that λ ( S, T ) = Λ b ⊗ ( S ⊗ T ) , ∀ S ∈ B 1 ( H ) , ∀ T ∈ B 1 ( J ) . (237) Mor e over, ∥ λ ∥ B = ∥ Λ b ⊗ ∥ B . Ther efor e, the fol lowing isomorphism of Banach sp ac es holds: B  B 1 ( H ) b ⊗ B 1 ( J ); S  ∼ = B ( B 1 ( H ) , B 1 ( J ); S ) . (238) This isomorphism is implemente d by the mapping B  B 1 ( H ) b ⊗ B 1 ( J ); S  ∋ Λ 7→ Λ ◦ b θ ∈ B ( B 1 ( H ) , B 1 ( J ); S ) , (239) wher e b θ : B 1 ( H ) × B 1 ( J ) → B 1 ( H ) b ⊗ B 1 ( J ) is the universal biline ar map on B 1 ( H ) × B 1 ( J ) . Pr o of. Apply Theorem 2.9 of [18], or Theorem 7.12 of [20], to the pro jectiv e tensor pro duct B 1 ( H ) b ⊗ B 1 ( J ). Definition 3.11. W e will call the application Λ b ⊗ : B 1 ( H ) b ⊗ B 1 ( J ) → S the c anonic al line arization of (or the c anonic al line ar map induced by) λ : B 1 ( H ) × B 1 ( J ) → S ; con versely , we call λ the c anonic al biline arization of Λ b ⊗ . R emark 3.20 . In the case where λ : B 1 ( H ) × B 1 ( J ) → S is naturally linearizable, b y relations (65) and (237) it is clear that the asso ciated induced linear maps Λ ⊗ and Λ b ⊗ coincide once restricted to the algebraic tensor pro duct B 1 ( H ) ˘ ⊗ B 1 ( J ), and this restriction is precisely the unique linear map Λ ˘ ⊗ : B 1 ( H ) ˘ ⊗ B 1 ( J ) → S of Prop osition 3.3. 52 Corollary 3.12. L et E : B 1 ( H ) → B 1 ( H ) , G : B 1 ( J ) → B 1 ( J ) b e b ounde d line ar maps. Then, ther e is a unique b ounde d line ar map E b ⊗ G : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ) b ⊗ B 1 ( J ) such that  E b ⊗ G  ( S ⊗ T ) = E ( S ) ⊗ G ( T ) , ∀ S ∈ B 1 ( H ) , ∀ T ∈ B 1 ( J ) , (240) and, mor e over,   E b ⊗ G   B = ∥ E ∥ [1] ∥ G ∥ [1] . Corollary 3.13. Supp ose that H and/or J ar e finite-dimensional Hilb ert sp ac es; pr e cisely, assume that M = min { dim( H ) , dim( J ) } < ∞ . Then, in this c ase, B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) ˘ ⊗ B 1 ( J ) , (241) a set e quality (or an isomorphism of line ar sp ac es) b eing understo o d, and any b ounde d biline ar map λ : B 1 ( H ) × B 1 ( J ) → S is natur al ly line arizable; mor e over — denoting by Λ ⊗ : B 1 ( H ) ⊗ B 1 ( J ) → S the natur al line arization of λ — we have that Λ ⊗ ( A ) = Λ b ⊗ ( A ) , for al l A ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) , and ∥ λ ∥ B = ∥ Λ b ⊗ ∥ B ≤ ∥ Λ ⊗ ∥ B ≤ N ∥ Λ b ⊗ ∥ B , wher e N = min  4 M , M 2  . (242) Pr o of. By Theorem 3.3, relation (241) holds and, moreov er, the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 are equiv alent on B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) ˘ ⊗ B 1 ( J ). Therefore, since, according to Theorem 3.5, there is a unique b ounded (wrt the norms ∥ · ∥ b ⊗ 1 and ∥ · ∥ S ) linear map Λ b ⊗ : B 1 ( H ) b ⊗ B 1 ( J ) → S satisfying λ ( S, T ) = Λ b ⊗ ( S ⊗ T ) , ∀ S ∈ B 1 ( H ) , ∀ T ∈ B 1 ( J ) (243) — that m ust coincide with the (unique) linear map Λ ˘ ⊗ : B 1 ( H ) ˘ ⊗ B 1 ( J ) → S of Prop osition 3.3 — then there is a unique linear map Λ : B 1 ( H ) ⊗ B 1 ( J ) → S such that Λ( S ⊗ T ) = Λ ˘ ⊗ ( S ⊗ T ) = Λ b ⊗ ( S ⊗ T ) = λ ( S, T ) , ∀ S ∈ B 1 ( H ) , ∀ T ∈ B 1 ( J ) , (244) whic h is bounded (wrt the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ S ). Thus, the bilinear map λ : B 1 ( H ) × B 1 ( J ) → S is naturally linearizable (Definit ion 3.2) and, b y Prop osition 3.3, and its (unique) natural linearization Λ ⊗ m ust b e precisely the map Λ. Hence, Λ ⊗ ( A ) = Λ ˘ ⊗ ( A ) = Λ b ⊗ ( A ), for all A ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ), and w e hav e that ∥ Λ b ⊗ ∥ B = ∥ λ ∥ B ≤ ∥ Λ ⊗ ∥ B . Moreov er, by relation (178), for every bipartite trace class op erator A ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) = B 1 ( H ⊗ J ), A  = 0, w e get the estimate ∥ Λ b ⊗ ( A ) ∥ ∥ A ∥ b ⊗ 1 ≤ ∥ Λ b ⊗ ( A ) ∥ ∥ A ∥ ⊗ 1 = ∥ Λ ⊗ ( A ) ∥ ∥ A ∥ ⊗ 1 ≤ N ∥ Λ b ⊗ ( A ) ∥ ∥ A ∥ b ⊗ 1 , (245) where N = min  4 M , M 2  , with M = min { dim( H ) , dim( J ) } < ∞ . No w, the lhs inequalit y in (245) is coherent with the previously obtained relation ∥ Λ b ⊗ ∥ B ≤ ∥ Λ ⊗ ∥ B , while, by the rhs inequality , we find out that ∥ Λ ⊗ ∥ B = sup A  =0 ∥ Λ ⊗ ( A ) ∥ ∥ A ∥ ⊗ 1 ≤ N sup A  =0 ∥ Λ b ⊗ ( A ) ∥ ∥ A ∥ b ⊗ 1 = N ∥ Λ b ⊗ ∥ B (246) as well, so that relation (242) holds true. In this case where min { dim( H ) , dim( J ) } < ∞ , the Banach space dual  B 1 ( H ) b ⊗ B 1 ( J )  ∗ of B 1 ( H ) b ⊗ B 1 ( J ) is c haracterized by Corollary 3.9; in the general case, w e hav e the following result: Corollary 3.14. Denoting by  B 1 ( H ) b ⊗ B 1 ( J )  ∗ = B  B 1 ( H ) b ⊗ B 1 ( J ); C  the Banach sp ac e dual of the cr oss tr ac e class B 1 ( H ) b ⊗ B 1 ( J ) , the fol lowing isomorphism of Banach sp ac es holds:  B 1 ( H ) b ⊗ B 1 ( J )  ∗ ∼ = F 1 ( H , J ) ≡ B ( B 1 ( H ) , B 1 ( J ); C ) . (247) 53 This isomorphism is implemente d by the mapping  B 1 ( H ) b ⊗ B 1 ( J )  ∗ ∋ Γ 7→ Γ ◦ b θ ∈ F 1 ( H , J ) , (248) wher e b θ : B 1 ( H ) × B 1 ( J ) → B 1 ( H ) b ⊗ B 1 ( J ) is the universal biline ar map on B 1 ( H ) × B 1 ( J ) . As a c onse quenc e, sinc e, for every cr oss tr ac e class op er ator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , we have that ∥ C ∥ b ⊗ 1 = max  | Γ( C ) | : Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , ∥ Γ ∥ B = 1  , (249) then, given any simple-tensor de c omp osition of C — namely, C = P k S k ⊗ T k , wher e the sum, whenever not finite, is supp ose d to c onver ge absolutely wrt the pr oje ctive norm ∥ · ∥ b ⊗ 1 , i.e., to b e such that P k ∥ S k ⊗ T k ∥ b ⊗ 1 = P k ∥ S k ∥ 1 ∥ T k ∥ 1 < ∞ — the fol lowing r elation holds: ∥ C ∥ b ⊗ 1 = max  | P k γ ( S k , T k ) | : γ ∈ F 1 ( H , J ) , ∥ γ ∥ (1) = 1  . (250) Pr o of. The isomorphism b etw een the Banach spaces  B 1 ( H ) b ⊗ B 1 ( J )  ∗ and F 1 ( H , J ) is a direct consequence of Theorem 3.5 (by taking S = C ). Moreo ver, for ev ery C ∈ B 1 ( H ) b ⊗ B 1 ( J ), the existence of a norming functional Γ 0 ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ suc h that ∥ Γ 0 ∥ B = 1 and ∥ C ∥ b ⊗ 1 = Γ 0 ( C ) is a well-kno wn consequence of the Hahn-Banach extension theorem (see, e.g., Corollary 2.3 of [47]). It follows that we ha ve: ∥ C ∥ b ⊗ 1 = max  | Γ( C ) | : Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , ∥ Γ ∥ B = 1  = max  | Γ( P k S k ⊗ T k ) | : Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , ∥ Γ ∥ B = 1  = max  | P k Γ( S k ⊗ T k ) | : Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , ∥ Γ ∥ B = 1  = max  | P k γ ( S k , T k ) | : γ ∈ F 1 ( H , J ) , ∥ γ ∥ (1) = 1  , (251) where C = P k S k ⊗ T k is any (absolutely ∥ · ∥ b ⊗ 1 -con vergen t) simple-tensor decomp osition of the cross trace class op erator C ∈ B 1 ( H ) b ⊗ B 1 ( J ). 4 Separabilit y , pseudo-mixtures and barycen tric decomp ositions After recalling some basic facts ab out the separable states in B 1 ( H ⊗ J ) — including the gen- eral definition of this class of bipartite states (see Definition 4.2 b elo w) — we will next prov e that they are cross states, and we will highlight some remark able connections of separability with the the pro jective norm ∥ · ∥ b ⊗ 1 on B 1 ( H ) b ⊗ B 1 ( J ) and with the Hermitian pro jectiv e norm | | | · | | | b ⊗ 1 on B 1 ( H ) R b ⊗ B 1 ( J ) R . In this section, w e will alwa ys assume that dim( H ) , dim( J ) ≥ 2, b ecause otherwise “there is no en tanglement” and, accordingly , separabilit y is an empty concept. 4.1 Pseudo-mixtures of pro duct s tates and discretely separable states The set D ( H ⊗ J ) b of cross states wrt to the bipartition H ⊗ J is a conv ex subset (of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) and) of B 1 ( H ⊗ J ), characterized, according to Prop osition 3.18, as the set of all p ositive trace class op erators on the Hilb ert space H ⊗ J that admit a decomp osition of the form D = X k t k ( ρ k ⊗ σ k ) , t k ∈ R ∗ , ρ k ∈ D ( H ) , σ k ∈ D ( J ) , (252) where 1 = tr( D ) = P k t k ≤ P k | t k | < ∞ , so that the — p ossibly , coun tably infinite — sum is absolutely con vergen t wrt the Hermitian pro jective norm (equiv alen tly , wrt the pro jective norm or the trace norm). Moreov er, for ev ery D ∈ D ( H ⊗ J ) b , we ha ve that 1 = tr( D ) = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ | | | D | | | b ⊗ 1 = inf  P k | t k | : P k t k ( ρ k ⊗ σ k ) = D  ≤ 2 ∥ D ∥ b ⊗ 1 , (253) 54 where the infim um is taken o ver all p ossible expansions of D of the form (252). Coheren tly with well-established literature on this topic [65], we call a densit y op erator D on H ⊗ J of the form (252) a pseudo-mixtur e of the product states { ρ k ⊗ σ k } ⊂ θ ( D ( H ) , D ( J )); in particular, D is said to b e a (statistical) mixtur e of this collection of pro duct states if it can b e expressed in the form (252) with { t k } ⊂ R + ∗ (i.e., if { t k } is a probability distribution). Therefore, every cr oss state on H ⊗ J c an b e expr esse d as a pseudo-mixtur e of a suitable c ol le ction of pr o duct states ; in particular, as a pseudo-mixture of pro duct states of the form π ⊗ ϖ , with π ∈ P ( H ) and ϖ ∈ P ( J ). In fact, the following result holds: Prop osition 4.1 (The pure-state signed decomp osition of a cross state) . Every cr oss density op er ator D ∈ D ( H ⊗ J ) b admits a pur e-state signe d de c omp osition of the form D = X l s l ( π l ⊗ ϖ l ) , s l ∈ R ∗ , π l ∈ P ( H ) , ϖ l ∈ P ( J ) , (254) wher e the — p ossibly c ountably infinite — sum is absolutely c onver gent wrt the Hermitian pr oje ctive norm (e quivalently, wrt the pr oje ctive norm or the t r ac e norm), and 1 = tr( D ) = X l s l = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ | | | D | | | b ⊗ 1 ≤ X l | s l | . (255) The de c omp osition (254) is | | | · | | | b ⊗ 1 -norming and, if D is | | | · | | | b ⊗ 1 -optimal ly de c omp osable (in p articular, Hermitian-optimal ly de c omp osable), then it admits a de c omp osition of the form (254) satisfying | | | D | | | b ⊗ 1 = P l | s l | (in p articular, | | | D | | | b ⊗ 1 = ∥ D ∥ b ⊗ 1 = P l | s l | ). If the cr oss state D ∈ D ( H ⊗ J ) b is not | | | · | | | b ⊗ 1 -optimal ly de c omp osable, then e ach exp ansion of D of the form (254) must c ontain b oth (strictly) p ositive and (strictly) ne gative sc alar c o efficients. Mor e over, for a cr oss state D ∈ D ( H ⊗ J ) b , e ach of c onditions (D1) – (D5) in Pr op osition 3.18 is e quivalent to the fol lowing: (D6) D is a statistic al mixtur e of tensor pr o ducts of pur e states; i.e., it admits a (she er) p ositive de c omp osition of the form D = X l p l ( π l ⊗ ϖ l ) , p l > 0  P l p l = 1  , π l ∈ P ( H ) , ϖ l ∈ P ( J ) . (256) In particular, if D ∈ D ( H ⊗ J ) b is p ositiv ely decomp osable (recall Definition 3.10), then the scalar co efficients { t k } in the expansion (252) can b e assumed to b e p ositive ; i.e., D can be expressed as a gen uine statistic al mixtur e of the form D = X k p k ( ρ k ⊗ σ k ) , p k > 0  P k p k = tr( D ) = 1  , ρ k ∈ D ( H ), σ k ∈ D ( J ), (257) whic h pro vides an optimal Hermitian standard decomp osition of D . It is clear that every tr ac e class op er ator D of the form (257) is a (p ositiv ely decomposable) cross state c haracterized, equiv alently , b y any of conditions (D2) – (D5) in Proposition 3.18 or, also, by condition (D6) in Proposition 4.1; moreo ver, ev ery sheer optimal Hermitian standard decomp osition of D — as w ell as ev ery sheer signed decomposition D = P k t k ( ρ k ⊗ σ k ) suc h that P k | t k | = 1 — m ust b e of the form (257) (with the conv ention that w e do not distinguish b et w een different represen tations of the same elemen tary tensor pro duct op erator in the decomp osition). 55 Definition 4.1. A state ς ∈ D ( H ⊗ J ) is said to b e discr etely sep ar able , or str ongly sep ar able , wrt the bipartition H ⊗ J if it is of the form ς = X k p k ( ρ k ⊗ σ k ) , with p k > 0, P k p k = 1, ρ k ∈ D ( H ), σ k ∈ D ( J ), (258) where the sum — whenever not finite — conv erges absolutely wrt each of the norms ∥ · ∥ ⊗ 1 , ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 ; otherwise stated, if it is a p ositiv ely decomp osable cross state (equiv alently , a cross state satisfying an y of conditions (D2) – (D5) in Proposition 3.18, or condition (D6) in Prop osition 4.1) so that it admits an optimal decomp osition in the form of a coun table con vex combination of product states and | | | ς | | | b ⊗ 1 = ∥ ς ∥ b ⊗ 1 = ∥ ς ∥ ⊗ 1 = tr( ς ) = 1. W e will denote b y D ( H ⊗ J ) ds the subset of D ( H ⊗ J ) consisting of all such states. A discretely separable state ς ∈ D ( H ⊗ J ) ds is said to be finitely sep ar able if it is of the form (258), where the sum is finite , and we will denote by D ( H ⊗ J ) f s the subset of D ( H ⊗ J ) ds consisting of all suc h states. Otherwise stated, D ( H ⊗ J ) f s : = co ( θ ( D ( H ) , D ( J ))) = D ( H ) ˘ ⊗ D ( J ), where we ha ve used Notation 3.2. Prop osition 4.2. D ( H ⊗ J ) ds is a c onvex subset of the cr oss tr ac e class B 1 ( H ) b ⊗ B 1 ( J ) . Mor e over, we have that B 1 ( H ) R b ⊗ B 1 ( J ) R = R + D ( H ⊗ J ) ds − R + D ( H ⊗ J ) ds =  r 1 ς 1 − r 2 ς 2 : r 1 , r 2 ∈ R + , ς 1 , ς 2 ∈ D ( H ⊗ J ) ds  (259) and B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) R b ⊗ B 1 ( J ) R + i  B 1 ( H ) R b ⊗ B 1 ( J ) R  = R + D ( H ⊗ J ) ds − R + D ( H ⊗ J ) ds + i  R + D ( H ⊗ J ) ds − R + D ( H ⊗ J ) ds  . (260) Let us put D ( H ⊗ J ) b ± : = D ( H ⊗ J ) b \ D ( H ⊗ J ) ds . (261) This set is c haracterized as the class of all cross states that c annot b e expressed as a discrete statistical mixture of pro duct states. Otherwise stated, D ( H ⊗ J ) b ± is the collection of all states whose signed decomp ositions must con tain b oth (strictly) p ositive and (s trictly) negative scalar co efficien ts. Note that, by Prop osition 4.1, D ( H ⊗ J ) b ± contains the set of all cross states that are not | | | · | | | b ⊗ 1 -optimally decomp osable. 4.2 Characterizing cross states in terms of discretely separable states Giv en a cross state D ∈ D ( H ⊗ J ) b , w e can re-elab orate a signed decomp osition of D as follows; w e put D = X k t k ( ρ k ⊗ σ k ) = X t k > 0 t k ( ρ k ⊗ σ k ) + X t k < 0 t k ( ρ k ⊗ σ k ) = α D + + (1 − α ) D − , (262) — with { t k } ⊂ R , { ρ k } ⊂ D ( H ), { σ k } ⊂ D ( J ) — where the sums, whenever not finite, conv erge absolutely wrt e ac h of the norms ∥ · ∥ ⊗ 1 , ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 (and with the obvious con ven tion that an ‘empt y sum’ is iden tically zero), and w e ha ve set α ≡ α ( { t k } ) : = X t k > 0 t k ≥ X k t k = tr( D ) = 1 , 1 − α = X t k < 0 t k ≤ 0 , (263) 56 D + ≡ D + ( { t k ( ρ k ⊗ σ k ) } ) : = α − 1 X t k > 0 t k ( ρ k ⊗ σ k ) ∈ D ( H ⊗ J ) ds , (264) D ( H ⊗ J ) ds ⊔ { 0 } ∋ D − ≡ D − ( { t k ( ρ k ⊗ σ k ) } ) : = ( (1 − α ) − 1 P t k < 0 t k ( ρ k ⊗ σ k ) if α > 1 0 if α = 1 . (265) Note that D + is a discretely separable state, whereas D − ∈ Pod  B 1 ( H ) R b ⊗ B 1 ( J ) R  is either a discretely separable state or, if α = 1 — i.e., if D = D + — identically zero. Moreov er, the co efficien t α ≡ α ( { t k } ) ≥ 1, as well as the positively decomposable cross trace class op erators D + ≡ D + ( { t k ( ρ k ⊗ σ k ) } ) and D − ≡ D − ( { t k ( ρ k ⊗ σ k ) } ), dep end on the specific signed decomp osition (262) c hosen for the cross state D ∈ D ( H ⊗ J ) b . Also note that b y (263) we ha ve: 2 α ( { t k } ) − 1 = X t k > 0 t k + X t k < 0 ( − t k ) = X k | t k | ≥ 1 , (266) where α ( { t k } ) = 1 ⇐ ⇒ D = D + ( { t k ( ρ k ⊗ σ k ) } ) ∈ D ( H ⊗ J ) ds . (267) R emark 4.1 . If D ∈ D ( H ⊗ J ) b ± : = D ( H ⊗ J ) b \ D ( H ⊗ J ) ds , then this cross state must b e of the form D = α D 1 + (1 − α ) D 2 , for some D 1 , D 2 ∈ D ( H ⊗ J ) ds and α ∈ ( −∞ , 0) ∪ (1 , + ∞ ); namely , D m ust b e a non-c onvex affine c ombination of tw o discretely separable states. Clearly , without loss of generality , we can assume that α ∈ (1 , + ∞ ). Moreo ver, since the (Hermitian) pro jective norm of a discretely separable state is equal to 1, we ha ve that | | | D | | | b ⊗ 1 ≤ α | | | D 1 | | | b ⊗ 1 + ( α − 1) | | | D 2 | | | b ⊗ 1 = 2 α − 1 , α > 1 . (268) The previous argumen ts lead us to formulate the following result: Prop osition 4.3. A cr oss state D ∈ D ( H ⊗ J ) b is either a discr etely sep ar able state — i.e., a state the form (258) — or else a non-c onvex affine c ombination of two discr etely sep ar able states, i.e., D = α D 1 + (1 − α ) D 2 ∈ D ( H ⊗ J ) b ± , α > 1 , D 1 , D 2 ∈ D ( H ⊗ J ) ds . (269) Mor e over, if D ∈ D ( H ⊗ J ) ds , then ∥ D ∥ ⊗ 1 = ∥ D ∥ b ⊗ 1 = | | | D | | | b ⊗ 1 = 1 . If, inste ad, D ∈ D ( H ⊗ J ) b ± : = D ( H ⊗ J ) b \ D ( H ⊗ J ) ds (270) — namely, if D is a cr oss state on H ⊗ J that is not discr etely sep ar able — then 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ | | | D | | | b ⊗ 1 = inf  2 α − 1 > 1 : α D 1 + (1 − α ) D 2 = D , D 1 , D 2 ∈ D ( H ⊗ J ) ds  . (271) Pr o of. By our previous discussion, w e only need to pro ve the second and the third assertions. In fact — as already noted in Definition 4.1 — if D ∈ D ( H ⊗ J ) ds , then, by the equiv alence of conditions (D1) – (D5) in Prop osition 3.18, w e ha ve, in particular, that ∥ D ∥ ⊗ 1 = ∥ D ∥ b ⊗ 1 = | | | D | | | b ⊗ 1 = 1. If, instead, D ∈ D ( H ⊗ J ) b ± , then 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤| | | D | | | b ⊗ 1 = inf  P k | t k | : P k t k ( ρ k ⊗ σ k ) = D  = inf  2 α ( { t k } ) − 1 > 1 : P k t k ( ρ k ⊗ σ k ) = D  = inf  2 α ( { t k } ) − 1 > 1 : α ( { t k } ) D + ( { t k ( ρ k ⊗ σ k ) } ) + (1 − α ( { t k } )) D − ( { t k ( ρ k ⊗ σ k ) } ) = D  = inf  2 α − 1 > 1 : α D 1 + (1 − α ) D 2 = D , D 1 , D 2 ∈ D ( H ⊗ J ) ds  . (272) 57 Note that the second line of (272) — i.e., the first equality therein — follows from relation (253) (the infim um being extended o ver all possible expansions of D of the form (252)) and the second equali ty from relation (266) (together with (267)); next, the third equality follo ws from decomp osition (262) (together with definitions (263)–(265)) and the last one from (268) (where D = α D 1 + (1 − α ) D 2 , with D 1 , D 2 ∈ D ( H ⊗ J ) ds ). R emark 4.2 . Clearly , if D = α D 1 + (1 − α ) D 2 — with D , D 1 , D 2 ∈ D ( H ⊗ J ) ds and α ≥ 1 — then 1 = | | | D | | | b ⊗ 1 ≤ α | | | D 1 | | | b ⊗ 1 + (1 − α ) | | | D 2 | | | b ⊗ 1 = 2 α − 1, where the inequality is saturated when α = 1. By this fact and b y Prop osition 4.3 — in particular, observing that if D ∈ D ( H ⊗ J ) b ± and D = α D 1 + (1 − α ) D 2 , with D 1 , D 2 ∈ D ( H ⊗ J ) ds , then α ∈ [0 , 1] (actually , we can assume, without loss of generality , that α ∈ (1 , + ∞ )) — for every cross state D ∈ D ( H ⊗ J ) b we ha ve: | | | D | | | b ⊗ 1 = inf  2 α − 1 ≥ 1 : α D 1 + (1 − α ) D 2 = D , D 1 , D 2 ∈ D ( H ⊗ J ) ds  . (273) W e stress that, for D ∈ D ( H ⊗ J ) ds , the infimum on the rhs of (273) is attained as a minimum (i.e., 2 α − 1 = 1), whereas, for D ∈ D ( H ⊗ J ) b ± : = D ( H ⊗ J ) b \ D ( H ⊗ J ) ds suc h that | | | D | | | b ⊗ 1 = 1, it cannot b e (since 2 α − 1 ∈ (1 , + ∞ )). W e will show that the latter class of cross states consists precisely of those states on H ⊗ J that are separable — according to the general definition of separabilit y that will b e recalled b elo w — but not discretely separable. 4.3 Separable bipartite states: the general case The definition of discr ete sep ar ability in tro duced in the previous subsection coincides with the notion of separability tout c ourt considered b y some authors (see, e.g., Sect. 9.5 of [8]), but is a particular case of a more general notion of separabilit y w e will now analyze. W e kno w that ev ery discr etely sep ar able state — i.e., ev ery trace class operator D on H ⊗ J of the form (257) — is a cross state. It is then natural to w onder whether the most gener al separable state is a cross state, as w ell. W e start by pro viding a formal definition of sep ar ability of bipartite states, a notion that seems to hav e b een first established in its full generality (and on the physical ground) by W erner [13]. Definition 4.2. The set D ( H ⊗ J ) se ⊂ D ( H ⊗ J ) of all sep ar able states wrt the bipartition H ⊗ J is defined as D ( H ⊗ J ) se : = cl ∥ · ∥ ⊗ 1  ς ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) : ς = P j p j ( ρ j ⊗ σ j ), ρ j ∈ D ( H ), σ j ∈ D ( J )  . (274) Here, the decomp osition of the densit y op erator ς ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) ranges ov er all (finite) conv ex com binations of pr o duct states of the form ρ ⊗ σ , with ρ ∈ D ( H ) and σ ∈ D ( J ). Otherwise stated, D ( H ⊗ J ) se coincides with the ∥ · ∥ ⊗ 1 -closed conv ex hull of the set of all pro duct states wrt the bipartition H ⊗ J ; i.e., recalling Notation 3.2, D ( H ⊗ J ) se = co ( θ ( D ( H ) , D ( J ))) = : D ( H ) ⊗ D ( J ) . (275) A state τ ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se is said to b e entangle d . It is quite natural to consider, in this con text, an analogous notion of separabilit y where the role of the trace norm ∥ · ∥ 1 is play ed, instead, by the pro jective norm ∥ · ∥ b ⊗ 1 . Definition 4.3. The set D ( H ⊗ J ) cs ⊂ B 1 ( H ) b ⊗ B 1 ( J ) of cr oss sep ar able states wrt the bipartition H ⊗ J is defined as D ( H ⊗ J ) cs : = cl ∥ · ∥ b ⊗ 1  ς ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) : ς = P j p j ( ρ j ⊗ σ j ), ρ j ∈ D ( H ), σ j ∈ D ( J )  . (276) 58 Otherwise stated, D ( H ⊗ J ) cs is the ∥ · ∥ b ⊗ 1 -closed con vex hull co b ⊗ ( θ ( D ( H ) , D ( J ))) of the set of all pro duct states wrt the bipartition H ⊗ J ; i.e., recalling Notation 3.3, D ( H ⊗ J ) cs = D ( H ) b ⊗ D ( J ) . (277) Prop osition 4.4. D ( H ⊗ J ) cs is a close d c onvex subset of the ∥ · ∥ b ⊗ 1 -close d c onvex set D ( H ⊗ J ) b of al l cr oss states wrt the bip artition H ⊗ J , and D ( H ⊗ J ) ds ⊂ cl ∥ · ∥ b ⊗ 1 ( D ( H ⊗ J ) ds ) = D ( H ⊗ J ) cs ⊂ D ( H ⊗ J ) se . (278) In p articular, in the c ase wher e at le ast one of the Hilb ert sp ac es of the bip artition H ⊗ J is finite-dimensional, D ( H ⊗ J ) cs = D ( H ⊗ J ) se . R emark 4.3 . Recall that the norms ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 are equiv alen t on B 1 ( H ) R b ⊗ B 1 ( J ) R . Therefore, in the definition of the set D ( H ⊗ J ) cs ⊂ D ( H ⊗ J ) b ⊂ B 1 ( H ) R b ⊗ B 1 ( J ) R — see (276) — the closur e can actually b e tak en wrt either of these norms. Prop osition 4.5. F or every cr oss sep ar able state ς ∈ D ( H ⊗ J ) cs , ∥ ς ∥ b ⊗ 1 = | | | ς | | | b ⊗ 1 = 1 . Recalling relation (50), we ha ve: D ( H ⊗ J ) se = co ( θ ( D ( H ) , D ( J ))) = co ( θ ( P ( H ) , P ( J ))) = : P ( H ) ⊗ P ( J ) . (279) W e will denote by P ( H ⊗ J ) se the set of all sep ar able pur e states on H ⊗ J , i.e., P ( H ⊗ J ) se : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) se . (280) 4.4 Separable states and barycen tric decomp ositions The precise relation b etw een the conv ex sets D ( H ⊗ J ) se and D ( H ⊗ J ) cs is not immediately clear in the genuinely infinite-dimensional setting (i.e., for dim( H ) = dim( J ) = ∞ ); we only know that D ( H ⊗ J ) cs ⊂ D ( H ⊗ J ) se . T o further in vestigate this relation — sp ecifically , to c heck whether D ( H ⊗ J ) cs = D ( H ⊗ J ) se in the most general case — w e first need to establish some preliminary results. T o this end, w e will use, in particular, basic facts regarding the in tegration of vector-v alued functions; see, e.g., the standard references [49, 62]. In fact, we will consider Bo chner inte gr als of functions that tak e v alues in a sep ar able Banach space (i.e., B 1 ( H ⊗ J ) or B 1 ( H ) b ⊗ B 1 ( J )). By a w ell-known result — the so-called Pettis Me asur ability The or em [49] — such a function is str ongly me asur able iff it is we akly me asur able ; to indicate this type of vector-v alued functions, w e will therefore synthetically use the term me asur able . It is, moreov er, a standard tec hnical fact that, whenev er in tegration wrt a finite measure (in particular, a probability measure) is inv olv ed, ev ery norm-b ounde d measurable function is Bo chner-inte gr able ; see, e.g., Prop osition 1.2.2 of [49]. Lemma 4.1. L et us endow Cartesian pr o duct D ( H ) × D ( J ) with the pr o duct top olo gy — D ( H ) and D ( J ) b eing endowe d with their standar d top olo gy, se e F act 2.7 — and the sets θ ( D ( H ) , D ( J )) , b θ ( D ( H ) , D ( J )) with their subsp ac e top olo gy (wrt the natur al norm top olo gy of the Banach sp ac es B 1 ( H ) ⊗ B 1 ( J ) = B 1 ( H ⊗ J ) and B 1 ( H ) b ⊗ B 1 ( J ) , r esp e ctively). Then, t hese sets ar e mutual ly home omorphic, b e c ause the bije ctive maps D ( H ) × D ( J ) ∋ ( ρ, σ ) 7→ ρ ⊗ σ ∈ θ ( D ( H ) , D ( J )) (281) and θ ( D ( H ) , D ( J )) ∋ ρ ⊗ σ 7→ ρ ⊗ σ ∈ b θ ( D ( H ) , D ( J )) (282) ar e home omorphisms. Mor e over, θ ( D ( H ) , D ( J )) and θ ( P ( H ) , P ( J )) ar e a norm-close d subsets of B 1 ( H ⊗ J ) . 59 Pr o of. By F act 3.4, the maps (281) and (282) are bijectiv e. Let us then pro ve that, in particular, they are homeomorphisms. First note that the Cartesian pro duct D ( H ) × D ( J ), endow ed with the pro duct top ology , is a second countable space, b ecause D ( H ), D ( J ) are endow ed with their (second coun table) standard topology , that coincides with the subspace top ology wrt the separa- ble Banach spaces B 1 ( H ) and B 1 ( J ), resp ectiv ely . Similarly , θ ( D ( H ) , D ( J )) and b θ ( D ( H ) , D ( J )) — once endo wed with the subspace top ology wrt the separable Banach spaces B 1 ( H ⊗ J ) and B 1 ( H ) b ⊗ B 1 ( J ), resp ectiv ely — are second countable spaces too. Therefore, a fortiori these spaces satisfy the first axiom of countabilit y , so that in the following w e can argue in terms of se quenc es (a map b et ween top ological spaces, with first coun table domain, is contin uous iff it is sequentially con tinuous). Now, a sequence { ( ρ n , σ n ) } ⊂ D ( H ) × D ( J ) con verges to some ( ρ, σ ) ∈ D ( H ) × D ( J ) iff lim n ∥ ρ − ρ n ∥ 1 = 0 and lim n ∥ σ − σ n ∥ 1 = 0. Moreov er, we ha ve that ∥ ρ ⊗ σ − ρ n ⊗ σ n ∥ ⊗ 1 ≤ ∥ ( ρ − ρ n ) ⊗ σ ∥ ⊗ 1 + ∥ ρ n ⊗ ( σ − σ n ) ∥ ⊗ 1 = ∥ ρ − ρ n ∥ 1 ∥ σ ∥ 1 + ∥ ρ n ∥ 1 ∥ σ − σ n ∥ 1 = ∥ ρ − ρ n ∥ 1 + ∥ σ − σ n ∥ 1 . (283) Therefore, the bijective map (281) is contin uous and, by the b oundedness of the partial traces tr J : B 1 ( H ) ⊗ B 1 ( J ) → B 1 ( H ) and tr H : B 1 ( H ) ⊗ B 1 ( J ) → B 1 ( J ), its inv erse — i.e., the mapping θ ( D ( H ) , D ( J )) ∋ ρ ⊗ σ 7→  ρ = tr J ( ρ ⊗ σ ) , σ = tr H ( ρ ⊗ σ )  ∈ D ( H ) × D ( J ) (284) — is con tinuous to o (the tw o co ordinate maps b eing con tinuous). Let us next to sho w that the bijection (282) is a homeomorphism, as w ell. T o this end, note that — since the trace norm ∥ · ∥ ⊗ 1 is ma jorized by the pro jective norm ∥ · ∥ b ⊗ 1 — for every sequence { ( ρ n , σ n ) } ⊂ D ( H ) × D ( J ), we hav e: ∥ ρ ⊗ σ − ρ n ⊗ σ n ∥ ⊗ 1 ≤ ∥ ρ ⊗ σ − ρ n ⊗ σ n ∥ b ⊗ 1 ≤ ∥ ( ρ − ρ n ) ⊗ σ ∥ b ⊗ 1 + ∥ ρ n ⊗ ( σ − σ n ) ∥ b ⊗ 1 = ∥ ρ − ρ n ∥ 1 ∥ σ ∥ 1 + ∥ ρ n ∥ 1 ∥ σ − σ n ∥ 1 = ∥ ρ − ρ n ∥ 1 + ∥ σ − σ n ∥ 1 . (285) Hence, lim n ∥ ρ ⊗ σ − ρ n ⊗ σ n ∥ ⊗ 1 = 0 ⇐ ⇒ lim n ∥ ρ ⊗ σ − ρ n ⊗ σ n ∥ b ⊗ 1 = 0, where for the direct implication “= ⇒ ” we ha ve also exploited the con tinuit y of the map (284) (otherwise stated, by the second inequalit y in (285) the map D ( H ) × D ( J ) ∋ ( ρ, σ ) 7→ ρ ⊗ σ ∈ b θ ( D ( H ) , D ( J )) is contin uous, and, comp osed with the contin uous map (284), giv es the map (282), whic h is then contin uous too). Let us now finally prov e that b oth the sets θ ( D ( H ) , D ( J )) and θ ( P ( H ) , P ( J )) are closed in B 1 ( H ⊗ J ). In fact, giv en any conv ergent sequence of pr o duct states { ρ n ⊗ σ n } ⊂ B 1 ( H ⊗ J ) — ∥ · ∥ ⊗ 1 – lim n ρ n ⊗ σ n = D ∈ B 1 ( H ⊗ J ); but note that, actually , D ∈ D ( H ⊗ J ), b ecause D ( H ⊗ J ) is a closed subset of B 1 ( H ⊗ J ) — by the contin uit y of the partial trace we ha v e that ρ ≡ tr J ( D ) = tr J  ∥ · ∥ ⊗ 1 – lim n ρ n ⊗ σ n  = ∥ · ∥ 1 – lim n tr J ( ρ n ⊗ σ n ) = ∥ · ∥ 1 – lim n ρ n ∈ D ( H ) (286) and, analogously , σ ≡ tr H ( D ) = ∥ · ∥ 1 – lim n σ n ∈ D ( J ). It follows that the sequence { ( ρ n , σ n ) } ⊂ D ( H ) × D ( J ) conv erges to ( ρ, σ ) ∈ D ( H ) × D ( J ); hence, by the contin uity of the map (281), we m ust ha ve that ∥ · ∥ ⊗ 1 – lim n ρ n ⊗ σ n = ρ ⊗ σ ∈ θ ( D ( H ) , D ( J )). Thus, θ ( D ( H ) , D ( J )) is a closed subset of B 1 ( H ⊗ J ). Moreov er, by the contin uit y of the pro duct of op erators in B 1 ( H ⊗ J ) (the trace class is a Banac h algebra), we see that ∥ · ∥ ⊗ 1 – lim n ρ n ⊗ σ n = ρ ⊗ σ = ⇒ ∥ · ∥ ⊗ 1 – lim n ( ρ n ⊗ σ n ) 2 = ( ρ ⊗ σ ) 2 = ρ 2 ⊗ σ 2 . (287) Therefore, if, in particular, { ρ n ⊗ σ n } ⊂ θ ( P ( H ) , P ( J )), then ρ ⊗ σ = ∥ · ∥ ⊗ 1 – lim n ρ n ⊗ σ n = ∥ · ∥ ⊗ 1 – lim n ( ρ n ⊗ σ n ) 2 = ( ρ ⊗ σ ) 2 , (288) so that ρ ⊗ σ ∈ θ ( P ( H ) , P ( J )). Hence, θ ( P ( H ) , P ( J )) is a closed subset of B 1 ( H ⊗ J ) to o. 60 R emark 4.4 . Endow ed with with their relative topology wrt B 1 ( H ⊗ J ), b oth θ ( D ( H ) , D ( J )) and θ ( P ( H ) , P ( J )) are Polish spaces, being closed subsets of the Polish space B 1 ( H ⊗ J ). Moreov er, b y Lemma 4.1, their P olish space top ology coincides with the top ology induced b y the pro jective norm ∥ · ∥ b ⊗ 1 . Otherwise stated, we can iden tify , also top ologically , the spaces θ ( D ( H ) , D ( J )) and b θ ( D ( H ) , D ( J )), as well as the spaces θ ( P ( H ) , P ( J )) and b θ ( P ( H ) , P ( J )). Therefore, the spaces θ ( D ( H ) , D ( J )) and b θ ( D ( H ) , D ( J )) p ossess the same natur al Bor el structur e (the smallest Borel structure con taining all the op en sets of their common top ology), wrt whic h they are (isomorphic) standar d Bor el sp ac es ; see Subsection 2.5, and references therein. Analogous facts hold for the spaces θ ( P ( H ) , P ( J )) and b θ ( P ( H ) , P ( J )) to o. Lemma 4.2. L et C b e a close d subset of D ( H ⊗ J ) . Then, denoting by P ( C ) the set of Bor el pr ob ability me asur es on C , we have (Bo chner inte gr als b eing understo o d): co ( C ) =  R C ξ d µ ( ξ ) : µ ∈ P ( C )  ; (289) i.e., the ∥ · ∥ ⊗ 1 -close d c onvex hul l of the set C c oincides with the set of barycen ters of al l Bor el pr ob ability me asur es on C . Pr o of. This is a well-kno wn result; see Lemma 1 of [28] (also see [29, 30]). Lemma 4.3. L et ( X , µ ) b e a Bor el pr ob ability me asur e sp ac e, and let f : X → B 1 ( H ⊗ J ) b e a me asur able function such that f ( X ) ⊂ θ ( D ( H ) , D ( J )) . Then, f is Bo chner-inte gr able and Z X f ( x ) d µ ( x ) ⊂ co ( f ( X )) ⊂ co ( θ ( D ( H ) , D ( J ))) = D ( H ⊗ J ) se , (290) wher e, on the lhs, a Bo chner inte gr al of B 1 ( H ⊗ J ) -value d functions is understo o d. A nalo gously, let g : X → B 1 ( H ) b ⊗ B 1 ( J ) b e a me asur able function such that g ( X ) ⊂ θ ( D ( H ) , D ( J )) ; then, g is Bo chner-inte gr able and Z X g ( x ) d µ ( x ) ⊂ co b ⊗ ( g ( X )) ⊂ co b ⊗ ( θ ( D ( H ) , D ( J ))) = D ( H ⊗ J ) cs , (291) wher e, on the lhs, a Bo chner inte gr al of B 1 ( H ) b ⊗ B 1 ( J ) -value d functions is understo o d. Mor e over, if f ( x ) = g ( x ) , for µ -almost al l x ∈ X , then the inte gr als in (290) and (291) ar e e qual; i.e., they c onver ge to the same density op er ator on H ⊗ J . Pr o of. Let us prov e the first assertion. Since here integration wrt a finite measure µ is in v olved, and the vector-v alued function f is norm-b ounded, then f is Bo c hner-in tegrable. Moreov er, by a w ell-known result — see [62], Corollary 8 in Section 2 of Chapter 2, or Prop osition 1.2.12 of [49] — the Bo c hner in tegral R X f ( x ) d µ ( x ) is con tained in the ∥ · ∥ ⊗ 1 -closed con vex hull co ( f ( X )). The pro of of the second assertion is analogous. Let us prov e the final assertion. T o this end, just recall that the linear immersion map j : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ⊗ J ) is b ounded (Prop osition 3.5), and therefore, since f ( x ) = j  g ( x )  =  j ◦ g  ( x ), for µ -almost all x ∈ X , b y a classical result of Hille (see Theorem 1.2.4 of [49]), we hav e that R X f ( x ) d µ ( x ) = R X  j ◦ g  ( x ) d µ ( x ) = j  R X g ( x ) d µ ( x )  ; i.e. — omitting, as usual, the immersion map j — the integrals in (290) and (291) are equal. Theorem 4.1. A state ς ∈ D ( H ⊗ J ) is sep ar able — i.e., ς ∈ D ( H ⊗ J ) se = D ( H ) ⊗ D ( J ) — iff ther e is a Bor el pr ob ability me asur e µ on θ ( D ( H ) , D ( J )) =  ρ ⊗ σ : ρ ∈ D ( H ) , σ ∈ D ( J )  such that ς = Z θ ( D ( H ) , D ( J )) ρ ⊗ σ d µ ( ρ ⊗ σ ) ; (292) 61 also, iff ther e is a Bor el pr ob ability me asur e ν on θ ( P ( H ) , P ( J )) =  π ⊗ ϖ : π ∈ P ( H ) , ϖ ∈ P ( J )  such that ς = Z θ ( P ( H ) , P ( J )) π ⊗ ϖ d ν ( π ⊗ ϖ ) . (293) In (292) and (293) , a Bo chner inte gr al of B 1 ( H ⊗ J ) -value d functions — or, e quivalently (i.e., with the inte gr al c onver ging to the same state ς ), a Bo chner inte gr al of B 1 ( H ) b ⊗ B 1 ( J ) -value d functions — is understo o d. Pr o of. By the final assertion of Lemma 4.1, θ ( D ( H ) , D ( J )) is a closed subset of B 1 ( H ⊗ J ). Therefore, we can apply Lemma 4.2, with C = θ ( D ( H ) , D ( J )), and conclude that if a state ς ∈ D ( H ⊗ J ) is separable — i.e., if ς ∈ D ( H ⊗ J ) se = co ( θ ( D ( H ) , D ( J ))) — then it is of the form (292), for some Borel probabilit y measure µ on θ ( D ( H ) , D ( J )), where the in tegral is a Bo c hner integral of B 1 ( H ⊗ J )-v alued functions. Conv ersely , if ς ∈ D ( H ⊗ J ) is of the form (292) — where a Bochner in tegral of B 1 ( H ⊗ J )-v alued functions is understo od — then, by the first assertion of Lemma 4.3, it is contained in ∥ · ∥ ⊗ 1 -closed conv ex h ull co ( θ ( D ( H ) , D ( J ))) = D ( H ⊗ J ) se . Let us prov e that the in tegral on the rhs of (292) can b e regarded as a Bo chner integral of B 1 ( H ) b ⊗ B 1 ( J )-v alued functions, as w ell. Indeed, by the first assertion of Lemma 4.1, the map θ ( D ( H ) , D ( J )) ∋ ρ ⊗ σ 7→ ρ ⊗ σ ∈ B 1 ( H ) b ⊗ B 1 ( J ) (294) is contin uous — hence, a fortiori , (strongly) µ -measurable — and Z θ ( D ( H ) , D ( J )) ∥ ρ ⊗ σ ∥ b ⊗ 1 d µ ( ρ ⊗ σ ) = Z θ ( D ( H ) , D ( J )) ∥ ρ ⊗ σ ∥ ⊗ 1 d µ ( ρ ⊗ σ ) = 1 < ∞ . (295) Hence, it is Bo c hner-integrable. Moreov er, by the final assertion Lemma 4.3, the Bo c hner integral in (292) con verges — wrt b oth the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 — to a unique state ς . Analogously , by Lemma 4.1, P ( H ⊗ J ) se = θ ( P ( H ) , P ( J )) is a closed subset of B 1 ( H ⊗ J ), and, restricting the domain of the map (294), w e see that the mapping θ ( P ( H ) , P ( J )) ∋ π ⊗ ϖ 7→ π ⊗ ϖ ∈ B 1 ( H ) b ⊗ B 1 ( J ) (296) is contin uous to o. Then arguing as ab o v e, w e conclude that ς ∈ D ( H ⊗ J ) se = co ( θ ( D ( H ) , D ( J ))) = co ( θ ( P ( H ) , P ( J ))) (297) — where we hav e used relation (279) — iff it is of the form (293), where, once again, the integral can b e regarded as a Bo c hner in tegral of B 1 ( H ⊗ J )-v alued functions or, equiv alen tly , as a Bo c hner in tegral of B 1 ( H ) b ⊗ B 1 ( J )-v alued functions. Once again, b y Lemma 4.3, the Bochner integral in (293) con verges — wrt b oth the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ b ⊗ 1 — to a unique state ς . W e will call the general expression (292) — or (293) — of a separable state ς ∈ D ( H ⊗ J ) se a b aryc entric de c omp osition of ς . It is, how ever, w orth observing that, by Theorem 4.1, a separable state also admits a decomposition that may b e regarded as a direct generalization of the (p ositiv e) decomp osition of a discretely separable state. Corollary 4.1. Ther e exists a Bor el isomorphism [0 , 1] ∋ x 7→  ρ ⊗ σ  ( x ) ∈ θ ( D ( H ) , D ( J )) . (298) Given any such a (me asur able) map, a state ς ∈ D ( H ⊗ J ) is sep ar able iff it is of the form ς = Z 1 0  ρ ⊗ σ  ( x ) d ζ ( x ) , (299) 62 for some Bor el pr ob ability me asur e ζ on the interval [0 , 1] . Her e, a Bo chner inte gr al of B 1 ( H ⊗ J ) - value d functions — or, e quivalently, a Bo chner inte gr al of B 1 ( H ) b ⊗ B 1 ( J ) -value d functions — is understo o d. A nalo gously, and with the same twofold p ossible interpr etation of the Bo chner inte gr al, ther e is a Bor el isomorphism [0 , 1] ∋ x 7→  π ⊗ ϖ  ( x ) ∈ θ ( P ( H ) , P ( J )) , (300) and, given any such a map, a state ς ∈ D ( H ⊗ J ) is sep ar able iff it is of the form ς = Z 1 0  π ⊗ ϖ  ( x ) d ξ ( x ) . (301) for some a Bor el pr ob ability me asur e ξ on [0 , 1] . Pr o of. Since θ ( D ( H ) , D ( J )) and θ ( P ( H ) , P ( J )) are Polish spaces (Remark 4.4), and they b oth ha ve the p ow er of the contin uum (for dim( H ⊗ J ) ≥ 2), they are Borel isomorphic to the unit in terv al [0 , 1] (F act 2.21). Now, let [0 , 1] ∋ x 7→ φ ( x ) ≡  ρ ⊗ σ  ( x ) ∈ θ ( D ( H ) , D ( J )) b e an y Borel isomorphism. F or ev ery Borel probabilit y measure ζ on [0 , 1], by the c hange-of-v ariables form ula (see, e.g., Prop osition 1.2.6 of [49]) we ha ve that Z 1 0  ρ ⊗ σ  ( x ) d ζ ( x ) = Z θ ( D ( H ) , D ( J )) ρ ⊗ σ d µ ( ρ ⊗ σ ) (302) where µ = φ ∗ ζ ; i.e., µ is the push-forw ard of the measure ζ : µ ( E ) : = ζ ( φ − 1 ( E )), for ev ery Borel subset E of θ ( D ( H ) , D ( J )). Note that, since φ is Borel isomorphism, the push-forward φ ∗ estab- lishes a bijection b et ween the set of all Borel probabilit y measures on the interv al [0 , 1] and the set of all Borel probability measures on θ ( D ( H ) , D ( J )). Then, by Theorem 4.1, the first assertion follo ws, and the pro of of the second assertion is analogous. Example 4.1. Let us supp ose that H = J = L 2 ([0 , 1]) ≡ L 2 ([0 , 1] , β ; C ), where β is the Leb esgue measure on the interv al [0 , 1] (i.e., d β ( x ) = d x ), and let  ϕ k = e i 2 π k ( · ) : k ∈ Z  b e the standard trigonometric basis in L 2 ([0 , 1]), so that the F ourier co efficien ts { c k ( ψ ) } k ∈ Z of any ψ ∈ L 2 ([0 , 1]) are giv en by c k ( ψ ) : = ⟨ ϕ k , ψ ⟩ = R 1 0 e − i 2 π kx ψ ( x ) d x , k ∈ Z . Now, denoting by T the circle group, the mapping T ∋ z ≡ e i 2 π ϑ 7→ V ( z ) : = X k ∈ Z z − k c ϕ k (303) — where c ϕ k ≡ | ϕ k ⟩ ⟨ ϕ k | , and V ( z ) is a unitary op erator on L 2 ([0 , 1]) (here defined via its sp ectral decomp osition) — is a strongly con tinuous unitary represen tation of the compact abelian group T . Precisely , T ∋ z 7→ V ( z ) is the r e gular r epr esentation [66] of T — the orthogonal sum of exactly one cop y of each unitary character of T — and, in fact, w e ha v e that  V  e i 2 π ϑ  ψ  ( x ) = ψ ([ x − ϑ ]), where ϑ ∈ [0 , 1), [ x − ϑ ] ∈ [0 , 1] and [ x − ϑ ] ≡ x − ϑ (mo d 1). Consider the asso ciated (diagonal or) inner tensor pr o duct r epr esentation [66] V ⊗ V of T in H ⊗ J = L 2 ([0 , 1]) ⊗ L 2 ([0 , 1]). Giv en an y pair of pure states π = c ψ 1 ≡ | ψ 1 ⟩ ⟨ ψ 1 | ∈ P ( H ) and ϖ = c ψ 2 ∈ P ( J ), w e can define a separable state ς ∈ D ( H ⊗ J ) se b y setting ς : = Z 1 0  a V  e i 2 π ϑ  ( π ⊗ ϖ )  d ϑ , (304) where, for ev ery trace class op erator A ∈ B 1 ( H ⊗ J ), a V  e i 2 π ϑ  A : =  V ⊗ V  e i 2 π ϑ  ∗ A  V ⊗ V  e i 2 π ϑ  ∈ B 1 ( H ⊗ J ) ; (305) 63 th us, in particular, a V  e i 2 π ϑ  ( π ⊗ ϖ ) ∈ θ ( P ( H ) , P ( J )). In formula (304), a Bo c hner in tegral of B 1 ( H ⊗ J )-v alued functions (equiv alently , b y the final assertion of Lemma 4.3, a Bochner integral of B 1 ( H ) b ⊗ B 1 ( J )-v alued functions) is understo od. Note that — b y Prop osition 4.1 of [67] — the map T ∋ z 7→ a V ( z ) is a str ongly c ontinuous isometric representation of the circle group T in the Banac h space B 1 ( H ⊗ J ). As a consequence, the mapping (which dep ends on b oth the represen tation V and the pure state π ⊗ ϖ ) [0 , 1] ∋ x 7→ φ ( x ) ≡  a V  e i 2 π x  ( π ⊗ ϖ )  ∈ θ ( P ( H ) , P ( J )) (306) — where θ ( P ( H ) , P ( J )) ⊂ B 1 ( H ⊗ J ) is endow ed with the subspace top ology — is contin uous; in particular, it is measurable. Therefore, one can define the push-forwar d me asur e ν : = φ ∗ β (d β ( x ) = d x ), that is a Borel probability measure on θ ( P ( H ) , P ( J )) (dep ending on V and on the state π ⊗ ϖ ), and by the c hange-of-v ariables formula, w e obtain the barycentric decomp osition ς = Z θ ( P ( H ) , P ( J )) π ⊗ ϖ d ν ( π ⊗ ϖ ) , ν = φ ∗ β , φ : [0 , 1] → θ ( P ( H ) , P ( J )) . (307) It can b e prov ed that if, for every k ∈ Z , c k ( ψ 1 ) : = ⟨ ϕ k , ψ 1 ⟩  = 0  = c k ( ψ 2 ) (i.e., if the state v ectors ψ 1 ∈ H and ψ 2 ∈ J hav e non-v anishing F ourier co efficien ts), then the separable state ς — defined b y (304) with π = c ψ 1 , ϖ = c ψ 2 , or, equiv alently , by (307) — is not discretely separable (see Theorem 3 of [28]). Therefore, if dim( H ) = dim( J ) = ∞ , then D ( H ⊗ J ) ds ⊊ D ( H ⊗ J ) se . Let us deriv e some further consequences of Theorem 4.1. Corollary 4.2. A sep ar able state ς ∈ D ( H ⊗ J ) se is pur e — i.e., a r ank-one pr oje ction — iff it is of the form (293) , wher e the pr ob ability me asur e ν is a Dir ac me asur e or, e quivalently, such that supp ( ν ) is a singleton set. Ther efor e, the set P ( H ⊗ J ) se of al l sep ar able pur e states on H ⊗ J is given explicitly by P ( H ⊗ J ) se = θ ( P ( H ) , P ( J )) =  π ⊗ ϖ : π ∈ P ( H ) , ϖ ∈ P ( J )  , (308) and D ( H ⊗ J ) se = co ( P ( H ⊗ J ) se ) . Mor e over, denoting by ext( D ( H ⊗ J ) se ) the set of al l extr eme p oints of the c onvex set D ( H ⊗ J ) se , we have that ext( D ( H ⊗ J ) se ) = θ ( P ( H ) , P ( J )) = P ( H ⊗ J ) se . (309) Pr o of. First note that a probability measure ν on θ ( P ( H ) , P ( J )) is a Dirac measure iff supp ( ν ) is a singleton set (b y Remark 2.4 and F act 2.20, b ecause supp ( ν ) is a second countable Hausdorff space). Therefore, given any ς ∈ D ( H ⊗ J ) se , let us prov e that ς is pure iff it is of the form (293), where ν = δ D , with { D } = supp ( ν ) ⊂ θ ( P ( H ) , P ( J )). Clearly , if the latter condition holds, then ς = D = π 0 ⊗ ϖ 0 , for some π 0 ∈ P ( H ) and ϖ 0 ∈ P ( J ). T o pro ve the reverse implication, supp ose that ς ∈ P ( H ⊗ J ) se and supp ( ν ) is not a singleton set. Then, since supp ( ν )  = ∅ (again by F act 2.20, θ ( P ( H ) , P ( J )) b eing second countable), there are at least two differ ent pur e states of the form π 1 ⊗ ϖ 1 and π 2 ⊗ ϖ 2 in supp ( ν ). It follows that there exist disjoint op en neighb orho o ds O , O ′ of π 1 ⊗ ϖ 1 and π 2 ⊗ ϖ 2 , resp ectively , so that ν ( O ) , ν ( O ′ ) > 0 and O ′ ⊂ O c (th us, for the closed set O c , ν ( O c ) > 0 to o). Hence, w e ha ve: ς = Z θ ( P ( H ) , P ( J ))  χ O ( π ⊗ ϖ ) + χ O c ( π ⊗ ϖ )  π ⊗ ϖ d ν ( π ⊗ ϖ ) = Z O π ⊗ ϖ d ν ( π ⊗ ϖ ) + Z O c π ⊗ ϖ d ν ( π ⊗ ϖ ) = ϵ D 1 + (1 − ϵ ) D 2 , ϵ ∈ (0 , 1) , D 1 , D 2 ∈ D ( H ⊗ J ) se . (310) 64 In the last line of (310), 0 < ϵ = ν ( O ) < 1, 0 < 1 − ϵ = ν ( O c ) < 1 and D 1 = ν ( O ) − 1 Z O π ⊗ ϖ d ν ( π ⊗ ϖ ) = Z θ ( P ( H ) , P ( J )) π ⊗ ϖ d µ 1 ( π ⊗ ϖ ) , (311) D 2 = Z θ ( P ( H ) , P ( J )) π ⊗ ϖ d µ 2 ( π ⊗ ϖ ) , (312) where µ 1 , µ 2 are the Borel probability measures on θ ( D ( H ) , D ( J )) determined by d µ 1 ( π ⊗ ϖ ) = ν ( O ) − 1 χ O ( π ⊗ ϖ ) d ν ( π ⊗ ϖ ) , d µ 2 ( π ⊗ ϖ ) = ν ( O c ) − 1 χ O c ( π ⊗ ϖ ) d ν ( π ⊗ ϖ ) . (313) Therefore, we find a contradiction, b ecause ς was assumed to b e an extreme p oin t of D ( H ⊗ J ), and w e must conclude that, if ς is a pure state, then the nonempt y set supp ( ν ) must be a singleton. Let us no w pro ve the second assertion. First note that, since D ( H ⊗ J ) se ⊂ D ( H ⊗ J ), then ext( D ( H ⊗ J )) ∩ D ( H ⊗ J ) se = : P ( H ⊗ J ) se ⊂ ext( D ( H ⊗ J ) se ) (b y F act 2.17). Next, supp ose that ς ∈ D ( H ⊗ J ) se \ P ( H ⊗ J ) se , and consider a barycentric decomp osition of ς of the form (293). Then, ν cannot be a Dirac measure and there m ust b e at least two differ ent pur e states of the form π 1 ⊗ ϖ 1 and π 2 ⊗ ϖ 2 in supp ( ν )  = ∅ . Arguing as ab ov e, we find that ς = ϵ D 1 + (1 − ϵ ) D 2 , ϵ ∈ (0 , 1), D 1 , D 2 ∈ D ( H ⊗ J ) se ; i.e., ς is not con tained in ext( D ( H ⊗ J ) se ) — hence, ext( D ( H ⊗ J ) se ) ⊂ P ( H ⊗ J ) se , as w ell — so that, actually , ext( D ( H ⊗ J ) se ) = P ( H ⊗ J ) se . W e are no w able to generalize the final assertion of Prop osition 4.4: The cross separable states wrt to the bipartition H ⊗ J are precisely the separable states tout c ourt in the gen uinely infinite- dimensional setting (dim( H ) = dim( J ) = ∞ ), as w ell; in fact, the following result holds. Corollary 4.3. The set D ( H ⊗ J ) se = D ( H ) ⊗ D ( J ) of al l sep ar able states is c ontaine d in the set D ( H ⊗ J ) b of cr oss states and c oincides with the set D ( H ⊗ J ) cs = D ( H ) b ⊗ D ( J ) of cr oss sep ar able states; in fact, we have: P ( H ) ⊗ P ( J ) = D ( H ) ⊗ D ( J ) = D ( H ) b ⊗ D ( J ) = P ( H ) b ⊗ P ( J ) ⊂ D ( H ⊗ J ) b . (314) Pr o of. W e hav e already shown that D ( H ⊗ J ) cs ⊂ D ( H ⊗ J ) se (Prop osition 4.4), and the rev erse inclusion holds to o. In fact, by Theorem 4.1, if a state ς ∈ D ( H ⊗ J ) is separable, then it is of the form (292), where the integral con verges wrt the pro jectiv e norm ∥ · ∥ b ⊗ 1 as w ell (to the same state), and hence, b y the second assertion of Lemma 4.3, if ς ∈ D ( H ⊗ J ) se , then it is contained in the ∥ · ∥ b ⊗ 1 -closed con vex h ull co b ⊗ ( θ ( D ( H ) , D ( J ))) = D ( H ⊗ J ) cs to o. Therefore, recalling relation (279), P ( H ) ⊗ P ( J ) = D ( H ) ⊗ D ( J ) = D ( H ⊗ J ) se = D ( H ⊗ J ) cs = D ( H ) b ⊗ D ( J ) ⊂ D ( H ⊗ J ) b . (315) It remains to pro ve that D ( H ) b ⊗ D ( J ) = P ( H ) b ⊗ P ( J ). Clearly , D ( H ) b ⊗ D ( J ) ⊃ P ( H ) b ⊗ P ( J ). Con versely , b y Theorem 4.1, ev ery separable state ς ∈ D ( H ⊗ J ) se = D ( H ) ⊗ D ( J ) = D ( H ) b ⊗ D ( J ) is of the form (293), where a Bochner integral of B 1 ( H ) b ⊗ B 1 ( J )-v alued functions is understo o d; hence, b y the first inclusion relation in (291) (Lemma 4.3), D ( H ) b ⊗ D ( J ) ⊂ co b ⊗ ( θ ( P ( H ) , P ( J ))) = : P ( H ) b ⊗ P ( J ) to o. Corollary 4.4. F or every cr oss state D ∈ D ( H ⊗ J ) b , we have that | | | D | | | b ⊗ 1 = inf  2 α − 1 ≥ 1 : α D 1 + (1 − α ) D 2 = D , D 1 , D 2 ∈ D ( H ⊗ J ) se  . (316) 65 Pr o of. Since D ( H ⊗ J ) ds ⊂ D ( H ⊗ J ) se , by relation (273), for every D ∈ D ( H ⊗ J ) b we hav e that | | | D | | | b ⊗ 1 ≥ inf  2 α − 1 ≥ 1 : α D 1 + (1 − α ) D 2 = D , D 1 , D 2 ∈ D ( H ⊗ J ) se  . (317) On the other hand, b y Corollary 4.3, D ( H ⊗ J ) se = D ( H ) ⊗ D ( J ) = D ( H ) b ⊗ D ( J ) = D ( H ⊗ J ) cs . Hence, b y Prop osition 4.4, D ( H ⊗ J ) se coincides with the | | | · | | | b ⊗ 1 -closure of the con vex set D ( H ⊗ J ) ds (the pro jectiv e norm and the Hermitian pro jective norm b eing equiv alen t), so that, if D = α D 1 ( α ) + (1 − α ) D 2 ( α ) , α ≥ 1 , D 1 , D 2 ∈ D ( H ⊗ J ) se , (318) then, for ev ery ϵ > 0, there are some D 1 ( α, ϵ ) , D 2 ( α, ϵ ) ∈ D ( H ⊗ J ) ds suc h that | | | D 1 ( α ) − D 1 ( α, ϵ ) | | | b ⊗ 1 < ϵ 2 α and | | | D 2 ( α ) − D 2 ( α, ϵ ) | | | b ⊗ 1 < ϵ 2( α − 1) . (319) It follo ws that | | | D | | | b ⊗ 1 = | | | α D 1 ( α ) + (1 − α ) D 2 ( α ) | | | b ⊗ 1 ≤ α | | | D 1 ( α ) − D 1 ( α, ϵ ) | | | b ⊗ 1 + α | | | D 1 ( α, ϵ ) | | | b ⊗ 1 + ( α − 1) | | | D 2 ( α ) − D 2 ( α, ϵ ) | | | b ⊗ 1 + ( α − 1) | | | D 2 ( α, ϵ ) | | | b ⊗ 1 < 2 α − 1 + ϵ , where, for obtaining the second inequalit y , we ha ve used relations (319) and the fact that | | | D 1 ( α, ϵ ) | | | b ⊗ 1 = 1 = | | | D 2 ( α, ϵ ) | | | b ⊗ 1 (b y Prop osition 4.3, since D 1 ( α, ϵ ) , D 2 ( α, ϵ ) ∈ D ( H ⊗ J ) ds ). By the previous estimate, we conclude that | | | D | | | b ⊗ 1 ≤ inf  2 α − 1 ≥ 1 : α D 1 + (1 − α ) D 2 = D , D 1 , D 2 ∈ D ( H ⊗ J ) se  , (320) as w ell. Hence, b y inequalities (317) and (320), we see that, actually , relation (316) holds true. Corollary 4.5. F or every cr oss state D ∈ D ( H ⊗ J ) b we have that | | | D | | | b ⊗ 1 ≥ ∥ D ∥ b ⊗ 1 ≥ 1 , and ς ∈ D ( H ⊗ J ) se = ⇒ | | | ς | | | b ⊗ 1 = ∥ ς ∥ b ⊗ 1 = 1 . (321) Pr o of. Indeed, for ev ery D ∈ D ( H ⊗ J ) b w e ha ve: | | | D | | | b ⊗ 1 ≥ ∥ D ∥ b ⊗ 1 ≥ ∥ D ∥ ⊗ 1 = 1. In particular, for ev ery state ς ∈ D ( H ⊗ J ) se = D ( H ⊗ J ) cs (Corollary 4.3), by Prop osition 4.5, ∥ ς ∥ b ⊗ 1 = | | | ς | | | b ⊗ 1 = 1. Alternativ ely , using the characterization (292) of a separable state, w e obtain the estimate 1 ≤ ∥ ς ∥ b ⊗ 1 ≤ | | | ς | | | b ⊗ 1 ≤ Z θ ( P ( H ) , P ( J )) | | | ρ ⊗ σ | | | b ⊗ 1 d µ ( ρ ⊗ σ ) = 1 , (322) i.e., | | | ς | | | b ⊗ 1 = ∥ ς ∥ b ⊗ 1 = 1. Here, we ha v e used the fact that the in tegral in (292) conv erges wrt the pro jective norm ∥ · ∥ b ⊗ 1 to o, and a w ell-known prop ert y of the Bo c hner in tegral. Corollary 4.6. Assume that b oth H and J ar e finite-dimensional: N ≡ dim( H ⊗ J ) < ∞ . Then, for every density op er ator D ∈ D ( H ⊗ J ) , the fol lowing facts ar e e quivalent: (S1) D ∈ D ( H ⊗ J ) f s , i.e., D is finitely sep ar able; (S2) D ∈ D ( H ⊗ J ) ds , i.e., D is discr etely sep ar able; (S3) D ∈ D ( H ⊗ J ) se , i.e., D is sep ar able; (S4) | | | D | | | b ⊗ 1 = 1 ; (S5) ∥ D ∥ b ⊗ 1 = 1 . 66 Ther efor e, if dim( H ⊗ J ) < ∞ , we have: D ( H ⊗ J ) f s = D ( H ⊗ J ) ds = D ( H ⊗ J ) se =  D ∈ D ( H ⊗ J ) : | | | D | | | b ⊗ 1 = 1  =  D ∈ D ( H ⊗ J ) : ∥ D ∥ b ⊗ 1 = 1  . (323) Mor e over, in this c ase, every sep ar able density op er ator ς ∈ D ( H ⊗ J ) se admits a finite c onvex de c omp osition of the form ς = K X k =1 p k ( π k ⊗ ϖ k ) , wher e K ≤ N 2 + 1 , p k > 0 ( P K k =1 p k = 1) , π k ∈ P ( H ) , ϖ k ∈ P ( J ) . (324) Pr o of. Clearly , (S1) = ⇒ (S2) = ⇒ (S3) . Moreov er, by Corollary 4.5, (S3) implies (S4) , and, since, for ev ery D ∈ D ( H ⊗ J ) b , | | | D | | | b ⊗ 1 ≥ ∥ D ∥ b ⊗ 1 ≥ ∥ D ∥ ⊗ 1 = 1, (S4) implies (S5) . Let us pro ve that (S5) = ⇒ (S3) = ⇒ (S1) , and hence the facts (S1) – (S5) are mutually equiv alent. T o prov e that (S5) implies (S3) , recall that, b y Prop osition 3.12, if b oth H and J are finite- dimensional, then ev ery linear op erator in B 1 ( H ) b ⊗ B 1 ( J ) is optimally decomp osable. Hence, b y the equiv alence of conditions (D1) and (D5) in Prop osition 3.18, we hav e: (S5) = ⇒ (S2) = ⇒ (S3) . Let us finally prov e that (S3) implies (S1) and decomp osition (324), simultaneously . In fact, in the case were N ≡ dim( H ⊗ J ) < ∞ , D ( H ⊗ J ) se is a (norm-b ounded and norm-closed, hence) ∥ · ∥ ⊗ 1 -compact con v ex subset of the N 2 -dimensional r e al linear space B ( H ⊗ J ) R = B 1 ( H ⊗ J ) R ; hence, by Carath ´ eo dory’s theorem [50], a separable density op erator ς on H ⊗ J can alw ays b e expressed as a conv ex combination of, at most, N 2 + 1 p oin ts of ext( D ( H ⊗ J ) se ), p oints that, by relation (309) in Corollary 4.2, must b e contained in the set P ( H ⊗ J ) se = θ ( P ( H ) , P ( J )) of all separable pure states. Thus, in particular, (S3) implies (S1) . 5 F urther characterizations of separable states In Subsection 4.3 (recall Corollary 4.5), w e hav e shown that if a bipartite state ς ∈ D ( H ⊗ J ) is separable — i.e., if ς ∈ D ( H ⊗ J ) se ⊂ B 1 ( H ) b ⊗ B 1 ( J ) — then ∥ ς ∥ b ⊗ 1 = 1. W e will no w prov e that the reverse implication holds true, as well, so that the set of all separable states (wrt the bipartition H ⊗ J ) is a ∥ · ∥ b ⊗ 1 -closed conv ex subset of B 1 ( H ) b ⊗ B 1 ( J ), completely characterized as the intersection of D ( H ⊗ J ) with a level set of the pro jectiv e norm (this result is already prov en in the finite-dimensional setting only; recall Corollary 4.6). Let us denote by D ( H ⊗ J ) S the intersection of the set D ( H ⊗ J ) with the unit spher e S  B 1 ( H ) b ⊗ B 1 ( J )  — equiv alen tly , since ∥ D ∥ b ⊗ 1 ≥ ∥ D ∥ ⊗ 1 = 1, for every D ∈ D ( H ⊗ J ) b , with the unit b al l B  B 1 ( H ) b ⊗ B 1 ( J )  : =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) : ∥ C ∥ b ⊗ 1 ≤ 1  — in B 1 ( H ) b ⊗ B 1 ( J ): D ( H ⊗ J ) S : = D ( H ⊗ J ) ∩ S  B 1 ( H ) b ⊗ B 1 ( J )  =  D ∈ D ( H ⊗ J ) b : ∥ D ∥ b ⊗ 1 = 1  =  D ∈ D ( H ⊗ J ) b : ∥ D ∥ b ⊗ 1 ≤ 1  = D ( H ⊗ J ) ∩ B  B 1 ( H ) b ⊗ B 1 ( J )  = : D ( H ⊗ J ) B . (325) The third equalit y in (325) implies that D ( H ⊗ J ) S = D ( H ⊗ J ) B is a c onvex subset (of D ( H ⊗ J ) b and) of D ( H ⊗ J ). In the following, we will use b oth notations — D ( H ⊗ J ) S or D ( H ⊗ J ) B — for indicating the same set; the former notation b eing preferred to emphasize the sharp v alue of the pro jectiv e norm, the latter b eing chosen to stress that w e are dealing with a conv ex set. With these notations, our claim is expressed b y the relation D ( H ⊗ J ) se = D ( H ⊗ J ) S = D ( H ⊗ J ) B . W e will also pro ve that D ( H ⊗ J ) se coincides with the intersection of D ( H ⊗ J ) with the unit 67 sphere S  B 1 ( H ) R b ⊗ B 1 ( J ) R  =  C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R : | | | C | | | b ⊗ 1 = 1  in the Hermitian trace class B 1 ( H ) R b ⊗ B 1 ( J ) R . F or technical reasons, to pro ve these claims it will b e conv enien t to regard the elemen ts of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) as functionals on a suitable unital C ∗ -algebra A ( H ⊗ J ) of b ounded op erators on the Hilb ert space H ⊗ J . This construction inv olv es v arious in termediate steps that will be outlined in Subsection 5.2, after defining the C ∗ -algebra A ( H ⊗ J ). 5.1 The relev an t op erator algebra In the case where dim( H ⊗ J ) = ∞ , w e will denote by A ( H ⊗ J ) the unitization [42] of the (non-unital) C ∗ -algebra C ( H ⊗ J ) of all c omp act op er ators on the Hilbert space H ⊗ J ; namely , w e consider the unital C ∗ -algebra A ( H ⊗ J ) : = C ( H ⊗ J ) + C I , (326) endo wed with the norm ∥ A + z I ∥ : = sup {∥ AB + z B ∥ ⊗ ∞ : B ∈ C ( H ⊗ J ) , ∥ B ∥ ⊗ ∞ = 1 } , that coincides with the restriction to A ( H ⊗ J ) of the standard op erator norm ∥ · ∥ ⊗ ∞ of B ( H ⊗ J ). Therefore, A ( H ⊗ J ) is, in a natural wa y , a (unital) C ∗ -subalgebra of the ambien t C ∗ -algebra B ( H ⊗ J ) (see Example 10.4.8 of [44]). In the case where L = dim( H ⊗ J ) < ∞ , we simply put A ( H ⊗ J ) ≡ C ( H ⊗ J ) = B ( H ⊗ J ) (say , the unital C ∗ -algebra of L × L complex matrices endo wed with the sp ectral norm). W e can endow the dual A ( H ⊗ J ) ∗ of the C ∗ -algebra A ( H ⊗ J ) with its w ∗ -top ology (weak ∗ top ology), i.e., with the initial top ology induced by the family of maps  [ [ K, · ] ] : K ∈ A ( H ⊗ J )  , [ [ K , · ] ] : A ( H ⊗ J ) ∗ → C , (327) where [ [ K, ξ ] ] ≡ ξ ( K ) ∈ C is the pairing b et ween K ∈ A ( H ⊗ J ) and ξ ∈ A ( H ⊗ J ) ∗ ; equiv alently , with the initial top ology induced by the family of semi-norms  | [ [ K, · ] ] | : K ∈ A ( H ⊗ J )  , where | [ [ K, · ] ] | : A ( H ⊗ J ) ∗ → R + . Endo wed with this top ology — that will b e called the A w ∗ -top olo gy — A ( H ⊗ J ) ∗ b ecomes a lo cally con v ex top ological vector space. A ( H ⊗ J ) ∗ con tains B 1 ( H ⊗ J ) as a distinguished linear subspace via the (injective, linear) immersion map f : B 1 ( H ⊗ J ) ∋ A 7→ ξ A ∈ A ( H ⊗ J ) ∗ (328) where the pairing b et ween the functional ξ A ∈ A ( H ⊗ J ) ∗ ( A ∈ B 1 ( H ⊗ J )) and a v ector K of A ( H ⊗ J ) is pro vided b y the trace, i.e., ξ A ( K ) ≡ [ [ K, ξ A ] ] : = tr( K A ). The map f is a linear isometry , b ecause — by the fact that, for every A ∈ B 1 ( H ⊗ J ) and K ∈ A ( H ⊗ J ), | ξ A ( K ) | ≤ ∥ A ∥ ⊗ 1 ∥ K ∥ ⊗ ∞ , and by F act 2.5 (i.e., C ( H ⊗ J ) is a norming subspace of B ( H ⊗ J ) = B 1 ( H ⊗ J ) ∗ ) — w e ha ve: ∥ ξ A ∥ B ≤ ∥ A ∥ ⊗ 1 = sup {| ξ A ( K ) | = | tr( AK ) | : K ∈ C ( H ⊗ J ) , ∥ K ∥ ⊗ ∞ = 1 } ≤ ∥ ξ A ∥ B . (329) Clearly , the rhs inequalit y is due to the fact that the supremum is taken ov er the unit ball of C ( H ⊗ J ) ⊂ A ( H ⊗ J ). Hence, actually , ∥ ξ A ∥ B = ∥ A ∥ ⊗ 1 . R emark 5.1 . F or every A ∈ B 1 ( H ⊗ J ) — taking into accoun t the equalit y in relation (329) and the fact that, b y the same relation, ∥ ξ A ∥ B = ∥ A ∥ ⊗ 1 — w e see that the functional f ( A ) = ξ A ∈ A ( H ⊗ J ) ∗ is a Hahn-Banac h extension of the functional  C ( H ⊗ J ) ∋ K 7→ tr( K A ) ∈ C  ∈ C ( H ⊗ J ) ∗ . The relativ e (or subspace) top ology on B 1 ( H ⊗ J ) ≡ f ( B 1 ( H ⊗ J )) wrt the A w ∗ -top ology — i.e., the initial top ology on B 1 ( H ⊗ J ), via the map f , associated with the A w ∗ -top ology on A ( H ⊗ J ) ∗ — will b e called the extende d w ∗ -top olo gy or, in short, the ew ∗ -top olo gy . Every ew ∗ -op en subset of B 1 ( H ⊗ J ) is of the form f − 1 ( O ), for some A w ∗ -op en subset O of A ( H ⊗ J ) ∗ . Therefore, endo wed with the ew ∗ -top ology , B 1 ( H ⊗ J ) is a top ological space homeomorphic to f ( B 1 ( H ⊗ J )) (equipp ed with its subspace top ology wrt the A w ∗ -top ology of A ( H ⊗ J )). A net { A i } in B 1 ( H ⊗ J ) con v erges to some A ∈ B 1 ( H ⊗ J ) wrt the ew ∗ -top ology iff lim i tr( K A i ) = tr( K A ), for all K ∈ A ( H ⊗ J ). 68 F act 5.1. By a w ell-known prop ert y of the closure of a set wrt the subspace top ology , for every subset X of B 1 ( H ⊗ J ), we ha ve that X ew ∗ = X A w ∗ ∩ B 1 ( H ⊗ J ) , (330) where X ew ∗ ≡ ew ∗ - cl ( X ), X A w ∗ ≡ A w ∗ - cl ( X ) denote the ew ∗ -closure and the A w ∗ -closure of X , resp ectiv ely , and all the obvious iden tifications via the immersion map f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ are understo o d; i.e., relation (330) is a shorthand form of f  X ew ∗  = f ( X ) A w ∗ ∩ f ( B 1 ( H ⊗ J )). Prop osition 5.1. L et X , Y b e nonempty subsets of B 1 ( H ⊗ J ) and A ( H ⊗ J ) ∗ , r esp e ctively. Then, for the ew ∗ -close d c onvex hul l of X and the A w ∗ -close d c onvex hul l of Y , we have that co ew ∗ ( X ) = ew ∗ - cl (co ( X )) , co A w ∗ ( Y ) = A w ∗ - cl (co ( Y )) . (331) Example 5.1. Assume that dim( H ⊗ J ) = ∞ . In this case, the zero v ector in B 1 ( H ⊗ J ) b elongs to the closure w ∗ - cl ( D ( H ⊗ J )) — wrt the w ∗ -top ology of B 1 ( H ⊗ J ) regarded as the Banach space dual of C ( H ⊗ J ) (via the trace functional) — of the conv ex set D ( H ⊗ J ). Indeed, giv en an y orthonormal basis { e k } ∞ k =1 in the Hilb ert space H ⊗ J , and putting D l : = l − 1 P l k =1 | e k ⟩ ⟨ e k | ∈ D ( H ⊗ J ), l ∈ N , for every A ∈ B 1 ( H ⊗ J ) we ha ve that lim l tr( D l A ) = lim l  l − 1 tr  ( P l k =1 | e k ⟩ ⟨ e k | ) A   = 0 , (332) b ecause lim l tr  ( P l k =1 | e k ⟩ ⟨ e k | ) A  = P ∞ k =1 ⟨ e k , A e k ⟩ = tr( A ). Since the trace class B 1 ( H ⊗ J ) is ∥ · ∥ ⊗ ∞ -dense in C ( H ⊗ J ), giv en any K ∈ C ( H ⊗ J ) and A ∈ B 1 ( H ⊗ J ), by the estimate | tr( D l K ) | ≤ | tr( D l ( K − A )) | + | tr( D l A ) | ≤ ∥ D l ( K − A ) ∥ ⊗ 1 + ∥ D l A ∥ ⊗ 1 ≤ ∥ D l ∥ ⊗ 1 ∥ K − A ∥ ⊗ ∞ + ∥ D l ∥ ⊗ ∞ ∥ A ∥ ⊗ 1 = ∥ K − A ∥ ⊗ ∞ + l − 1 ∥ A ∥ ⊗ 1 , (333) it follo ws that relation (332) actually holds for every A ∈ C ( H ⊗ J ). Therefore, w ∗ – lim l D l = 0 ∈ w ∗ - cl ( D ( H ⊗ J )). By this fact, it also follo ws that, for any D ∈ D ( H ⊗ J ) and s ∈ [0 , 1], putting D l ( s ) : = s D + (1 − s ) D l ∈ D ( H ⊗ J ), w e ha ve that w ∗ – lim l D l ( s ) = s D ∈ w ∗ - cl ( D ( H ⊗ J )); hence: co ( D ( H ⊗ J ) ∪ { 0 } ) = { s D : s ∈ [0 , 1] , D ∈ D ( H ⊗ J ) } ⊂ w ∗ - cl ( D ( H ⊗ J )). Moreo ver, b y relation (332), whic h holds true for every A ∈ C ( H ⊗ J ), one can easily c heck that A w ∗ – lim l f ( D l ) = Ξ ∈ A ( H ⊗ J ) ∗ \ f ( B 1 ( H ⊗ J )), where Ξ is the p ositiv e linear functional on A ( H ⊗ J ) determined b y setting Ξ( K ) = 0, for all K ∈ C ( H ⊗ J ), and Ξ( I ) = 1. As a consequence, the sequence { D l } ∞ l =1 ⊂ D ( H ⊗ J ) do es not admit a limit in B 1 ( H ⊗ J ) wrt the ew ∗ -top ology . The following result justifies the term “extended w ∗ -top ology” in tro duced ab o ve. Prop osition 5.2. Supp ose that dim( H ⊗ J ) = ∞ . Then, the ew ∗ -top olo gy is strictly str onger — i.e., strictly finer — than the w ∗ -top olo gy of B 1 ( H ⊗ J ) , r e gar de d as the Banach sp ac e dual of C ( H ⊗ J ) via the tr ac e functional; i.e., the w ∗ -top olo gy is a strict subtop olo gy of the ew ∗ -top olo gy. Mor e over, D ( H ⊗ J ) is ew ∗ -close d, but not w ∗ -close d; f ( D ( H ⊗ J )) is not A w ∗ -close d. In spite of the fact that, when dim( H ⊗ J ) = ∞ , the ew ∗ -top ology is strictly stronger than the w ∗ -top ology of B 1 ( H ⊗ J ), the asso ciated subspace top ologies on D ( H ⊗ J ) do c oincide . Prop osition 5.3. The r elative top olo gies on D ( H ⊗ J ) wrt the w ∗ -top olo gy and the ew ∗ -top olo gy of B 1 ( H ⊗ J ) b oth c oincide with the standar d top olo gy — recall F act 2.7 — of D ( H ⊗ J ) . 69 Pr o of. By Prop osition 5.2, the relativ e ew ∗ -top ology on D ( H ⊗ J ) is stronger than the relative w ∗ - top ology on D ( H ⊗ J ) inherited from B 1 ( H ⊗ J ). The latter, in turn, is stronger than the relativ e top ology induced on D ( H ⊗ J ) b y the weak op erator top ology of B ( H ⊗ J ); i.e., the standard top ology . Con versely , the relativ e top ology on D ( H ⊗ J ) wrt the trace norm (the Sc hatten 1-norm ∥ · ∥ ⊗ 1 ) top ology — i.e., again, the standard top ology of D ( H ⊗ J ) — is stronger than the relative ew ∗ -top ology , whic h is stronger than the relative w ∗ -top ology . Hence, all these top ologies must coincide with the standard top ology of D ( H ⊗ J ). By Prop osition 5.3, w e get to an imp ortan t p oin t, i.e., Theorem 5.1 b elo w. Lemma 5.1. With the obvious identific ations co ew ∗ ( θ ( P ( H ) , P ( J ))) ≡ f  co ew ∗ ( θ ( P ( H ) , P ( J )))  , θ ( P ( H ) , P ( J )) ≡ f ( θ ( P ( H ) , P ( J ))) and D ( H ⊗ J ) ≡ f ( D ( H ⊗ J )) , the fol lowing r elation holds: co ew ∗ ( θ ( P ( H ) , P ( J ))) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ) . (334) Pr o of. By F act 5.1 and relations (331), and with suitable iden tifications via the immersion map f , w e ha ve: co ew ∗ ( θ ( P ( H ) , P ( J ))) = ew ∗ - cl  co ( θ ( P ( H ) , P ( J )))  = A w ∗ - cl  co ( θ ( P ( H ) , P ( J )))  ∩ B 1 ( H ⊗ J ) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ B 1 ( H ⊗ J ) . (335) W e need to pro ve that co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ B 1 ( H ⊗ J ) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ). In fact, let { D i } ⊂ co ( θ ( P ( H ) , P ( J ))) ⊂ D ( H ⊗ J ) se b e a net of (finitely) separable states conv erging — wrt the A w ∗ -top ology — to A ≡ [ [ · , ξ A ] ] ∈ B 1 ( H ⊗ J ) ≡ f ( B 1 ( H ⊗ J )) ⊂ A ( H ⊗ J ) ∗ (with obvious iden tifications via the immersion map f ); i.e., lim i tr( K D i ) = lim i [ [ K, ξ D i ] ] = [ [ K, ξ A ] ] = tr( K A ), for all K ∈ A ( H ⊗ J ). Now, by imposing this condition for every K ∈ A ( H ⊗ J ), with K ≥ 0 (in particular, for every rank-one pro jection on H ⊗ J ), we conclude that A ≥ 0, and, moreo ver, with K = I , w e see that tr( A ) = 1. Thus, A = A w ∗ – lim i D i ∈ D ( H ⊗ J ), and the pro of is complete. Theorem 5.1. The c onvex set of al l sep ar able states wrt the bip artition H ⊗ J c an b e describ e d as a close d c onvex hul l wrt the ew ∗ -top olo gy of B 1 ( H ⊗ J ) , i.e., D ( H ⊗ J ) se = co ew ∗ ( θ ( P ( H ) , P ( J ))) . (336) Pr o of. By relation (279), w e ha ve: D ( H ⊗ J ) se = co ( θ ( D ( H ) , D ( J ))) = co ( θ ( P ( H ) , P ( J ))). Let us prov e that co ( θ ( P ( H ) , P ( J ))) = co ew ∗ ( θ ( P ( H ) , P ( J ))). In fact, by Prop osition 5.3, the relativ e top ology on D ( H ⊗ J ) wrt the ew ∗ -top ology of B 1 ( H ⊗ J ) coincides with the standard top ology of D ( H ⊗ J ), and hence with the relative top ology on D ( H ⊗ J ) wrt the trace norm top ology of B 1 ( H ⊗ J ). Therefore, since co ew ∗ ( θ ( P ( H ) , P ( J ))) = ew ∗ - cl  co ( θ ( P ( H ) , P ( J )))  (recall the first of relations (331)), b y a w ell-known prop ert y of the closure of a set wrt the subspace top ology , w e ha v e that co ew ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ) = co ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ) = co ( θ ( P ( H ) , P ( J ))). It just remains to notice that, b y Lemma 5.1, co ew ∗ ( θ ( P ( H ) , P ( J ))) ⊂ D ( H ⊗ J ). 5.2 Our strategy F or the reader’s con v enience, we no w briefly outline our plan for deriving, in the rest of this section, a complete c haracterization of the set D ( H ⊗ J ) se in terms of the pro jectiv e norms. As a b ypro duct, w e will also obtain v arious interesting c haracterizations of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ). 70 1. As a starting p oint, we kno w that D ( H ⊗ J ) se ⊂ D ( H ⊗ J ) S = D ( H ⊗ J ) B (see Corollary 4.5 and relation (325)), and we wan t to prov e, among other facts, that the rev erse inclusion relation holds to o. T o this end, we need to establish some further technical results. 2. W e then in tro duce t wo mutually related norms ∥ · ∥ γ and ∥ · ∥ Γ — respectively , on the spaces B 1 ( H ) b ⊗ B 1 ( J ) and B ( H ⊗ J ) — and w e sho w that these norms are dominated, resp ectiv ely , b y the pro jective norm ∥ · ∥ b ⊗ 1 (Prop osition 5.4 b elo w) and by the standard operator norm ∥ · ∥ ⊗ ∞ (Prop osition 5.5). W e next obtain more precise characterizations of these norms. 3. It is immediate to realize that, in the finite-dimensional setting, ∥ · ∥ γ is pr e cisely (another w ay of defining) the pro jective norm ∥ · ∥ b ⊗ 1 of B 1 ( H ) b ⊗ B 1 ( J ); see Remark 5.2 b elo w. With no assumption on the dimension of the Hilb ert spaces H and J , the norm ∥ · ∥ Γ can b e regarded as a natural extension to B ( H ⊗ J ) of the inje ctive norm [17–20] of the algebraic tensor pro duct C ( H ) ˘ ⊗ C ( J ), or, equiv alently , of the injective norm of B ( H ) ˘ ⊗ B ( J ); see Remark 5.4. Exploiting standard dualit y relations b etw een the injective and the pro jective norms, w e then prov e that the pro jective tensor pro duct B 1 ( H ) b ⊗ B 1 ( J ) is isomorphic to the dual of the Banac h space completion of the algebraic tensor pro duct C ( H ) ˘ ⊗ C ( J ) wrt to the injectiv e norm ∥ · ∥ Γ (i.e., the injective tensor pro duct C ( H ) ˇ ⊗ C ( J ) of C ( H ) and C ( J )). This fact en tails that ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 in the infinite-dimensional setting to o; see Theorem 5.2 below. Therefore, the norm ∥ · ∥ γ ma y be regarded as a ‘useful reform ulation’ of the pro jective norm. 4. As previously observed, the restriction of the norm ∥ · ∥ Γ to the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) coincides with the injective norm ∥ · ∥ ˇ ⊗ ∞ of B ( H ) ˘ ⊗ B ( J ) (Remark 5.4). Then, further exploring the duality relations inv olving the cross trace class, it turns out that the injectiv e tensor product B ( H ) ˇ ⊗ B ( J ) — namely , the Banac h space completion of the normed space ( B ( H ) ˘ ⊗ B ( J ) , ∥ · ∥ Γ = ∥ · ∥ ˇ ⊗ ∞ ) — is (isomorphic to) a closed subspace of the Banach space dual  B 1 ( H ) b ⊗ B 1 ( J )  ∗ of B 1 ( H ) b ⊗ B 1 ( J ); see Theorem 5.3 b elow. In particular, in the case where at least one of the Hilb ert spaces H and J is finite-dimensional, the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 are equiv alent on B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) (a set equality being understo od); moreo ver, in this case, the linear space B ( H ⊗ J ), endo w ed with the norm ∥ · ∥ Γ , is a Banach space — which is isomorphic to the injective tensor pro duct B ( H ) ˇ ⊗ B ( J ), and a r enorming [64] of the standard Banac h space ( B ( H ⊗ J ) , ∥ · ∥ ⊗ ∞ ) of b ounded op erators — and can b e iden tified with the Banac h space dual of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ). 5. Let us no w restrict our fo cus on the cross trace class regarded as a set of b ounded functionals. The tec hnical reason wh y this picture is of in terest to us is related to the p eculiar prop erties of a weak ∗ algebra [46]. As previously noted, the elements of B 1 ( H ) b ⊗ B 1 ( J ) can b e identified with the b ounded functionals on the injective tensor product C ( H ) ˇ ⊗ C ( J ). This fact is crucial, e.g., for proving the coincidence of the norms ∥ · ∥ γ and ∥ · ∥ b ⊗ 1 . But, according to Remark 5.6 and Prop osition 5.6 b elo w, if dim( H ⊗ J ) = ∞ , the mentioned identification is not suitable to reach our final target of proving that D ( H ⊗ J ) se = D ( H ⊗ J ) S . 6. Recall that, in Subsection 5.1, we hav e regarded the elements of the trace class B 1 ( H ⊗ J ) — hence, in particular, of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) — as b ounded functionals on the C ∗ -algebra A ( H ⊗ J ), via the linear isometry f : B 1 ( H ⊗ J ) ∋ A 7→ ξ A ∈ A ( H ⊗ J ) ∗ . In addition, by means of the norms ∥ · ∥ γ and ∥ · ∥ Γ (and of the identification of the norms ∥ · ∥ γ and ∥ · ∥ b ⊗ 1 ), the elemen ts of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) can also be regarded as sp e cial b ounded functionals on A ( H ⊗ J ). In fact, the Banach space B 1 ( H ) b ⊗ B 1 ( J ) can b e iden tified with (i.e., is isomorphic to) a linear subspace A ( H ⊗ J ) b of the dual A ( H ⊗ J ) ∗ of A ( H ⊗ J ), provided that this subspace is equipp ed with a suitable norm ∥ · ∥ 𭟋 ; i.e., there 71 is an isomorphism of Banach spaces b f : B 1 ( H ) b ⊗ B 1 ( J ) → A ( H ⊗ J ) b ⊂ A ( H ⊗ J ) ∗ , that is a (domain and co domain) restriction of the immersion map f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ . Moreo ver, it turns out that the Banac h space  A ( H ⊗ J ) b , ∥ · ∥ 𭟋  ∼ = B 1 ( H ) b ⊗ B 1 ( J ) is a closed subspace of the dual A ( H ⊗ J ) ∗ Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  ∗ of the normed space A ( H ⊗ J ) Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  ; in particular, the norm ∥ · ∥ 𭟋 is the restriction to the subspace A ( H ⊗ J ) b of the dual norm ∥ · ∥ ∗ Γ of ∥ · ∥ Γ , the latter being regarded as a norm on the linear space A ( H ⊗ J ) (see Prop osition 5.7 b elo w). Summarizing, we ha ve two further natural c haracterizations of the elements of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) as b ounded functionals. 7. A t this stage, one ma y ask whether w e should consider the cr oss states D ( H ⊗ J ) b — i.e., our main ob jects of in terest — as (p ositiv e) functionals on the C ∗ -algebra A ( H ⊗ J ) or, rather, on the closely related normed space A ( H ⊗ J ) Γ . In this regard, a remark able fact is the follo wing (see Remark 5.8 below): A ( H ⊗ J ) ∗ Γ is, in a natural wa y , a linear subspace of A ( H ⊗ J ) ∗ , and, moreov er, the σ  A ( H ⊗ J ) ∗ Γ , A ( H ⊗ J ) Γ  -top ology coincides with the subspace topology on A ( H ⊗ J ) ∗ Γ ⊂ A ( H ⊗ J ) ∗ wrt the σ ( A ( H ⊗ J ) ∗ , A ( H ⊗ J ))-top ology (the A w ∗ -top ology). Therefore, for our main purposes, we may think of the elemen ts of D ( H ⊗ J ) b as functionals in A ( H ⊗ J ) b (i.e., as elements of the set b f ( D ( H ⊗ J ) b ) ⊂ A ( H ⊗ J ) b ⊂ A ( H ⊗ J ) ∗ Γ ) or, e quivalently , as functionals in A ( H ⊗ J ) ∗ (i.e., as elements of f ( D ( H ⊗ J )) ⊂ A ( H ⊗ J ) ∗ ), b ecause, ev entually , the A w ∗ -top ology , and the asso ciated subspace topology on B 1 ( H ⊗ J ) ≡ f ( B 1 ( H ⊗ J )) (the ew ∗ -top ology), will turn out to b e the relev ant mathematical to ol. 8. By the previous reasoning, the conv ex set D ( H ⊗ J ) se ⊂ B 1 ( H ) b ⊗ B 1 ( J ) of all separable states wrt the bipartition H ⊗ J can ev en tually b e iden tified with a con vex subset of A ( H ⊗ J ) ∗ . W e can then use the topological prop erties of the space A ( H ⊗ J ) ∗ as the dual of the C ∗ -algebra A ( H ⊗ J ) to prov e that, denoting b y co ew ∗ ( θ ( P ( H ) , P ( J ))) the ew ∗ -closed con vex hull of θ ( P ( H ) , P ( J )) (equiv alently , by the first of relations (331), the closure of co ( θ ( P ( H ) , P ( J ))) wrt the ew ∗ -top ology), w e hav e: D ( H ⊗ J ) S = D ( H ⊗ J ) B ⊂ co ew ∗ ( θ ( P ( H ) , P ( J ))), and hence — recalling also that D ( H ⊗ J ) se = co ew ∗ ( θ ( P ( H ) , P ( J ))) (Theorem 5.1) — actually , D ( H ⊗ J ) se = D ( H ⊗ J ) S (see Theorem 5.4 b elo w). As a byproduct, it turns out that D ( H ⊗ J ) se can also b e describ ed as the in tersection of the set D ( H ⊗ J ) b of all cross states with the unit sphere in the Hermitian trace class B 1 ( H ) R b ⊗ B 1 ( J ) R ; see Corollary 5.1 b elo w. It is worth stressing, ho wev er, that regarding the elements of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) as b ounded functionals on the normed space A ( H ⊗ J ) Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  — or, also, on the normed space B ( H ⊗ J ) Γ ≡  B ( H ⊗ J ) , ∥ · ∥ Γ  (see Remark 5.7 b elow) — will b e fundamen tal, in Section 6, for introducing the so-called entanglement function . 5.3 Tw o useful norms and duality relations for the cross trace class With ev ery b ounded op erator L ∈ B ( H ⊗ J ), we can associate a linear functional Γ L on the Banac h space B 1 ( H ) b ⊗ B 1 ( J ), defined by Γ L : B 1 ( H ) b ⊗ B 1 ( J ) ∋ C 7→ tr( C L ) ∈ C . (337) The functional Γ L is b ounded, b ecause | tr( C L ) | ≤ ∥ C ∥ ⊗ 1 ∥ L ∥ ⊗ ∞ ≤ ∥ C ∥ b ⊗ 1 ∥ L ∥ ⊗ ∞ , and w e ha ve: ∥ Γ L ∥ B : =  | tr( C L ) | : C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ∥ C ∥ b ⊗ 1 = 1  ≤ ∥ L ∥ ⊗ ∞ ; (338) in particular, Γ L ( S ⊗ T ) = tr(( S ⊗ T ) L ) = γ L ( S, T ) , S ∈ B 1 ( H ) , T ∈ B 1 ( J ) , (339) 72 where γ L : B 1 ( H ) × B 1 ( J ) → C , γ L ∈ F 1 ( H , J ), is the canonical bilinearization of the b ounded linear functional Γ L ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ (Definition 3.11, with S = C ). Note that, b y Theorem 3.5, the norm ∥ Γ L ∥ B of the linear functional Γ L : B 1 ( H ) b ⊗ B 1 ( J ) → C satisfies the follo wing imp ortan t relation: ∥ Γ L ∥ B = ∥ γ L ∥ (1) = sup {| tr(( S ⊗ T ) L ) | : S ⊗ T ∈ θ ( B 1 ( H ) , B 1 ( J )) , ∥ S ∥ 1 = ∥ T ∥ 1 = 1 } . (340) F or every cross trace class op erator C ∈ B 1 ( H ) b ⊗ B 1 ( J ), w e can now in tro duce the quan tity ∥ C ∥ γ : = sup {| Γ K ( C ) | = | tr( C K ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B = 1 } . (341) Prop osition 5.4. F ormula (341) defines a norm ∥ · ∥ γ on the cr oss tr ac e class B 1 ( H ) b ⊗ B 1 ( J ) , which dominates (the r estriction of ) the tr ac e norm and is dominate d by the pr oje ctive norm; i.e., for every C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ γ ≤ ∥ C ∥ b ⊗ 1 = sup  | Γ( C ) | : Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , ∥ Γ ∥ B = 1  . (342) Pr o of. Let us pro ve that formula (341) defines a norm. Clearly , ∥ C ∥ γ ≥ 0 and, for every c ∈ C , ∥ c C ∥ γ = | c | ∥ C ∥ γ . Moreov er, ∥ C ∥ γ = 0 ⇐ ⇒ C = 0, b ecause, if ∥ C ∥ γ = 0, then, in particular, 0 = | tr( C | a ⟩ ⟨ c | ) | = |⟨ c , C a ⟩| , for all a , c ∈ H ⊗ J ( | a ⟩ ⟨ c | ∈ C ( H ⊗ J )); hence, C = 0. It remains to pro ve the triangle inequality; in fact, for all C 1 , C 2 ∈ B 1 ( H ) b ⊗ B 1 ( J ), w e hav e: ∥ C 1 + C 2 ∥ γ = sup {| Γ K ( C 1 + C 2 ) | = | tr(( C 1 + C 2 ) K ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B = 1 } ≤ sup {| Γ K ( C 1 ) | + | Γ K ( C 2 ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B = 1 } ≤ sup {| Γ K ( C 1 ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B = 1 } + sup {| Γ K ( C 2 ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B = 1 } = ∥ C 1 ∥ γ + ∥ C 2 ∥ γ . (343) Let us now sho w that ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ γ . Indeed, by F act 2.5 and, next, b y the inequality ∥ Γ K ∥ B ≤ ∥ K ∥ ⊗ ∞ (for any K ∈ C ( H ⊗ J )), w e hav e that ∥ C ∥ ⊗ 1 = sup {| tr( C K ) | : K ∈ C ( H ⊗ J ) , ∥ K ∥ ⊗ ∞ = 1 } = sup {| Γ K ( C ) | = | tr( C K ) | : K ∈ C ( H ⊗ J ) , ∥ K ∥ ⊗ ∞ ≤ 1 } ≤ sup {| Γ K ( C ) | : K ∈ C ( H ⊗ J ) , ∥ Γ K ∥ B ≤ 1 } ≤ sup {| Γ K ( C ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B ≤ 1 } = ∥ C ∥ γ . (344) Finally , recalling relation (249), it is clear that ∥ C ∥ γ ≤ ∥ C ∥ b ⊗ 1 , as w ell, and relation (342) holds. Note that, for every b ounded op erator L ∈ B ( H ⊗ J ), w e can put ∥ L ∥ Γ : = ∥ Γ L ∥ B ≤ ∥ L ∥ ⊗ ∞ . (345) Prop osition 5.5. The map ∥ · ∥ Γ : B ( H ⊗ J ) ∋ L 7→ ∥ Γ L ∥ B ∈ R + is a norm — dominate d by the op er ator norm ∥ · ∥ ⊗ ∞ — and, for every L ∈ B ( H ⊗ J ) , we have that ∥ L ∥ Γ : = sup  | tr( C L ) | : C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ∥ C ∥ b ⊗ 1 = 1  = sup {| tr(( S ⊗ T ) L ) | : S ⊗ T ∈ θ ( B 1 ( H ) , B 1 ( J )) , ∥ S ∥ 1 = ∥ T ∥ 1 = 1 } = sup {|⟨ ϕ ⊗ ψ , L ( η ⊗ χ ) ⟩ H⊗J | : ϕ, η ∈ H , ψ , χ ∈ J , ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } = ∥ L ∗ ∥ Γ . (346) 73 Mor e over, for every A ∈ B ( H ) and B ∈ B ( J ) , we have that ∥ A ⊗ B ∥ Γ = ∥ A ∥ ∞ ∥ B ∥ ∞ = ∥ A ⊗ B ∥ ⊗ ∞ , and, for every p air U , V of unitary op er ators on H and J , r esp e ctively, ∥ L ∥ Γ = ∥ ( U ⊗ V ) L ( U ⊗ V ) ∗ ∥ Γ . (347) Final ly, the algebr aic tensor pr o duct F ( H ) ˘ ⊗ F ( J ) is ∥ · ∥ Γ -dense in C ( H ) ˘ ⊗ C ( J ) . Pr o of. F rom definition (345), it is clear that ∥ · ∥ Γ is a semi-norm. Moreo ver, since the linear mapping B ( H ⊗ J ) ∋ L 7→ Γ L ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ (348) is injectiv e, this semi-norm is also p oint-separating; i.e., actually , a norm. F or the c haracterization of ∥ L ∥ Γ on the second line of (346), recall relation (340). Let us next derive the characterization of ∥ L ∥ Γ on the third line of (346). F or an y S ∈ S ( B 1 ( H )) and T ∈ S ( B 1 ( J )), by the singular v alue decomp osition of a trace class op erator (Remark 2.1), we hav e that S = ∥ · ∥ 1 – P k s k | η k ⟩ ⟨ ϕ k | and T = ∥ · ∥ 1 – P l t l | χ l ⟩ ⟨ ψ l | , where s k , t l > 0 ( P k s k = ∥ S ∥ 1 = 1 = ∥ T ∥ 1 = P l t l ), and { η k } , { ϕ k } ⊂ H , { χ l } , { ψ l } ⊂ J are orthonormal systems. Hence, for ev ery L ∈ B ( H ⊗ J ), we obtain the estimate | tr(( S ⊗ T ) L ) | =   P kl s k t l ⟨ ϕ k ⊗ ψ l , L ( η k ⊗ χ l ) ⟩ H⊗J   ≤ P kl s k t l |⟨ ϕ k ⊗ ψ l , L ( η k ⊗ χ l ) ⟩ H⊗J | ≤ sup kl |⟨ ϕ k ⊗ ψ l , L ( η k ⊗ χ l ) ⟩ H⊗J | ≤ sup {|⟨ ϕ ⊗ ψ , L ( η ⊗ χ ) ⟩ H⊗J | : ϕ, η ∈ H , ψ , χ ∈ J , ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } , ∀ S ∈ S ( B 1 ( H )) , ∀ T ∈ S ( B 1 ( J )) . (349) Moreo ver, recalling relation (340), we ha ve that ∥ L ∥ Γ = ∥ Γ L ∥ B = sup {| tr(( S ⊗ T ) L ) | : S ⊗ T ∈ θ ( B 1 ( H ) , B 1 ( J )) , ∥ S ∥ 1 = ∥ T ∥ 1 = 1 } ≥ sup {|⟨ ϕ ⊗ ψ , L ( η ⊗ χ ) ⟩ H⊗J | : ϕ, η ∈ H , ψ , χ ∈ J , ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } ≥ sup {| tr(( S ⊗ T ) L ) | : S ⊗ T ∈ θ ( S ( B 1 ( H )) , S ( B 1 ( J ))) } , (350) where the first inequalit y holds by set restriction to the rank-one partial isometries of the form | η ⟩ ⟨ ϕ | ⊗ | χ ⟩ ⟨ ψ | = | η ⊗ χ ⟩ ⟨ ϕ ⊗ ψ | , whereas the second one follo ws immediately from the estimate (349). But the supremum on the second line of (350) and the supremum on the fourth line are actually tak en o ver the same set, so that relation (346) holds true. By an y of the v arious equiv alent expressions in (346) of ∥ L ∥ Γ , it is also clear that ∥ L ∥ Γ = ∥ L ∗ ∥ Γ , and by the final expression relation (347) follows immediately . Therefore, for ev ery A ∈ B ( H ) and B ∈ B ( J ), we ha v e: ∥ A ⊗ B ∥ Γ = sup {|⟨ ϕ ⊗ ψ , ( A ⊗ B )( η ⊗ χ ) ⟩ H⊗J | : ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } = sup {|⟨ ϕ, Aη ⟩ H | |⟨ ψ , B χ ⟩ J | : ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } = sup {|⟨ ϕ, Aη ⟩ H | : ∥ ϕ ∥ = ∥ η ∥ = 1 } sup {|⟨ ψ , B χ ⟩ J | : ∥ ψ ∥ = ∥ χ ∥ = 1 } = ∥ A ∥ ∞ ∥ B ∥ ∞ , (351) where the last equality follo ws by F act 2.1. Finally , let { A k } K k =1 , { E k } K k =1 ⊂ B ( H ), { B k } K k =1 , { F k } K k =1 ⊂ B ( J ) ( K ∈ N ) b e K -tuples of b ounded op erators. Note that ∥ P k A k ⊗ B k − P k E k ⊗ F k ∥ Γ ≤ P k ∥ A k ⊗ B k − E k ⊗ F k ∥ Γ ≤ P k ( ∥ ( A k − E k ) ⊗ B k ∥ Γ + ∥ E k ⊗ ( B k − F k ) ∥ Γ ) = P k ( ∥ A k − E k ∥ ∞ ∥ B k ∥ ∞ + ∥ E k ∥ ∞ ∥ B k − F k ∥ ∞ ), where for the last equality w e ha ve used relation (351). By this estimate, it is clear that — since the linear subspaces of finite rank op erators F ( H ), F ( J ) are ∥ · ∥ ∞ -dense in C ( H ) and C ( J ), resp ectiv ely — the algebraic tensor pro duct F ( H ) ˘ ⊗ F ( J ) is ∥ · ∥ Γ -dense in C ( H ) ˘ ⊗ C ( J ). 74 R emark 5.2 . In the case where dim( H ⊗ J ) < ∞ , we hav e that the finite-dimensional linear space A ( H ⊗ J ) = C ( H ⊗ J ) = B ( H ⊗ J ) — once endow ed with the appropriate norm, that is, B ( H ⊗ J ) ∋ L 7→ ∥ L ∥ Γ ≡ ∥ Γ L ∥ B — can be iden tified with the Banac h space dual  B 1 ( H ) b ⊗ B 1 ( J )  ∗ of the pro jectiv e trace class (recall (338)), and hence, by definition (341) and b y the final equation in (342), w e see that ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 . W e will so on prov e that the equalit y of the t wo norms actually holds in the infinite-dimensional setting to o (see the first assertion of Theorem 5.2 b elo w). R emark 5.3 . Identifying B 1 ( H ), B 1 ( J ) with the Banac h space duals of C ( H ) and C ( J ), respectively , from the expression of the norm ∥ · ∥ Γ on the second line of (346) it is clear that this norm can b e regarded as a natural extension to B ( H ⊗ J ) of the inje ctive norm — see, e.g., Chapter I of [17], Chapter 3 of [18], or Chapter 7 of [20] — defined on the algebraic tensor pro duct C ( H ) ˘ ⊗ C ( J ). R emark 5.4 . The restriction of the norm ∥ · ∥ Γ : B ( H ⊗ J ) → R + to the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) of the Banach spaces B ( H ) and B ( J ) coincides with the injective norm ∥ · ∥ ˇ ⊗ ∞ of B ( H ) ˘ ⊗ B ( J ). In fact, as noted in Remark 3.17, for ev ery P j A j ⊗ B j ∈ B ( H ) ˘ ⊗ B ( J ), w e ha ve that   P j A j ⊗ B j   ˇ ⊗ ∞ : = sup    P j s ( A j ) t ( B j )   : s ∈ B ( H ) ∗ , t ∈ B ( J ) ∗ , ∥ s ∥ B = ∥ t ∥ B = 1  = sup    P j tr( S A j ) tr( T B j )   : S ∈ B 1 ( H ) , T ∈ B 1 ( J ) , ∥ S ∥ 1 = ∥ T ∥ 1 = 1  =   P j A j ⊗ B j   Γ , (352) where we ha ve used the fact that the unit balls B ( B 1 ( H )) and B ( B 1 ( J )) — regarded as subsets of B 1 ( H ) ∗∗ and B 1 ( J ) ∗∗ — are norming sets for B 1 ( H ) ∗ = B ( H ) and B 1 ( J ) ∗ = B ( J ), resp ectiv ely (see Subsection 4.1 in Chapter I of [17], or formula (3.3) in Section 3.1 of [18]), and next the c haracterization of   P j A j ⊗ B j   Γ as on the second line of (346). Therefore, the Banac h space completion of B ( H ) ˘ ⊗ B ( J ) wrt the norm ∥ · ∥ Γ is precisely the inje ctive tensor pr o duct [17–20] B ( H ) ˇ ⊗ B ( J ). Since the norm ∥ · ∥ Γ is ma jorized b y the standard op erator norm ∥ · ∥ ⊗ ∞ (recall (345)), it turns out that B ( H ) ⊗B ( J ) ⊂ B ( H ) ˇ ⊗ B ( J ), where B ( H ) ⊗ B ( J ) is the natural (or spatial) tensor pro duct of B ( H ) and B ( J ), i.e., the Banach space completion of the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) wrt the norm ∥ · ∥ ⊗ ∞ of B ( H ⊗ J ), and B ( H ) ˘ ⊗ B ( J ) = B ( H ) ⊗ B ( J ) = B ( H ⊗ J ) iff min { dim( H ) , dim( J ) } < ∞ (recall Remark 3.5, and references therein). Also note that the injective norm of C ( H ) ˘ ⊗ C ( J ) coincides with the restriction of the injective norm ∥ · ∥ ˇ ⊗ ∞ of B ( H ) ˘ ⊗ B ( J ) and (thus) of the norm ∥ · ∥ Γ . This fact can b e verified by comparing relation (352) with the definition of the injective norm of C ( H ) ˘ ⊗ C ( J ) (see Chapter 3 of [18], or Chapter 7 of [20]), where the duals of C ( H ), C ( J ) are identified via the trace with the Banac h spaces B 1 ( H ) and B 1 ( J ), resp ectiv ely . Alternatively , one can use the fact that the injectiv e tensor pro duct C ( H ) ˇ ⊗ C ( J ) is isomorphic to a closed subspace of B ( H ) ˇ ⊗ B ( J ) — b y virtue of Prop osition 3.2 of [18] (as observ ed in the discussion follo wing this result therein), or by Theorem 7.19 of [20] — b ecause C ( H ), C ( J ) are (closed) subspaces of B ( H ) and B ( J ), resp ectiv ely , where the isomorphism is obtained by con tinuously extending the natural immersion of C ( H ) ˘ ⊗ C ( J ), endow ed with its natural injective norm, into B ( H ) ˇ ⊗ B ( J ). In the light of the preceding tw o remarks, with a slight abuse, we will call the norm ∥ · ∥ Γ the inje ctive norm of B ( H ⊗ J ). By combining the previous results with fundamental dualit y relations b et ween the pr oje ctive and the inje ctive norms [17, 18], we obtain the follo wing imp ortan t fact: Theorem 5.2. The norm ∥ · ∥ γ on B 1 ( H ) b ⊗ B 1 ( J ) , define d by (341) , c oincides with the pr oje ctive 75 norm ∥ · ∥ b ⊗ 1 , and, in fact, for every C ∈ B 1 ( H ) b ⊗ B 1 ( J ) we have: ∥ C ∥ γ : = sup {| tr( C K ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = sup {| tr( C K ) | : K ∈ C ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = sup {| tr( C K ) | : K ∈ C ( H ) ˘ ⊗ C ( J ) , ∥ K ∥ Γ = 1 } = sup {| tr( C F ) | : F ∈ F ( H ) ˘ ⊗ F ( J ) , ∥ F ∥ Γ = 1 } = sup {| tr( C L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } = sup  | Γ( C ) | : Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ , ∥ Γ ∥ B = 1  = ∥ C ∥ b ⊗ 1 . (353) Mor e over, the inje ctive norm of C ( H ) ˘ ⊗ C ( J ) c oincides with the (r estriction of the) norm ∥ · ∥ Γ define d by (345) , and the isomorphism of Banach sp ac es B 1 ( H ) b ⊗ B 1 ( J ) ∼ =  C ( H ) Γ ⊗ C ( J )  ∗ (354) holds, wher e C ( H ) Γ ⊗ C ( J ) ≡ C ( H ) ˇ ⊗ C ( J ) is the inje ctive tensor pr o duct of the Banach sp ac es C ( H ) and C ( J ) , i.e., the Banach sp ac e c ompletion of the algebr aic tensor pr o duct C ( H ) ˘ ⊗ C ( J ) wrt (the r estriction of ) the norm ∥ · ∥ Γ . This isomorphism is implemente d by the mapping B 1 ( H ) b ⊗ B 1 ( J ) ∋ C 7→ κ C ∈  C ( H ) Γ ⊗ C ( J )  ∗ , (355) wher e κ C : C ( H ) Γ ⊗ C ( J ) → C is a b ounde d line ar functional determine d by κ C ( K ) = tr( C K ) , for every K ∈ C ( H ) ˘ ⊗ C ( J ) . Pr o of. The trace classes B 1 ( H ), B 1 ( J ) can b e iden tified — via the trace functional — with the Banac h space duals of C ( H ) and C ( J ), resp ectively; moreov er, the Banac h spaces B 1 ( H ) and B 1 ( J ) ha ve b oth the approximation prop ert y and the Radon-Nik o d´ ym prop ert y (recall F act 2.12). Therefore, by a classical result — see, e.g., the theorem in Subsection 16.6, Chapter I I of [17], or Theorem 5.33 of [18] — the cross trace class B 1 ( H ) b ⊗ B 1 ( J ), regarded as a pro jectiv e tensor pro duct, is isomorphic to the Banac h space dual of the injectiv e tensor pro duct C ( H ) ˇ ⊗ C ( J ) (the Banac h space completion of the algebraic tensor pro duct C ( H ) ˘ ⊗ C ( J ) wrt the injectiv e norm). Note that, b y Remark 5.4, the injective norm of C ( H ) ˘ ⊗ C ( J ) coincides with the restriction of both the injective norm ∥ · ∥ ˇ ⊗ ∞ of B ( H ) ˘ ⊗ B ( J ) and the norm ∥ · ∥ Γ , i.e., the so-called injective norm of B ( H ⊗ J ). The men tioned isomorphism of Banach spaces is induced b y the natural pairing — see Subsection 16.6, Chapter I I of [17] — tr( S A )tr( T B ) = tr(( S ⊗ T )( A ⊗ B )) b et w een the elemen tary tensors S ⊗ T and A ⊗ B , with S ∈ B 1 ( H ) = C ( H ) ∗ , S ∈ B 1 ( J ) = C ( J ) ∗ , A ∈ C ( H ) and B ∈ C ( J ); i.e., by the mapping  B 1 ( H ) ˘ ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1  ∋ P k S k ⊗ T k 7→ P k tr(( S k ⊗ T k )( · )) ∈  C ( H ) ˘ ⊗ C ( J ) , ∥ · ∥ Γ  ∗ , (356) where, with a slight abuse of notation, ∥ · ∥ Γ is the restriction to the algebraic tensor pro duct C ( H ) ˘ ⊗ C ( J ) of the injective norm of B ( H ⊗ J ) (i.e., the injective norm of C ( H ) ˘ ⊗ C ( J )) and ( C ( H ) ˘ ⊗ C ( J ) , ∥ · ∥ Γ ) ∗ =  C ( H ) Γ ⊗ C ( J )  ∗ ≡  C ( H ) ˇ ⊗ C ( J )  ∗ . (357) Therefore, denoting b y κ C the b ounded linear functional on C ( H ) Γ ⊗ C ( J ), corresp onding to the cross trace class op erator C via the isomorphism (355), we ha ve that κ C ( K ) = tr( C K ) , ∀ C ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) , ∀ K ∈ C ( H ) ˘ ⊗ C ( J ) . (358) 76 Let us pro ve that this relation extends to every cross trace class op erator C ∈ B 1 ( H ) b ⊗ B 1 ( J ). Indeed, if { C n } n ∈ N ⊂ B 1 ( H ) ˘ ⊗ B 1 ( J ) is an y ∥ · ∥ b ⊗ 1 -con vergen t sequence — ∥ · ∥ b ⊗ 1 – lim n C n = C , for some C ∈ B 1 ( H ) b ⊗ B 1 ( J ) — then κ C ( K ) = lim n κ C n ( K ) = lim n tr( C n K ) = tr( C K ) , ∀ K ∈ C ( H ) ˘ ⊗ C ( J ) . (359) Here, the first equalit y holds b ecause of the isomorphism (355), the second one by relation (358), while for the last equality w e ha ve: lim n tr( C n K ) = tr  ∥ · ∥ ⊗ 1 – lim n C n K  = tr  ∥ · ∥ b ⊗ 1 – lim n C n K  = tr( C K ) (the trace norm ∥ · ∥ ⊗ 1 b eing ma jorized b y the pro jective norm ∥ · ∥ b ⊗ 1 ). By relation (359), for ev ery C ∈ B 1 ( H ) b ⊗ B 1 ( J ), the following relation holds: ∥ C ∥ b ⊗ 1 = sup {| κ C ( K ) | = | tr( C K ) | : K ∈ C ( H ) ˘ ⊗ C ( J ) , ∥ K ∥ Γ = 1 } ≤ ∥ C ∥ γ . (360) Here, the inequality follo ws from definition (341), taking in to account that ∥ K ∥ Γ : = ∥ Γ K ∥ B . Now, b y Prop osition 5.4, ∥ C ∥ b ⊗ 1 ≥ ∥ C ∥ γ , as well, so that, actually , ∥ · ∥ b ⊗ 1 = ∥ · ∥ γ ; in particular, for every C ∈ B 1 ( H ) b ⊗ B 1 ( J ) w e hav e: ∥ C ∥ γ = sup {| κ C ( K ) | = | tr( C K ) | : K ∈ C ( H ) ˘ ⊗ C ( J ) , ∥ K ∥ Γ = 1 } = ∥ C ∥ b ⊗ 1 . (361) Finally , b y relation (361) — and taking into account that A ( H ⊗ J ) ⊃ C ( H ⊗ J ) ⊃ C ( H ) ˘ ⊗ C ( J ), that F ( H ) ˘ ⊗ F ( J ) is ∥ · ∥ Γ -dense in C ( H ) ˘ ⊗ C ( J ) (by the final assertion of Prop osition 5.5) and that C ( H ) ˘ ⊗ C ( J ) ⊂ B ( H ⊗ J ) ⊂ ( B 1 ( H ) b ⊗ B 1 ( J )  ∗ — it is also clear that the v arious equiv alen t expressions (353) of the norm ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 hold true. R emark 5.5 . In the case where C ∈ B 1 ( H ) R b ⊗ B 1 ( J ) R , in the v arious equiv alent expressions (353) of ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 w e can assume that the op erators K , F and L therein (b elonging to their resp ectiv e spaces) are selfadjoint . Indeed, e.g., putting K ℜ : = 1 2 ( K + K ∗ ), we ha ve that ∥ C ∥ b ⊗ 1 = ∥ C ∥ γ : = sup {| tr( C K ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = sup {|ℜ e(tr( C K )) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = sup {| tr( C K ℜ ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = sup {| tr( C K ) | : K ∈ A ( H ⊗ J ) , K = K ∗ , ∥ K ∥ Γ = 1 } = sup { tr( C K ) : K ∈ A ( H ⊗ J ) , K = K ∗ , ∥ K ∥ Γ = 1 } . (362) Here, the second equality is due to the fact that ∥ K ∥ Γ = ∥ z K ∥ Γ , for ev ery z ∈ T , while, for the third equality , notice that ℜ e(tr( C K )) = 1 2 (tr( C K ) + tr( C K ) ∗ ) = 1 2 (tr( C K ) + tr( C K ∗ )) = tr( C K ℜ ), where we hav e used the fact that C = C ∗ and the cyclic property of the trace. Next, for the fourth equalit y , note that ∥ K ℜ ∥ Γ ≤ 1 2 ( ∥ K ∥ Γ + ∥ K ∗ ∥ Γ ) = ∥ K ∥ Γ , whic h w ould imply that the suprem um on the third line is smaller that the suprem um on the fourth line; but, comparing the latter with the suprem um on the first line, we see that the reverse inequality holds to o, and we actually hav e an equalit y . Finally , the fifth line obtains noting that | tr( C K ) | = max { tr( C K ) , tr( C ( − K )) } therein. In the ligh t of Theorem 5.2, we can now endo w the Banach space B 1 ( H ) b ⊗ B 1 ( J ) with the σ  B 1 ( H ) b ⊗ B 1 ( J ) , C ( H ) ˇ ⊗ C ( J )  -top ology , i.e., the weak ∗ top ology p ertaining to the Banac h space dual B 1 ( H ) b ⊗ B 1 ( J ) of the injective tensor pro duct C ( H ) ˇ ⊗ C ( J ). In the follo wing, w e will call this top ology the cr oss w ∗ -top olo gy of B 1 ( H ) b ⊗ B 1 ( J ) or, in short, the cw ∗ -top olo gy . Since C ( H ) ˘ ⊗ C ( J ) is norm-dense in C ( H ) ˇ ⊗ C ( J ), by F act 2.8, a ∥ · ∥ b ⊗ 1 -b ounde d net { C i } ⊂ B 1 ( H ) b ⊗ B 1 ( J ) will con verge to some C ∈ B 1 ( H ) b ⊗ B 1 ( J ) wrt the cw ∗ -top ology iff lim i tr( C i K ) = tr( C K ), for all K ∈ C ( H ) ˘ ⊗ C ( J ); or, equiv alently , iff lim i tr( C i ( K 1 ⊗ K 2 )) = tr( C ( K 1 ⊗ K 2 )) , ∀ K 1 ∈ C ( H ) , ∀ K 2 ∈ C ( J ) . (363) 77 R emark 5.6 . Assume that dim( H ⊗ J ) = ∞ . In this case, the zero v ector b elongs to the cw ∗ -closure D ( H ⊗ J ) B cw ∗ ≡ cw ∗ - cl ( D ( H ⊗ J ) B ) (364) of the con v ex set D ( H ⊗ J ) B = D ( H ⊗ J ) S . In fact, giv en an y pair of orthonormal bases { ϕ m } m ∈ I , { ϕ n } n ∈ J in H and J , respectively — where I , J are coun table index sets, at least one of whic h can b e c hosen to be N — let { e k } ∞ k =1 denote the orthonormal basis in the Hilb ert space H ⊗ J obtained b y an y ordering of the set { ϕ m ⊗ ψ n : ( m, n ) ∈ I × J } . Then, arguing as in Example 5.1, w e put ς l : = l − 1 l X k =1 | e k ⟩ ⟨ e k | ∈ co ( θ ( P ( H ) , P ( J ))) ⊂ D ( H ⊗ J ) f s : = co ( θ ( D ( H ) , D ( J ))) ⊂ D ( H ⊗ J ) B , l ∈ N , (365) and for ev ery A ∈ B 1 ( H ⊗ J ) we ha ve that lim l tr( ς l A ) = lim l  l − 1 tr  ( P l k =1 | e k ⟩ ⟨ e k | ) A   = 0 , (366) b ecause lim l tr  ( P l k =1 | e k ⟩ ⟨ e k | ) A  = P ∞ k =1 ⟨ e k , A e k ⟩ = tr( A ). Since the trace class B 1 ( H ⊗ J ) is ∥ · ∥ ⊗ ∞ -dense in C ( H ⊗ J ), giv en any K ∈ C ( H ⊗ J ) and A ∈ B 1 ( H ⊗ J ), by the estimate | tr( ς l K ) | ≤ | tr( ς l ( K − A )) | + | tr( ς l A ) | ≤ ∥ ς l ( K − A ) ∥ ⊗ 1 + ∥ ς l A ∥ ⊗ 1 ≤ ∥ ς l ∥ ⊗ 1 ∥ K − A ∥ ⊗ ∞ + ∥ ς l ∥ ⊗ ∞ ∥ A ∥ ⊗ 1 = ∥ K − A ∥ ⊗ ∞ + l − 1 ∥ A ∥ ⊗ 1 , (367) it follo ws that relation (366) actually holds for every A ∈ C ( H ⊗ J ); hence, in particular, for ev ery A ∈ C ( H ) ˘ ⊗ C ( J ). Therefore, cw ∗ – lim l ς l = 0 ∈ co cw ∗ ( θ ( P ( H ) , P ( J ))) ⊂ cw ∗ - cl ( D ( H ⊗ J ) B ). By this fact, it also follo ws that, for an y D ∈ D ( H ⊗ J ) B and s ∈ [0 , 1], putting C l : = s D + (1 − s ) ς l , we ha ve that cw ∗ – lim l C l = s D ; hence: co ( D ( H ⊗ J ) B ∪ { 0 } ) = { s D : s ∈ [0 , 1] , D ∈ D ( H ⊗ J ) B } ⊂ cw ∗ - cl ( D ( H ⊗ J ) B ). Prop osition 5.6. The c onvex set D ( H ⊗ J ) B is a cw ∗ -pr e c omp act subset of B 1 ( H ) b ⊗ B 1 ( J ) , that is close d — henc e, c omp act — iff dim( H ⊗ J ) < ∞ . In the c ase wher e dim( H ⊗ J ) = ∞ , the cw ∗ -c omp act set cw ∗ - cl ( D ( H ⊗ J ) B ) satisfies the r elation D ( H ⊗ J ) se ⊊ co cw ∗ ( θ ( P ( H ) , P ( J ))) ⊂ co cw ∗ ( D ( H ⊗ J ) B ∪ { 0 } ) = cw ∗ - cl ( D ( H ⊗ J ) B ) ⊂ B  B 1 ( H ) b ⊗ B 1 ( J )  + , (368) wher e B  B 1 ( H ) b ⊗ B 1 ( J )  + : =  C ∈ B 1 ( H ) b ⊗ B 1 ( J ) : C ≥ 0 , ∥ C ∥ b ⊗ 1 ≤ 1  is the c onvex set of al l p ositive cr oss tr ac e class op er ators b elonging to the unit b al l of B 1 ( H ) b ⊗ B 1 ( J ) . Pr o of. T o pro ve the first claim, observ e that D ( H⊗J ) B is a subset of the unit ball in B 1 ( H ) b ⊗ B 1 ( J ), whic h, by the Banach-Alaoglu theorem (Theorem 2.6.18 of [46]), is compact wrt its w eak ∗ top ol- ogy , i.e., the cross w ∗ -top ology . Therefore, D ( H ⊗ J ) B is precompact wrt the cw ∗ -top ology . In the case where dim( H ⊗ J ) < ∞ , the cw ∗ -top ology coincides with any norm top ology on the finite- dimensional vector space B 1 ( H ) b ⊗ B 1 ( J ), and hence D ( H ⊗ J ) B is a cw ∗ -closed set, because it is the intersection of the norm-closed sets D ( H ⊗ J ) and B  B 1 ( H ) b ⊗ B 1 ( J )  . Assume no w that dim( H ⊗ J ) = ∞ . By Remark 5.6, D ( H ⊗ J ) B is not cw ∗ -closed, and its cw ∗ -closure con tains co cw ∗ ( θ ( P ( H ) , P ( J ))), whic h, since the cw ∗ -top ology is a subtop ology of the ∥ · ∥ b ⊗ 1 -top ology , con tains co b ⊗ ( θ ( D ( H ) , D ( J ))) = D ( H ⊗ J ) se (Corollary 4.3); moreo ver, 0 ∈ 78 co cw ∗ ( θ ( P ( H ) , P ( J ))). Thus, D ( H ⊗ J ) se ⊊ co cw ∗ ( θ ( P ( H ) , P ( J ))) ⊂ co cw ∗ ( D ( H ⊗ J ) B ∪ { 0 } ), where the last inclusion relation is due to the fact that θ ( P ( H ) , P ( J )) ⊂ D ( H ⊗J ) B . In Remark 5.6, w e also argue that co ( D ( H ⊗J ) B ∪ { 0 } ) ⊂ cw ∗ - cl ( D ( H⊗ J ) B ); hence, we ha ve: cw ∗ - cl ( D ( H⊗ J ) B ) ⊂ cw ∗ - cl (co ( D ( H ⊗ J ) B ∪ { 0 } )) = co cw ∗ ( D ( H ⊗ J ) B ∪ { 0 } ) ⊂ cw ∗ - cl ( D ( H ⊗ J ) B ), where the first inclusion relation is ob vious and the equalit y holds b y Remark 2.2. Let us finally pro v e that the compact subset cw ∗ - cl ( D ( H ⊗ J ) B ) of the unit ball in B 1 ( H ) b ⊗ B 1 ( J ) consists of p ositive op erators. T o this end, it is sufficient to to observe that, if { D i } is a net in D ( H ⊗ J ) B that conv erges, wrt the cw ∗ -top ology , to some C ∈ B 1 ( H ) b ⊗ B 1 ( J ) — i.e., lim i tr( D i K ) = tr( C K ), for all K ∈ C ( H ) ˘ ⊗ C ( J ) — then, in particular, w e ha ve: 0 ≤ lim i ⟨ a , D i a ⟩ H⊗J = lim i tr( D i | a ⟩ ⟨ a | ) = tr( C | a ⟩ ⟨ a | ) = ⟨ a , C a ⟩ H⊗J , ∀ a ∈ H ˘ ⊗ J ; (369) hence, ⟨ a , C a ⟩ H⊗J ≥ 0, for all a ∈ H ⊗ J . Th us, cw ∗ - cl ( D ( H ⊗ J ) B ) ⊂ B  B 1 ( H ) b ⊗ B 1 ( J )  + . According to Remark 5.6 and Prop osition 5.6, in the case where dim( H ⊗ J ) = ∞ , the cross w ∗ - top ology is too weak to be useful for our purp ose of suitably c haracterizing the set D ( H ⊗ J ) se ; e.g., 0 ∈ co cw ∗ ( θ ( P ( H ) , P ( J ))) ⊋ D ( H ⊗ J ) se . It will turn out that, instead, the extended w ∗ -top ology of B 1 ( H ⊗ J ) — or, equiv alently , the standard topology of D ( H ⊗ J ) — do es the job. In this regard, as it will b e clear so on, a crucial p oin t is that, for our main purposes, w e may consider the elemen ts of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) as bounded functionals on the C ∗ -algebra  A ( H ⊗ J ) , ∥ · ∥ ⊗ ∞  (that allows us to define the ew ∗ -top ology of B 1 ( H ⊗ J )) — or, equiv alently , on the normed space ( A ( H ⊗ J ) , ∥ · ∥ Γ ) — rather than on the Banach space C ( H ) ˇ ⊗ C ( J ) (as it w ould be natural in the ligh t of Theorem 5.2); see Prop osition 5.7 and Remark 5.8 b elo w. Before moving forw ard in this direction, for the sake of completeness, it is w orth observing that the injective tensor pro duct B ( H ) ˇ ⊗ B ( J ) can b e iden tified with a closed subspace of the Banach space dual of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ). Moreov er — in the case where at least one of the Hilbert spaces of the bipartition H ⊗ J is finite-dimensional — the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 on B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) are m utually equiv alen t (recall Theorem 3.3), and, analogously , the norms ∥ · ∥ Γ and ∥ · ∥ ⊗ ∞ on B ( H ⊗ J ) are equiv alen t; in addition, in this case we can identify the dual space of B 1 ( H ) b ⊗ B 1 ( J ) with B ( H ) ˇ ⊗ B ( J ), and the latter space can be realized precisely as the renorming ( B ( H ⊗ J ) , ∥ · ∥ Γ ) of the Banach space ( B ( H ⊗ J ) , ∥ · ∥ ⊗ ∞ ). In fact, the follo wing result holds: Theorem 5.3. The inje ctive tensor pr o duct B ( H ) ˇ ⊗ B ( J ) is (isomorphic to) a close d subsp ac e of the Banach sp ac e dual  B 1 ( H ) b ⊗ B 1 ( J )  ∗ of B 1 ( H ) b ⊗ B 1 ( J ) . Assume now that M = min { dim( H ) , dim( J ) } < ∞ . Then, the norms ∥ · ∥ ⊗ 1 , and ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 ar e e quivalent on B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) (a set e quality b eing understo o d). Inde e d, for every C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) , the fol lowing ine qualities hold: ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 ≤ N ∥ C ∥ ⊗ 1 , wher e N = min  4 M , M 2  ; (370) in p articular, for the selfadjoint tr ac e class op er ators and for the density op er ators on H ⊗ J , the fol lowing str onger estimates hold: ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 ≤ 2 M ∥ C ∥ ⊗ 1 , ∀ C ∈ B 1 ( H ⊗ J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R , (371) 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ γ = ∥ D ∥ b ⊗ 1 ≤ M , ∀ D ∈ D ( H ⊗ J ) = D ( H ⊗ J ) b . (372) Mor e over, the algebr aic, the natur al (or sp atial) and the inje ctive tensor pr o ducts of B ( H ) and B ( J ) al l c oincide, as line ar sp ac es, with B ( H ⊗ J ) , i.e., B ( H ) ˘ ⊗ B ( J ) = B ( H ) ⊗ B ( J ) = B ( H ) ˇ ⊗ B ( J ) = B ( H ⊗ J ) , (373) 79 and the inje ctive norm ∥ · ∥ Γ of B ( H ⊗ J ) is e quivalent to the the op er ator norm ∥ · ∥ ⊗ ∞ ; pr e cisely, for every b ounde d op er ator L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ ≤ ∥ L ∥ ⊗ ∞ ≤ N ∥ L ∥ Γ , wher e N = min  4 M , M 2  . (374) The dual sp ac e  B 1 ( H ) b ⊗ B 1 ( J )  ∗ of B 1 ( H ) b ⊗ B 1 ( J ) c an b e identifie d — as a line ar sp ac e and via the tr ac e functional — with B ( H ⊗ J ) = B ( H ) ˘ ⊗ B ( J ) , endowe d with the inje ctive norm ∥ · ∥ Γ ; in p articular, every functional Γ ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ is of the form Γ( C ) = J X j =1 tr  C ( A j ⊗ B j )  , ∀ C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) , (375) for some J -tuples of b ounde d op er ators { A j } J j =1 ⊂ B ( H ) and { B j } J j =1 ⊂ B ( J ) , with J ≤ K = M 2 . Final ly, for every C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) , we have: ∥ C ∥ b ⊗ 1 = max {| tr( C L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } . (376) (In comparison with the fourth line of (353), here we ha ve a maximum rather than a supr emum .) Pr o of. Let us prov e the first assertion. By Remark 5.4, the restriction of the norm ∥ · ∥ Γ to the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) coincides with the injective norm ∥ · ∥ ˇ ⊗ ∞ of B ( H ) ˘ ⊗ B ( J ). Therefore, for ev ery L ∈ B ( H ) ˘ ⊗ B ( J ), ∥ L ∥ ˇ ⊗ ∞ = ∥ L ∥ Γ = ∥ Γ L ∥ B : =  | tr( C L ) | : C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ∥ C ∥ b ⊗ 1 = 1  . (377) Hence, the map ( B ( H ) ˘ ⊗ B ( J ) , ∥ · ∥ ˇ ⊗ ∞ ) ∋ L 7→ tr(( · ) L ) ∈  B 1 ( H ) b ⊗ B 1 ( J )  ∗ is a linear isometry that extends uniquely to an isometry of B ( H ) ˇ ⊗ B ( J ) into ( B 1 ( H ) b ⊗ B 1 ( J )  ∗ . Otherwise stated, B ( H ) ˇ ⊗ B ( J ) is isomorphic to a closed subspace of  B 1 ( H ) b ⊗ B 1 ( J )  ∗ . Supp ose that, in particular, M = min { dim( H ) , dim( J ) } < ∞ . Let us prov e the second assertion. In fact, in the case where M = min { dim( H ) , dim( J ) } < ∞ , b y Theorem 3.3, the linear spaces B 1 ( H ) b ⊗ B 1 ( J ) and B 1 ( H ⊗ J ) coincide, and the asso ciated norms ∥ · ∥ b ⊗ 1 and ∥ · ∥ ⊗ 1 are equiv alen t; sp ecifically — dep ending on which case applies — they satisfy relations (178), (179) or (180). Moreov er, b y Theorem 5.2, for every trace class operator C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ), we hav e that ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 . Hence, relations (370), (371) and (372) hold true. T o prov e the third assertion, note that, by Corollary 3.9 (recall also Remark 3.5), in this case the dual space  B 1 ( H ) b ⊗ B 1 ( J )  ∗ can b e identified — as a linear space — with B ( H ) ˘ ⊗ B ( J ) = B ( H ) ⊗ B ( J ) = B ( H ⊗ J ) (and every b ounded linear functional Γ on B 1 ( H ) b ⊗ B 1 ( J ) can b e expressed in the form (375)), endow ed with a norm that is equiv alent to the op erator norm ∥ · ∥ ⊗ ∞ , and, by definition ( ∥ L ∥ Γ : = ∥ Γ L ∥ B ), is precisely the norm ∥ · ∥ Γ , which is the injectiv e norm of B ( H ) ˘ ⊗ B ( J ) = B ( H ⊗ J ) (Remark 5.4). Now, since, for every C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ), w e hav e that ∥ C ∥ ⊗ 1 ≤ ∥ C ∥ b ⊗ 1 ≤ N ∥ C ∥ ⊗ 1 — by standard Banach space p olarit y argumen ts (see, e.g., Sect. 3.1 of [64]) — the corresp onding (equiv alent) dual norms ∥ · ∥ ⊗ ∞ and ∥ · ∥ Γ m ust v erify relation (374). Moreov er, since the t wo norms ∥ · ∥ ⊗ ∞ and ∥ · ∥ Γ are equiv alent, the algebraic tensor pro duct B ( H ) ˘ ⊗ B ( J ) m ust b e complete wrt the injectiv e norm ∥ · ∥ Γ to o; hence, we hav e that B ( H ) ˘ ⊗ B ( J ) = B ( H ) ˇ ⊗ B ( J ) (set equalit y), as well, and relation (373) holds true. Finally , b y our previous iden tification of the dual space  B 1 ( H ) b ⊗ B 1 ( J )  ∗ with a renorming of the Banach space B ( H ⊗ J ), relation (376) holds to o. 80 5.4 The cross trace class op erators as suitable functionals The norm ∥ · ∥ Γ allo ws us to relate the pro jectiv e norm ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 of B 1 ( H ) b ⊗ B 1 ( J ) to a norm on a suitable class of b ounded functionals on A ( H ⊗ J ). Precisely , we select those functionals of A ( H ⊗ J ) ∗ that are induced b y cross trace class operators; namely , we introduce the linear subspace A ( H ⊗ J ) b of A ( H ⊗ J ) ∗ defined by A ( H ⊗ J ) b : =  ξ C ≡ [ [ · , ξ C ] ] ∈ A ( H ⊗ J ) ∗ : C ∈ B 1 ( H ) b ⊗ B 1 ( J )  . (378) with [ [ K , ξ C ] ] ≡ ξ C ( K ) = tr( C K ) ∈ C denoting the canonical pairing b et w een K ∈ A ( H ⊗ J ) and ξ C ∈ A ( H ⊗ J ) ∗ . Recalling the isometry f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ defined by (328), we see that A ( H ⊗ J ) b = f  B 1 ( H ) b ⊗ B 1 ( J )  . F or ev ery C ∈ B 1 ( H ) b ⊗ B 1 ( J ), b y definitions (341) and (345), and by Theorem 5.2, w e hav e that ∥ C ∥ b ⊗ 1 = ∥ C ∥ γ = sup {| Γ K ( C ) | = | tr( C K ) | : K ∈ A ( H ⊗ J ) , ∥ Γ K ∥ B = 1 } = sup {| ξ C ( K ) | = | tr( C K ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = : ∥ ξ C ∥ 𭟋 . (379) W e then obtain a norm ∥ · ∥ 𭟋 on the linear space A ( H ⊗ J ) b = f  B 1 ( H ) b ⊗ B 1 ( J )  ⊂ A ( H ⊗ J ) ∗ of all b ounded functionals on A ( H ⊗ J ) of the form ξ C ≡ [ [ · , ξ C ] ], with C ∈ B 1 ( H ) b ⊗ B 1 ( J ); see Prop osition 5.7 b elow. Note that, since the norm ∥ · ∥ Γ is dominated by the op erator norm ∥ · ∥ ⊗ ∞ (and f is an isometry), ∥ ξ C ∥ 𭟋 ≥ ∥ ξ C ∥ B : = sup  | ξ C ( K ) | = | tr( C K ) | =    f ( C )  ( K )   : K ∈ A ( H ⊗ J ) , ∥ K ∥ ⊗ ∞ = 1  = ∥ C ∥ ⊗ 1 , ∀ ξ C ∈ A ( H ⊗ J ) b ⊂ A ( H ⊗ J ) ∗ . (380) Prop osition 5.7. By setting ∥ ξ C ∥ 𭟋 : = sup {| ξ C ( K ) | = | tr( C K ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } , for every C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , we obtain a norm ∥ · ∥ 𭟋 on the line ar sp ac e A ( H ⊗ J ) b ⊂ A ( H ⊗ J ) ∗ . This norm satisfies the fol lowing r elation: ∥ C ∥ ⊗ 1 ≤ ∥ ξ C ∥ 𭟋 = ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 , ∀ C ∈ B 1 ( H ) b ⊗ B 1 ( J ) . (381) Then, by the rhs e quality ab ove, the bije ctive line ar map b f :  B 1 ( H ) b ⊗ B 1 ( J ) , ∥ · ∥ b ⊗ 1  ∋ C 7→ ξ C ∈  A ( H ⊗ J ) b , ∥ · ∥ 𭟋  (382) is an isometry, and, as a c onse quenc e,  A ( H ⊗ J ) b , ∥ · ∥ 𭟋  is a Banach sp ac e, isomorphic to the cr oss tr ac e class B 1 ( H ) b ⊗ B 1 ( J ) ∼ =  C ( H ) ˇ ⊗ C ( J )  ∗ ; pr e cisely, it is a close d subsp ac e of the Banach sp ac e A ( H ⊗ J ) ∗ Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  ∗ , (383) i.e., the dual of the norme d sp ac e A ( H ⊗ J ) Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  . In p articular, the norm ∥ · ∥ 𭟋 is the r estriction to the subsp ac e A ( H ⊗ J ) b of the dual norm ∥ · ∥ ∗ Γ of ∥ · ∥ Γ . In the c ase wher e M = min { dim( H ) , dim( J ) } < ∞ , A ( H ⊗ J ) Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  is — up to normalization of the norm, i.e., r eplacing the norm ∥ · ∥ Γ with N 2 ∥ · ∥ Γ , wher e N = min  4 M , M 2  — a (unital) Banach algebr a, which is a r enorming of the C ∗ -algebr a  A ( H ⊗ J ) , ∥ · ∥ ⊗ ∞  . Mor e over, in this c ase, A ( H ⊗ J ) b = f ( B 1 ( H ⊗ J )) (set e quality), and the Banach subsp ac e  A ( H ⊗ J ) b , ∥ · ∥ 𭟋  of A ( H ⊗ J ) ∗ Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  ∗ is a r enorming of the close d subsp ac e f ( B 1 ( H ⊗ J )) ∼ = B 1 ( H ⊗ J ) ∼ = C ( H ⊗ J ) ∗ (384) of A ( H ⊗ J ) ∗ ; sp e cific al ly, for every C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) (set e quality), we have: ∥ ξ C ∥ B = ∥ C ∥ ⊗ 1 ≤ ∥ ξ C ∥ 𭟋 = ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 ≤ N ∥ C ∥ ⊗ 1 = N ∥ ξ C ∥ B , (385) wher e ∥ ξ C ∥ B is the norm of ξ C as a functional on  A ( H ⊗ J ) , ∥ · ∥ ⊗ ∞  and N = min  4 M , M 2  . 81 Pr o of. Since, by (379), ∥ ξ C ∥ 𭟋 = ∥ C ∥ γ = ∥ C ∥ b ⊗ 1 , C ∈ B 1 ( H ) b ⊗ B 1 ( J ), and, moreo ver, the mapping B 1 ( H ) b ⊗ B 1 ( J ) ∋ C 7→ ξ C ≡ [ [ · , ξ C ] ] = f ( C ) ∈ A ( H ⊗ J ) b = f  B 1 ( H ) b ⊗ B 1 ( J )  ⊂ A ( H ⊗ J ) ∗ is a linear bijection, we conclude that ∥ · ∥ 𭟋 : A ( H ⊗ J ) b → R + is indeed a norm, and the bijective linear map (382) is an isometry . Note that, by (140), relation (381) holds. By (379), it is also clear that  A ( H ⊗ J ) b , ∥ · ∥ 𭟋  is a (closed) subspace of the Banach space  A ( H ⊗ J ) , ∥ · ∥ Γ  ∗ , and the norm ∥ · ∥ 𭟋 is precisely the restriction to the subspace A ( H ⊗ J ) b of the dual norm ∥ · ∥ ∗ Γ of ∥ · ∥ Γ . Assume no w that at least one of the Hilb ert spaces H and J is finite-dimensional, i.e., that M = min { dim( H ) , dim( J ) } < ∞ . Then, b y the third assertion of Theorem 5.3, the injective norm ∥ · ∥ Γ of B ( H ⊗ J ) is equiv alen t to the op erator norm ∥ · ∥ ⊗ ∞ . Hence, the restrictions of these norms to A ( H ⊗ J ) are equiv alent norms to o, so that A ( H ⊗ J ) Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  is a Banac h space, b ecause  A ( H ⊗ J ) , ∥ · ∥ ⊗ ∞  is a Banac h space. Sp ecifically , A ( H ⊗ J ) Γ is — up to normalization of its norm — a (unital) Banac h algebra since ∥ K 1 K 2 ∥ Γ ≤ ∥ K 1 ∥ ⊗ ∞ ∥ K 2 ∥ ⊗ ∞ ≤ N 2 ∥ K 1 ∥ Γ ∥ K 2 ∥ Γ , with N = min  4 M , M 2  , and hence  A ( H ⊗ J ) , N 2 ∥ · ∥ Γ  is a unital Banach algebra and a renorming of the C ∗ -algebra  A ( H ⊗ J ) , ∥ · ∥ ⊗ ∞  . Moreov er, by the second assertion of Theorem 5.3, we hav e that B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ⊗ J ) (set equalit y), and the norms ∥ · ∥ ⊗ 1 and ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 on B 1 ( H ⊗ J ) are equiv alent (precisely , they satisfy relation (371)). The Banac h space B 1 ( H ⊗ J ) ≡  B 1 ( H ⊗ J ) , ∥ · ∥ ⊗ 1  can b e identified, via the trace, with the dual of C ( H ⊗ J ), and the Banac h space B 1 ( H ⊗ J ) ∼ = C ( H ⊗ J ) ∗ can be iden tified with a closed subspace of A ( H ⊗ J ) ∗ ; in fact, recalling Remark 5.1, w e hav e: B 1 ( H ⊗ J ) ∼ = C ( H ⊗ J ) ∗ ∼ = f ( B 1 ( H ⊗ J )). In conclusion, if min { dim( H ) , dim( J ) } < ∞ , then A ( H ⊗ J ) b = f  B 1 ( H ) b ⊗ B 1 ( J )  = f ( B 1 ( H ⊗ J )), and  A ( H ⊗ J ) b = f ( B 1 ( H ⊗ J )) , ∥ · ∥ 𭟋  ∼ =  B 1 ( H ⊗ J ) , ∥ · ∥ γ = ∥ · ∥ b ⊗ 1  is a renorming of the closed subspace f ( B 1 ( H ⊗ J )) ∼ = B 1 ( H ⊗ J ) ∼ = C ( H ⊗ J ) ∗ of A ( H ⊗ J ) ∗ ; precisely , by (371) and by the fact that f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ is an isometry , for every C ∈ B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ), relation (385) holds. W e stress that the isomorphism of Banach spaces (382) is a map restriction (wrt b oth the domain and the co domain) of the linear isometry f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ , see (328). R emark 5.7 . F or every C ∈ B 1 ( H ) b ⊗ B 1 ( J ) the linear functional ξ C ∈ A ( H ⊗ J ) b ⊂ A ( H ⊗ J ) ∗ Γ admits a natural extension to a b ounded linear functional Ξ C on the normed space B ( H ⊗ J ) Γ ≡ ( B ( H ⊗ J ) , ∥ · ∥ Γ ), defined b y Ξ C ( L ) : = tr( C L ), for all L ∈ B ( H ⊗ J ). In fact, the functional Ξ C is b ounded, b ecause | tr( C L ) | ≤ ∥ C ∥ ⊗ 1 ∥ L ∥ ⊗ ∞ ≤ ∥ C ∥ b ⊗ 1 ∥ L ∥ ⊗ ∞ , and, moreov er, by relation (353), ∥ Ξ C ∥ B : = sup {| tr( C L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } = ∥ C ∥ b ⊗ 1 = ∥ ξ C ∥ 𭟋 = ∥ ξ C ∥ ∗ Γ . R emark 5.8 . W e are no w regarding the elements of the cross trace class B 1 ( H ) b ⊗ B 1 ( J ) as b ounded functionals on the normed space A ( H ⊗ J ) Γ ≡ ( A ( H ⊗ J ) , ∥ · ∥ Γ ), rather than on the Banac h space C ( H ) ˇ ⊗ C ( J ) (as it would b e natural according to Theorem 5.2), or on the C ∗ -algebra A ( H ⊗ J ) ≡  A ( H ⊗ J ) , ∥ · ∥ ⊗ ∞  (that allows us to in tro duce the A w ∗ -top ology). Ho wev er, by F act 2.10, with the ob vious iden tifications V ≡ A ( H ⊗ J ), ∥ · ∥ ≡ ∥ · ∥ Γ and | | | · | | | ≡ ∥ · ∥ ⊗ ∞ , w e ha ve that A ( H ⊗ J ) ∗ Γ is, in a natural wa y , a linear subspace of A ( H ⊗ J ) ∗ , and the σ  A ( H ⊗ J ) ∗ Γ , A ( H ⊗ J )  -top ology — or, with a more precise notation, the σ  A ( H ⊗ J ) ∗ Γ , A ( H ⊗ J ) Γ  -top ology — coincides with the subspace top ology on A ( H ⊗ J ) ∗ Γ ⊂ A ( H ⊗ J ) ∗ wrt the σ ( A ( H ⊗ J ) ∗ , A ( H ⊗ J ))-top ology , i.e., the A w ∗ -top ology . Therefore, in the follo wing, we may think of the cross states D ( H ⊗ J ) b as functionals in A ( H ⊗ J ) b (i.e., as elemen ts of the set b f ( D ( H ⊗ J ) b ) ⊂ A ( H ⊗ J ) b ) or, equiv alen tly , as functionals in A ( H ⊗ J ) ∗ (i.e., as elements of the set f ( D ( H ⊗ J )) ⊂ A ( H ⊗ J ) ∗ ), b ecause the A w ∗ -top ology will turn out to b e the relev ant one. 82 5.5 The main result: c haracterization of separable states Let us now further consider the intersection D ( H ⊗ J ) B of the conv ex set D ( H ⊗ J ) b of all cross states with the unit ball in B 1 ( H ) b ⊗ B 1 ( J ); see (325). As previously noted, D ( H ⊗ J ) B is a con vex set and, b y the the fact that ∥ · ∥ γ = ∥ · ∥ b ⊗ 1 , by third equality in (325) and b y Corollary 4.5, w e ha ve: D ( H ⊗ J ) B : =  D ∈ D ( H ⊗ J ) b : ∥ D ∥ γ = ∥ D ∥ b ⊗ 1 ≤ 1  = D ( H ⊗ J ) S ⊃ D ( H ⊗ J ) se . (386) By the definition of the norms ∥ · ∥ γ and ∥ · ∥ 𭟋 (see (341) and (379), resp ectively), and b y the fact that the norm ∥ · ∥ 𭟋 is the restriction of the dual norm ∥ · ∥ ∗ Γ of ∥ · ∥ Γ to the Banach subspace A ( H ⊗ J ) b of A ( H ⊗ J ) ∗ Γ (Prop osition 5.7), we ha ve: S ( H ⊗ J ) : = b f ( D ( H ⊗ J ) B ) =  ξ D ∈ b f ( D ( H ⊗ J ) b ) : ∥ D ∥ b ⊗ 1 ≤ 1  =  ξ D ∈ b f ( D ( H ⊗ J ) b ) : ∥ ξ D ∥ 𭟋 = ∥ ξ D ∥ ∗ Γ ≤ 1  =  ξ D ∈ b f ( D ( H ⊗ J ) b ) : | ξ D ( K ) | : = | tr( D K ) | ≤ ∥ K ∥ Γ , ∀ K ∈ A ( H ⊗ J )  ⊂ A ( H ⊗ J ) b ⊂ A ( H ⊗ J ) ∗ Γ ⊂ A ( H ⊗ J ) ∗ , (387) where, for the last inclusion relation, recall Remark 5.8. W e will also consider the following con vex subset of A ( H ⊗ J ) ∗ : e S ( H ⊗ J ) : =  ξ ∈ A ( H ⊗ J ) ∗ : ξ ≥ 0, ξ ( I ) = 1, and | ξ ( K ) | ≤ ∥ K ∥ Γ , ∀ K ∈ A ( H ⊗ J )  =  ξ ∈ A ( H ⊗ J ) ∗ Γ : ξ ≥ 0, ξ ( I ) = 1 and ∥ ξ ∥ ∗ Γ ≤ 1  ⊃ S ( H ⊗ J ) . (388) Here, ξ ≥ 0 (positivity) means that ξ ( K ) ≥ 0, for all p ositive K ∈ A ( H ⊗ J ). Clearly , e S ( H ⊗ J ) is a con vex subset of the con vex set of all normalized, p ositiv e functionals on the C ∗ -algebra A ( H ⊗ J ). Regarding definition (388), we stress the following fact. F or every ξ ∈ A ( H ⊗ J ) ∗ , the conditions ξ ≥ 0 and ξ ( I ) = 1 imply that ∥ ξ ∥ B : = sup {| ξ ( K ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ ⊗ ∞ = 1 } = 1 — see, e.g., Corollary 3.3.4 of [42] — but not, in general, that | ξ ( K ) | ≤ ∥ K ∥ Γ , for all K ∈ A ( H ⊗ J ) (or ev en that ξ ∈ A ( H ⊗ J ) ∗ Γ , in the case where dim( H ) = dim( J ) = ∞ ). Indeed, the latter condition is satisfied precisely by those (p ositiv e) functionals on A ( H ⊗ J ) that are contained in the unit ball of the linear subspace A ( H ⊗ J ) ∗ Γ of A ( H ⊗ J ) ∗ . In the follo wing, for the sake of notational conciseness, it will b e conv enient to identify the set D ( H ⊗ J ) B ⊂ D ( H ⊗ J ) b with the conv ex subset S ( H ⊗ J ) of A ( H ⊗ J ) ∗ ; i.e., with the image, via the isomorphism of Banach spaces b f : B 1 ( H ) b ⊗ B 1 ( J ) → A ( H ⊗ J ) b (see (382)), or via the isometry f (see (328)), of the conv ex set D ( H ⊗ J ) B . More generally , (an y subset of ) B 1 ( H ⊗ J ) will b e iden tified with (a corresponding subset of ) the linear subspace f ( B 1 ( H ⊗ J )) of A ( H ⊗ J ) ∗ . A t this p oint, it is also worth recalling that the ew ∗ -top ology on B 1 ( H ⊗ J ) is defined as the initial top ology induced b y the map f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ , A ( H ⊗ J ) ∗ b eing endo w ed with the A w ∗ -top ology (see Subsection 5.1). R emark 5.9 . By F act 2.9 — with the iden tifications: S ≡ A ( H ⊗ J ) Γ , regarded as a normed space endo w ed with the norm ∥ · ∥ Γ , whic h is ob viously a ∥ · ∥ Γ -dense linear subspace of its Banac h space completion V ≡ A ( H ⊗ J ) Γ , and B ≡ e S ( H ⊗ J ), which is a ∥ · ∥ ∗ Γ -b ounded subset of S ∗ ≡ A ( H ⊗ J ) ∗ Γ = A ( H ⊗ J ) Γ ∗ ≡ V ∗ — we conclude that the relativ e top ology on e S ( H ⊗ J ) wrt the σ  A ( H ⊗ J ) ∗ Γ , A ( H ⊗ J ) Γ  -top ology coincides with the relativ e top ology on e S ( H ⊗ J ) wrt the σ ( A ( H ⊗ J ) ∗ Γ , A ( H ⊗ J ) Γ )-top ology; i.e., b y Remark 5.8, with the relative topology on e S ( H ⊗ J ) wrt the A w ∗ -top ology of A ( H ⊗ J ) ∗ . All of which is to sa y that, as far as top ological arguments are concerned in the follo wing, the functionals in e S ( H ⊗ J ) can be regarded, in terchangeably , as b ounded functionals on the spaces A ( H ⊗ J ) (endow ed with the norm ∥ · ∥ ⊗ ∞ ), A ( H ⊗ J ) Γ or A ( H ⊗ J ) Γ . 83 T o prov e the main theorem, it remains to establish a further tec hnical fact. Lemma 5.2. The c onvex set S ( H ⊗ J ) is a pr e c omp act subset of A ( H ⊗ J ) ∗ wrt the A w ∗ -top olo gy, and, in fact, with the obvious identific ations via the immersion map f , we have that S ( H ⊗ J ) ≡ D ( H ⊗ J ) B ⊂ co ew ∗ ( θ ( P ( H ) , P ( J ))) ≡ f  co ew ∗ ( θ ( P ( H ) , P ( J )))  ⊂ co A w ∗ ( θ ( P ( H ) , P ( J ))) = e S ( H ⊗ J ) , (389) wher e the A w ∗ -close d c onvex hul l co A w ∗ ( θ ( P ( H ) , P ( J ))) = e S ( H ⊗ J ) of the set θ ( P ( H ) , P ( J )) ≡ f ( θ ( P ( H ) , P ( J ))) (390) is a A w ∗ -c omp act c onvex subset of A ( H ⊗ J ) ∗ . Mor e over, identifying B 1 ( H ⊗ J ) , D ( H ⊗ J ) with f ( B 1 ( H ⊗ J )) and f ( D ( H ⊗ J )) , r esp e ctively, we have: co ew ∗ ( θ ( P ( H ) , P ( J ))) ≡ f  co ew ∗ ( θ ( P ( H ) , P ( J )))  = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ B 1 ( H ⊗ J ) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ) = e S ( H ⊗ J ) ∩ D ( H ⊗ J ) . (391) Pr o of. As previously noted (see (387) and (388)), S ( H ⊗ J ) : = b f ( D ( H ⊗ J ) B ) = f ( D ( H ⊗ J ) B ) is con tained in the con vex set of all states ξ of the C ∗ -algebra A ( H ⊗ J ) suc h that | ξ ( K ) | ≤ ∥ K ∥ Γ , for all K ∈ A ( H ⊗ J ); i.e., w e hav e: S ( H ⊗ J ) =  ξ D ∈ b f ( D ( H ⊗ J ) b ) : | ξ D ( K ) | : = | tr( D K ) | ≤ ∥ K ∥ Γ , ∀ K ∈ A ( H ⊗ J )  = { ξ D ∈ A ( H ⊗ J ) ∗ : D ∈ D ( H ⊗ J ) b , | ξ D ( K ) | ≤ ∥ K ∥ Γ , ∀ K ∈ A ( H ⊗ J ) } ⊂ { ξ ∈ A ( H ⊗ J ) ∗ : ξ ≥ 0, ξ ( I ) = 1, and | ξ ( K ) | ≤ ∥ K ∥ Γ , ∀ K ∈ A ( H ⊗ J ) } = : e S ( H ⊗ J ) . (392) By Theorem 6.3 of [25] — also see the pro of of this theorem ibidem , and suitably define the subset V of H ⊗ J therein as V : = { ϕ ⊗ ψ : ϕ ∈ H , ψ ∈ J , ∥ ϕ ∥ = ∥ ψ ∥ = 1 } — the con vex subset e S ( H ⊗ J ) of A ( H ⊗ J ) ∗ is A w ∗ -compact and coincides with the A w ∗ -closed conv ex hull of the set θ ( P ( H ) , P ( J )) ≡ f ( θ ( P ( H ) , P ( J ))), so that S ( H ⊗ J ) ⊂ e S ( H ⊗ J ) = co A w ∗  f ( θ ( P ( H ) , P ( J )))  . (393) Th us, the A w ∗ -closure of the set S ( H ⊗ J ) is contained in the A w ∗ -compact set e S ( H ⊗ J ); hence, it is A w ∗ -compact too, and S ( H ⊗ J ) is a precompact subset of A ( H ⊗ J ) ∗ wrt the A w ∗ -top ology . No w, by F act 5.1, co ew ∗ ( θ ( P ( H ) , P ( J ))) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ B 1 ( H ⊗ J ), and actually , b y Lemma 5.1, co ew ∗ ( θ ( P ( H ) , P ( J ))) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ), so that relation (391) holds true. T o complete the pro of, just note that D ( H ⊗ J ) B ⊂ co ew ∗ ( θ ( P ( H ) , P ( J ))). In fact, b y relations (391) and (393), w e argue that S ( H ⊗ J ) ≡ D ( H ⊗ J ) B = D ( H ⊗ J ) B ∩ D ( H ⊗ J ) ⊂ e S ( H ⊗ J ) ∩ D ( H ⊗ J ) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ) = co ew ∗ ( θ ( P ( H ) , P ( J ))) , (394) all obvious identifications via the immersion map f being understo o d. Hence, we finally obtain relation (389). 84 A t this p oin t, ev entually , the con v ex set D ( H ⊗ J ) se can be characterized as the intersection of the conv ex set D ( H ⊗ J ) b of all cross states with the unit ball wrt the norm ∥ · ∥ b ⊗ 1 = ∥ · ∥ γ . Theorem 5.4. The set D ( H ⊗ J ) se admits the fol lowing char acterization: D ( H ⊗ J ) se = D ( H ⊗ J ) S = D ( H ⊗ J ) B = b f − 1 ( S ( H ⊗ J )) =  D ∈ D ( H ⊗ J ) b : ∥ ξ D ∥ ∗ Γ = 1  . (395) Mor e over, D ( H ⊗ J ) se ≡ b f ( D ( H ⊗ J ) se ) = f ( D ( H ⊗ J ) se ) = S ( H ⊗ J ) is a A w ∗ -pr e c omp act subset of A ( H ⊗ J ) ∗ , b e c ause D ( H ⊗ J ) se = co ew ∗ ( θ ( P ( H ) , P ( J ))) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ B 1 ( H ⊗ J ) = co A w ∗ ( θ ( P ( H ) , P ( J ))) ∩ D ( H ⊗ J ) = e S ( H ⊗ J ) ∩ D ( H ⊗ J ) (396) — the obvious identific ations θ ( P ( H ) , P ( J )) ≡ f ( θ ( P ( H ) , P ( J ))) , B 1 ( H ⊗ J ) ≡ f ( B 1 ( H ⊗ J )) and D ( H ⊗ J ) ≡ f ( D ( H ⊗ J )) b eing understo o d — wher e co A w ∗ ( θ ( P ( H ) , P ( J ))) = e S ( H ⊗ J ) (397) is a A w ∗ -c omp act c onvex subset of A ( H ⊗ J ) ∗ . Pr o of. T aking into accoun t relation (386), to prov e (395) it is sufficien t to show that the reverse inclusion D ( H ⊗ J ) S = D ( H ⊗J ) B = b f − 1 ( S ( H ⊗ J )) ⊂ D ( H ⊗ J ) se holds to o. In fact, b y Lemma 5.2 (see, in particular, relation (389)), the conv ex set S ( H ⊗ J ) is contained in the ew ∗ -closed conv ex h ull of the set θ ( P ( H ) , P ( J )), so that S ( H ⊗ J ) ≡ D ( H ⊗ J ) B ⊂ co ew ∗ ( θ ( P ( H ) , P ( J ))) = D ( H ⊗ J ) se . (398) Here, the equalit y after the inclusion relation holds by Theorem 5.1, and natural iden tifications via the linear isometry f : B 1 ( H ⊗ J ) → A ( H ⊗ J ) ∗ — or, equiv alently , via the isomorphism of Banach spaces b f — are understo o d. At this p oint, regarding the last equality in relation (395), just note that S ( H ⊗ J ) : = b f ( D ( H ⊗ J ) B ) = b f ( D ( H ⊗ J ) se ) ≡ D ( H ⊗ J ) se , and hence D ( H ⊗ J ) se = b f − 1 ( S ( H ⊗ J )) =  D ∈ D ( H ⊗ J ) b : ∥ ξ D ∥ 𭟋 ≤ 1  =  D ∈ D ( H ⊗ J ) b : ∥ ξ D ∥ 𭟋 = 1  =  D ∈ D ( H ⊗ J ) b : ∥ ξ D ∥ ∗ Γ = 1  , (399) where, for the last tw o equalities, recall that 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 = ∥ ξ D ∥ 𭟋 = ∥ ξ D ∥ ∗ Γ (Prop osi- tion 5.7). By relation (391), the characterization (396) of D ( H ⊗ J ) se = co ew ∗ ( θ ( P ( H ) , P ( J ))) follo ws immediately . Finally , recall that, b y Lemma 5.2, co A w ∗ ( θ ( P ( H ) , P ( J ))) = e S ( H ⊗ J ) — with θ ( P ( H ) , P ( J )) ≡ f ( θ ( P ( H ) , P ( J ))) — is a A w ∗ -c omp act con vex subset of A ( H ⊗ J ) ∗ . Corollary 5.1. The set D ( H ⊗ J ) se admits the fol lowing further char acterization, in terms of the Hermitian pr oje ctive norm: D ( H ⊗ J ) se =  D ∈ D ( H ⊗ J ) b : | | | D | | | b ⊗ 1 = 1  . (400) 85 Pr o of. Observ e, indeed, that D ( H ⊗ J ) se ⊂ D ( H ⊗ J ) ∩ S  B 1 ( H ) R b ⊗ B 1 ( J ) R  =  D ∈ D ( H ⊗ J ) b : | | | D | | | b ⊗ 1 = 1  = D ( H ⊗ J ) ∩ B  B 1 ( H ) R b ⊗ B 1 ( J ) R  ⊂ D ( H ⊗ J ) ∩ B  B 1 ( H ) b ⊗ B 1 ( J )  = D ( H ⊗ J ) S = D ( H ⊗ J ) se . (401) In the previous argument, the first inclusion relation, follo ws from Corollary 4.5, and the second equalit y is a direct consequence of the fact that | | | D | | | b ⊗ 1 ≥ 1, for all D ∈ D ( H ⊗ J ) b ; whereas the second inclusion relation follows from the fact that B  B 1 ( H ) R b ⊗ B 1 ( J ) R  ⊂ B  B 1 ( H ) b ⊗ B 1 ( J )  , b ecause the pro jective norm ∥ · ∥ b ⊗ 1 is dominated on B 1 ( H ) R b ⊗ B 1 ( J ) R b y the Hermitian pro jectiv e norm | | | · | | | b ⊗ 1 . Finally , we ha ve used relation (395) in Theorem 5.4. 5.6 The cross norm criterion in any Hilb ert space dimension It is now worth collecting some of our main findings concerning the characterization of separable states in a unique statement. As previously noted, in the case where min { dim( H ) , dim( J ) } < ∞ our results can b e compared with the results obtained b y Rudolph [21,22] (who considered the finite- dimensional setting only) and b y Arv eson [25] (who considered m ultipartite systems, as well). Here, w e hav e extended these results to include the genuinely infinite-dimensional setting — i.e., the case where dim( H ) = dim( J ) = ∞ — and therefore we call the asso ciated separabilit y criterion the Extende d Cr oss Norm Criterion (ECNC). Theorem 5.5 (The Extended Cross Norm Criterion) . L et us supp ose, at first, that at le ast one of the Hilb ert sp ac es H , J is finite-dimensional: M ≡ min { dim( H ) , dim( J ) } < ∞ . Then, we have that B 1 ( H ⊗ J ) = B 1 ( H ) b ⊗ B 1 ( J ) = B 1 ( H ) ˘ ⊗ B 1 ( J ) , (402) B 1 ( H ⊗ J ) R = B 1 ( H ) R b ⊗ B 1 ( J ) R = B 1 ( H ) R ˘ ⊗ B 1 ( J ) R and D ( H ⊗ J ) = D ( H ⊗ J ) b (403) — wher e al l e qualities have to b e understo o d as set e qualities — and the norms ∥ · ∥ ⊗ 1 (tr ac e norm) and ∥ · ∥ b ⊗ 1 (pr oje ctive norm) ar e e quivalent on B 1 ( H ⊗ J ) , wher e as their r estrictions to the r e al Banach sp ac e B 1 ( H ⊗ J ) R ar e e quivalent to the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 . These norms, evaluate d on the density op er ators D ( H ⊗ J ) — i.e., the ∥ · ∥ ⊗ 1 -normalize d p ositive tr ac e class op er ators — satisfy the r elations 1 = ∥ D ∥ ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 ≤ M and 1 ≤ ∥ D ∥ b ⊗ 1 ≤ | | | D | | | b ⊗ 1 ≤ 2 ∥ D ∥ b ⊗ 1 ≤ 2 M , ∀ D ∈ D ( H ⊗ J ) . (404) Mor e over, given a density op er ator D ∈ D ( H ⊗ J ) , the fol lowing facts ar e e quivalent: • D ∈ D ( H ⊗ J ) se , i.e., D is a sep ar able state. • ∥ D ∥ b ⊗ 1 = 1 , i.e., D is normalize d wrt the pr oje ctive norm ∥ · ∥ b ⊗ 1 . • | | | D | | | b ⊗ 1 = 1 , i.e., D is normalize d wrt the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 . In p articular, if N ≡ dim( H ⊗ J ) < ∞ , then every sep ar able density op er ator ς ∈ D ( H ⊗ J ) se admits a finite c onvex de c omp osition of the form ς = K X k =1 p k ( π k ⊗ ϖ k ) , wher e K ≤ N 2 + 1 , p k > 0 ( P K k =1 p k = 1) , π k ∈ P ( H ) , ϖ k ∈ P ( J ) . (405) 86 Supp ose now that dim( H ) = dim( J ) = ∞ . Then, we have that B 1 ( H ⊗ J ) ⊋ B 1 ( H ) b ⊗ B 1 ( J ) ⊋ B 1 ( H ) ˘ ⊗ B 1 ( J ) , (406) B 1 ( H ⊗ J ) R ⊋ B 1 ( H ) R b ⊗ B 1 ( J ) R ⊋ B 1 ( H ) R ˘ ⊗ B 1 ( J ) R and D ( H ⊗ J ) ⊋ D ( H ⊗ J ) b , (407) — wher e al l r elations have to b e understo o d mer ely as strict set c ontainments — and the norms ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 ar e unb ounde d on the c onvex set D ( H ⊗ J ) b of al l cr oss states on H ⊗ J . Mor e over, given a density op er ator D ∈ D ( H ⊗ J ) , the fol lowing facts ar e e quivalent: • D ∈ D ( H ⊗ J ) se , i.e., D is a sep ar able state. • D ∈ D ( H ⊗ J ) b and ∥ D ∥ b ⊗ 1 = 1 , i.e., D is a cr oss state, and normalize d wrt the pr oje ctive norm ∥ · ∥ b ⊗ 1 . • D ∈ D ( H ⊗ J ) b and | | | D | | | b ⊗ 1 = 1 , i.e., D is a cr oss state, and normalize d wrt the Hermitian pr oje ctive norm | | | · | | | b ⊗ 1 . Final ly, for any dimension of the Hilb ert sp ac es H and J , and for every entangle d cr oss state D ∈ D ( H ⊗ J ) b — i.e., for every D ∈ D ( H ⊗ J ) b \ D ( H ⊗ J ) se — | | | D | | | b ⊗ 1 ≥ ∥ D ∥ b ⊗ 1 > 1 . Pr o of. F or the first assertion of the theorem, see Theorem 3.3, and, for the inequalities (404), see, in particular, relations (180) and (183) therein. F or the equiv alence of all the v arious c haracterizations of the set D ( H ⊗ J ) se of separable states, see Theorem 5.4 and Corollary 5.1, that hold for any dimension of the Hilb ert spaces H , J of the bipartition. In particular, for N ≡ dim( H ⊗ J ) < ∞ , see Corollary 4.6. F o cusing now on the case where dim( H ) = dim( J ) = ∞ , relations (406) and (407) follow from Corollary 3.11, and, for the assertion concerning the un b oundedness of the norms ∥ · ∥ b ⊗ 1 and | | | · | | | b ⊗ 1 on D ( H ⊗ J ) b , see Corollary 3.10. Finally , with no assumption on the dimension of the Hilb ert spaces H and J , for every cross state D ∈ D ( H ⊗ J ) b , | | | D | | | b ⊗ 1 ≥ ∥ D ∥ b ⊗ 1 ≥ 1 (Corollary 4.5), and, for every entangle d cross state D ∈ D ( H ⊗ J ) b , ∥ D ∥ b ⊗ 1  = 1; hence, for every D ∈ D ( H ⊗ J ) b \ D ( H ⊗ J ) se , | | | D | | | b ⊗ 1 ≥ ∥ D ∥ b ⊗ 1 > 1. 6 The pro jectiv e norm and the en tanglemen t function F or every densit y op erator D ∈ D ( H ⊗ J ), let us define the quantit y E ( D ) : = sup {| tr( D L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } , (408) where ∥ · ∥ Γ is the injective norm of B ( H ⊗ J ); see (346). Considering the case where L = I on the rhs of (408), we see that 1 ≤ E ( D )( ≤ + ∞ ). W e then obtain an extended real-v alued function E : D ( H ⊗ J ) ∋ D 7→ E ( D ) ∈ [1 , + ∞ ] , (409) that will b e called the entanglement function . W e will pro ve that the minim um v alue E ( D ) = 1 is actually attained by the function E on certain density op erators D , and — in the gen uinely infinite-dimensional setting where dim( H ) = dim( J ) = ∞ — the v alue + ∞ is attained to o. Therefore, in the following, where appropriate, standard arithmetics and ordering of the extended real n umber system should b e understo o d. The reader will also notice that, for ev ery cross state D ∈ D ( H ⊗ J ) b , E ( D ) is precisely the norm ∥ Ξ D ∥ B = ∥ D ∥ b ⊗ 1 of the b ounded linear functional Ξ D : B ( H ⊗ J ) Γ ∋ L 7→ tr( D L ) ∈ C , where B ( H ⊗ J ) Γ denotes the normed space  B ( H ⊗ J ) , ∥ · ∥ Γ  (recall Remark 5.7). Sp ecifically — in the case where min { dim( H ) , dim( J ) } < ∞ , and hence, by 87 Theorem 3.3, D ( H ⊗ J ) = D ( H ⊗ J ) b — the en tanglement function E is just the restriction to D ( H ⊗ J ) of the pro jective norm ∥ · ∥ b ⊗ 1 . If dim( H ) = dim( J ) = ∞ , instead, D ( H ⊗ J ) b ⊊ D ( H ⊗ J ) (Corollary 3.11), and E coincides with ∥ · ∥ b ⊗ 1 on the cross states D ( H ⊗ J ) b , whereas it takes the v alue + ∞ on all those states that are not cross states. In fact, the follo wing result holds: Theorem 6.1. The extende d r e al-value d entanglement function E : D ( H ⊗ J ) → [1 , + ∞ ] satisfies the fol lowing pr op erties: (E1) E ( tD 1 + (1 − t ) D 2 ) ≤ t E ( D 1 ) + (1 − t ) E ( D 2 ) , for al l t ∈ [0 , 1] and D 1 , D 2 ∈ D ( H ⊗ J ) ; (E2) E ( D ) = ∥ D ∥ b ⊗ 1 , for al l D ∈ D ( H ⊗ J ) b ; (E3) E ( D ) = + ∞ , for al l D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b , i.e., for al l density op er ators D on H ⊗ J that ar e not cr oss states (which form a nonempty set iff dim( H ) = dim( J ) = ∞ ); (E4) E is lower semic ontinuous wrt the standar d top olo gy of D ( H ⊗ J ) ; (E5) E ( D ) = 1 if D ∈ D ( H ⊗ J ) se ( D separable) , wher e as E ( D ) > 1 if D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se ( D entangled) ; (E6) E ( D ) = sup { tr( D F ) : F ∈ F ( H ) ˘ ⊗ F ( J ) , F = F ∗ , ∥ F ∥ Γ = 1 } ; (E7) for every D ∈ D ( H ⊗ J ) , and every p air U , V of unitary op er ators on H and J , r esp e ctively, E ( D ) = E (( U ⊗ V ) D ( U ⊗ V ) ∗ ) ; (410) (E8) for every r ∈ [1 , + ∞ ) , the sublevel set D ( H ⊗ J ) r : = { D ∈ D ( H ⊗ J ) : E ( D ) ≤ r } of E is close d in D ( H ⊗ J ) , wrt the standar d top olo gy of D ( H ⊗ J ) . Pr o of. Property (E1) holds by the fact that, for every t ∈ [0 , 1], every L ∈ B ( H ⊗ J ) and any D 1 , D 2 ∈ D ( H ⊗ J ), we hav e: | tr(( tD 1 + (1 − t ) D 2 ) L ) | ≤ t | tr( D 1 L ) | + (1 − t ) | tr( D 2 L ) | ; hence, if D = tD 1 + (1 − t ) D 2 , then E ( D ) : = sup {| tr( D L ) | : L ∈ A ( H ⊗ J ) , ∥ L ∥ Γ = 1 } ≤ sup { t | tr( D 1 L ) | + (1 − t ) | tr( D 2 L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } ≤ t E ( D 1 ) + (1 − t ) E ( D 2 ) . (411) Next, by definition (408), and by Theorem 5.2 (see relation (353)), prop ert y (E2) holds to o. Let us now pro ve (E3) . W e reason by con tradiction. Supp osing that dim( H ) = dim( J ) = ∞ — otherwise, by Theorem 3.3, D ( H ⊗ J ) = D ( H ⊗ J ) b — let D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b , and assume that E ( D ) < + ∞ . Then, since C ( H ) ˘ ⊗ C ( J ) ⊂ B ( H ⊗ J ), we ha ve: + ∞ > E ( D ) : = sup {| tr( D L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } ≥ sup {| tr( D K ) | : K ∈ C ( H ) ˘ ⊗ C ( J ) , ∥ K ∥ Γ = 1 } . (412) Therefore, the mapping C ( H ) ˘ ⊗ C ( J ) ∋ K 7→ tr( DK ) ∈ C is a b ounded functional on the normed space ( C ( H ) ˘ ⊗ C ( J ) , ∥ · ∥ Γ ), that (uniquely) extends to a b ounded functional on the the Banach space completion of this normed space, namely , to the injectiv e tensor pro duct C ( H ) ˇ ⊗ C ( J ). By Theorem 5.2 — see (354) — the isomorphism of Banac h spaces B 1 ( H ) b ⊗ B 1 ( J ) ∼ =  C ( H ) ˇ ⊗ C ( J )  ∗ holds, this isomorphism b eing implemented by the map κ C (definition (355)). Th us, there m ust b e a cross trace class op erator C ∈ B 1 ( H ) b ⊗ B 1 ( J ) such that tr( D K ) = κ C ( K ) = tr( C K ), for all K ∈ C ( H ) ˘ ⊗ C ( J ). But this conclusion would imply that, actually , D = C — e.g., b ecause 88 tr( D ( | ϕ k ⟩ ⟨ ϕ l | ⊗ | ψ m ⟩ ⟨ ψ n | )) = tr( C ( | ϕ k ⟩ ⟨ ϕ l | ⊗ | ψ m ⟩ ⟨ ψ n | )) = ⟨ ϕ l ⊗ ψ n , C ( ϕ k ⊗ ψ m ) ⟩ , where { ϕ k } k ∈ N , { ϕ m } m ∈ N are orthonormal bases in H and J , resp ectiv ely — whic h leads us to a con tradiction wrt our initial assumption that D ∈ B 1 ( H ) b ⊗ B 1 ( J ). T o prov e (E4) , it is sufficient to note that the entanglemen t function E : D ( H ⊗ J ) → [1 , + ∞ ] is defined as the p oin twise supremum o ver a collection of functions of the form D ( H ⊗ J ) ∋ D 7→ | tr( D L ) | ∈ R , L ∈ B ( H ⊗ J ). Since B ( H ⊗ J ) = B 1 ( H ⊗ J ) ∗ , all these functions are contin uous — hence, low er semicontin uous — wrt the relative top ology on D ( H ⊗ J ) induced by the (trace) norm top ology of B 1 ( H ⊗ J ), i.e., wrt the standard top ology of D ( H ⊗ J ) (F act 2.7). Therefore, by a well-kno wn result (see, e.g., Proposition 7.11 of [52]), E is low er semicontin uous wrt the standard top ology of D ( H ⊗ J ). Let us prov e (E5) . By Theorem 5.4, D ( H ⊗ J ) se = D ( H ⊗ J ) S , i.e., for D ∈ D ( H ⊗ J ) b , E ( D ) = ∥ D ∥ b ⊗ 1 = 1 iff D ∈ D ( H ⊗ J ) se . Therefore, E ( D ) > 1 iff D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se ; in particular, E ( D ) = ∥ D ∥ b ⊗ 1 > 1, if D ∈ D ( H ⊗ J ) b \ D ( H ⊗ J ) se (see Theorem 5.5), and E ( D ) = + ∞ , if D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b (by prop ert y (E3) of E ). W e next prov e prop ert y (E6) . Arguing as in Remark 5.5, but with A ( H ⊗ J ) replaced with F ( H ) ˘ ⊗ F ( J ), one shows that (E6) holds true for ev ery D ∈ D ( H ⊗ J ) b , because, in this case, E ( D ) = ∥ D ∥ b ⊗ 1 . Let us now supp ose that, instead, D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . By the pro of of (E3) , we know that the linear functional C ( H ) ˘ ⊗ C ( J ) ∋ K 7→ tr( D K ) ∈ C is not b ounded in this case, and therefore — recalling final assertion of Prop osition 5.5 — it cannot b e b ounded on the ∥ · ∥ Γ -dense linear subspace F ( H ) ˘ ⊗ F ( J ) of C ( H ) ˘ ⊗ C ( J ) (and hence of C ( H ) ˘ ⊗ C ( J )), as w ell. Therefore, we ha ve: sup {| tr( D F ) | : F ∈ F ( H ) ˘ ⊗ F ( J ) , ∥ F ∥ Γ = 1 } = + ∞ = E ( D ) . (413) A t this p oin t, we can reason as in Remark 5.5; i.e., considering, for every F ∈ F ( H ) ˘ ⊗ F ( J ), the selfadjoin t comp onen t F ℜ : = 1 2 ( F + F ∗ ) ∈ F ( H ) ˘ ⊗ F ( J ) of F , w e ha ve that + ∞ = E ( D ) = sup {| tr( D F ) | : F ∈ F ( H ) ˘ ⊗ F ( J ) , ∥ F ∥ Γ = 1 } = sup {|ℜ e(tr( D F )) | : F ∈ F ( H ) ˘ ⊗ F ( J ) , ∥ F ∥ Γ = 1 } = sup {| tr( D F ℜ ) | : F ∈ F ( H ) ˘ ⊗ F ( J ) , ∥ F ∥ Γ = 1 } = sup {| tr( D F ) | : F ∈ F ( H ) ˘ ⊗ F ( J ) , F = F ∗ , ∥ F ∥ Γ = 1 } = sup { tr( D F ) : F ∈ F ( H ) ˘ ⊗ F ( J ) , F = F ∗ , ∥ F ∥ Γ = 1 } . (414) Here, the second equality follo ws from the fact that ∥ F ∥ Γ = ∥ z F ∥ Γ , for every z ∈ T , while, for the third equalit y , notice that ℜ e(tr( DF )) = 1 2 (tr( D F ) + tr( D F ) ∗ ) = 1 2 (tr( D F ) + tr( D F ∗ )) = tr( D F ℜ ), where we ha ve used the fact that D = D ∗ and the cyclic prop ert y of the trace. F or the fourth equalit y , observe that ∥ F ℜ ∥ Γ ≤ 1 2 ( ∥ F ∥ Γ + ∥ F ∗ ∥ Γ ) = ∥ F ∥ Γ , which would imply that the supremum on the third line is smaller that the suprem um on the fourth line; but, comparing the latter with the suprem um on the first line, w e see that the rev erse inequality holds to o, and then we hav e an equalit y . The fifth line obtains just noting that | tr( D F ) | = max { tr( D F ) , tr( D ( − F )) } therein. Let us prov e property (E7) . Since tr(( U ⊗ V ) D ( U ⊗ V ) ∗ F ) = tr( D ( U ⊗ V ) ∗ F ( U ⊗ V )), for ev ery F ∈ F ( H ) ˘ ⊗ F ( J ), and, moreov er, the set { F ∈ F ( H ) ˘ ⊗ F ( J ) : F = F ∗ , ∥ F ∥ Γ = 1 } is inv ariant wrt the lo cal unitary transformation F 7→ ( U ⊗ V ) ∗ F ( U ⊗ V ) (recall that, by relation (347), ∥ F ∥ Γ = ∥ ( U ⊗ V ) ∗ F ( U ⊗ V ) ∥ Γ ), this is a straigh tforward consequence of prop erty (E6) . Let us finally pro v e prop ert y (E8) . In fact, b y (E4) , the en tanglemen t function E : D ( H ⊗ J ) → [1 , + ∞ ] is lo wer semicon tin uous wrt the standard topology of D ( H ⊗ J ); i.e., for ev ery r ∈ [1 , + ∞ ), the sup erlev el set { D ∈ D ( H ⊗ J ) : E ( D ) > r } of E is op en, or, equiv alen tly , its complement — the sublev el set D ( H ⊗ J ) r — is closed in D ( H ⊗ J ), wrt the standard top ology of D ( H ⊗ J ). 89 R emark 6.1 . Note that the entanglemen t function E : D ( H ⊗ J ) → [1 , + ∞ ] can b e extended, in a natural wa y to the whole bipartite trace class B 1 ( H ⊗ J ). T o the b est of our knowledge, the first prop osal of an entanglemen t measure based on the pro jective norm is due to Rudolph [68], who considered a finite-dimensional setting. A general entanglemen t function of the kind w e are considering here w as defined, later on, by Arveson — see Part 2 of [25] — who also pro ved that, in the case where at most one of the Hilb ert spaces of the bipartition (or, more generally , of a m ultipartition) is infinite-dimensional, the en tanglement function coincides with the pro jectiv e norm. Here, we hav e prov ed that, in the gen uinely infinite-dimensional setting, the entanglemen t function coincides with the pro jective norm when r estricte d to cr oss states , whereas it assumes the v alue + ∞ pr e cisely on all other states. R emark 6.2 . Note that { F ∈ F ( H ) ˘ ⊗ F ( J ) : F = F ∗ } = F ( H ) R ˘ ⊗ F ( J ) R , where F ( H ) R , F ( J ) R are the linear spaces of finite-rank selfadjoin t operators on H and J . In fact, the set on the lhs of the equalit y contains the set on the rhs. Conv ersely , if F = P k G k ⊗ H k ∈ F ( H ) ˘ ⊗ F ( J ) and F = F ∗ , then F = 1 2 ( F + F ∗ ) = 1 4 P k  G k + G ∗ k  ⊗  H k + H ∗ k  + 1 4 P k  i  G k − G ∗ k  ⊗  − i  H k − H ∗ k  ∈ F ( H ) R ˘ ⊗ F ( J ) R . Therefore, b y prop ert y (E6) in Theorem 6.1, we ha ve: E ( D ) = sup { tr( D F ) : F ∈ F ( H ) R ˘ ⊗ F ( J ) R , ∥ F ∥ Γ = 1 } , ∀ D ∈ D ( H ⊗ J ) . (415) R emark 6.3 . Observe that, for every densit y op erator D ∈ D ( H ⊗ J ), we ha ve: E ( D ) = sup {| tr( D K ) | : K ∈ A ( H ⊗ J ) , ∥ K ∥ Γ = 1 } = sup {| tr( D K ) | : K ∈ A ( H ⊗ J ) , K = K ∗ , ∥ K ∥ Γ = 1 } = sup { tr( D K ) : K ∈ A ( H ⊗ J ) , K = K ∗ , ∥ K ∥ Γ = 1 } . (416) These relations follo w immediately from definition (408) and prop ert y (E6) in Theorem 5.2, simply noting that { F ∈ F ( H ) ˘ ⊗ F ( J ) : F = F ∗ } ⊂ { K ∈ A ( H ⊗ J ) : K = K ∗ } ⊂ A ( H ⊗ J ) ⊂ B ( H ⊗ J ), and, for obtaining the third equalit y , that, if K = K ∗ , then | tr( D K ) | = max { tr( D K ) , tr( D ( − K )) } . By the same reasoning, the suprem um in (408) can b e tak en o ver the selfadjoin t part B ( H ⊗ J ) R of B ( H ⊗ J ) only , and, in this case, one may replace | tr( D L ) | with tr( D L ) therein. R emark 6.4 . Let st( B ( H ⊗ J )) b e the conv ex set of al l states of the C ∗ -algebra B ( H ⊗ J ), i.e., the set of all normalized, p ositiv e (hence, b ounded) functionals on B ( H ⊗ J ) [6, 10, 11, 42]: st( B ( H ⊗ J ))) : = { ℧ ∈ B ( H ⊗ J ) ∗ : ℧ ≥ 0 , ℧ ( I ) = 1 } . (417) The entanglemen t function E : D ( H ⊗ J ) → [1 , + ∞ ] admits a natural extension, i.e., E e : st( B ( H ⊗ J )) ∋ ℧ 7→ E e ( ℧ ) ∈ [1 , + ∞ ] , (418) where, for ev ery ℧ ∈ st( B ( H ⊗ J )), we set E e ( ℧ ) : = sup {| ℧ ( B ) | : B ∈ B ( H ⊗ J ) , ∥ B ∥ Γ = 1 } = sup {| ℧ ( B ) | : B ∈ B ( H ⊗ J ) R , ∥ B ∥ Γ = 1 } . (419) The second line ab o ve obtains arguing as in Remark 5.5, and taking into account the well-kno wn fact that ℧ ( B ∗ ) = ℧ ( B ) ∗ , for every p ositiv e functional on the C ∗ -algebra B ( H ⊗ J ) (see, e.g., Theorem 3.3.2 of [42]). It is worth observing that, if dim( H ⊗ J ) = ∞ , in the definition of the extended entanglemen t function E e one may not replace the C ∗ -algebra B ( H ⊗ J ) with, say , the smaller algebra C ∗ -algebra A ( H ⊗ J ), as it can b e done for the entanglemen t function E (recall Remark 6.3). Indeed, there are states of the C ∗ -algebra B ( H ⊗ J ) — which form the con v ex set 90 st( B ( H ⊗ J )) s of singular states — that v anish iden tically on the compact operators C ( H ⊗ J ). Precisely , ev ery state ℧ ∈ st( B ( H ⊗ J )) can be uniquely decomp osed into a conv ex com bination ℧ = p ℧ D + (1 − p ) ℧ s , p ∈ [0 , 1], where ℧ D = tr( D ( · )), for some D ∈ D ( H ⊗ J ), is a normal state of the C ∗ -algebra B ( H ⊗ J ), whereas ℧ s ∈ st( B ( H ⊗ J )) s is a singular state; see Section 3, in Chapter 8 of [10], and Prop osition 10.4.3 of [44]. R emark 6.5 . Along lines similar to those follow ed for introducing the entanglemen t function E , one can define an Hermitian entanglement function H : D ( H ⊗ J ) → [1 , + ∞ ], whic h is an extended real-v alued entanglemen t function such that H ( D ) = | | | D | | | b ⊗ 1 , for all cross states D ∈ D ( H ⊗ J ) b , and H ( D ) = + ∞ , for all other states D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . Corollary 6.1. The c onvex subset D ( H ⊗ J ) b of D ( H ⊗ J ) admits the fol lowing char acterization: D ( H ⊗ J ) b =  D ∈ D ( H ⊗ J ) : ξ D = tr( D ( · )) ∈ A ( H ⊗ J ) ∗ Γ  . (420) Mor e over, for every D ∈ D ( H ⊗ J ) b , ∥ ξ D ∥ ∗ Γ = E ( D ) = ∥ D ∥ b ⊗ 1 , and henc e D ( H ⊗ J ) se =  D ∈ D ( H ⊗ J ) : ξ D ∈ A ( H ⊗ J ) ∗ Γ and ∥ ξ D ∥ ∗ Γ = 1  . (421) Pr o of. By Remark 6.3, and by p oin ts (E2) and (E3) in Theorem 6.1, giv en a density op erator D ∈ D ( H ⊗ J ), the asso ciated functional ξ D : A ( H ⊗ J ) Γ ∋ K 7→ tr( D K ) ∈ C is b ounded iff D ∈ D ( H ⊗ J ) b ; moreov er, for every D ∈ D ( H ⊗ J ) b , ∥ ξ D ∥ ∗ Γ = E ( D ) = ∥ D ∥ b ⊗ 1 , and hence, by p oin t (E5) in Theorem 6.1, the c haracterization (421) of D ( H ⊗ J ) se holds. In the case where at least one of the Hilb ert spaces H and J is finite-dimensional, the c haracteri- zations (395) and (421) of D ( H ⊗ J ) se are equiv alent, because, in this case, D ( H ⊗ J ) b = D ( H ⊗ J ) and, moreo v er, the Banach space A ( H ⊗ J ) ∗ Γ ≡  A ( H ⊗ J ) , ∥ · ∥ Γ  ∗ is a renorming of the closed subspace f ( B 1 ( H ⊗ J )) ∼ = B 1 ( H ⊗ J ) of A ( H ⊗ J ) ∗ ; whereas, in the genuinely infinite- dimensional case — i.e., if dim( H ) = dim( J ) = ∞ — D ( H ⊗ J ) b ⊊ D ( H ⊗ J ), and, for every D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b , the linear functional ξ D : A ( H ⊗ J ) Γ → C is not b ounded. Therefore, in the latter case, relation (421) pro vides a stronger c haracterization of separable states. The entanglemen t function has a regular b ehavior wrt to restriction to (or extension from) subspaces of the Hilb ert spaces of the bipartition: Prop osition 6.1. L et V , W b e close d subsp ac es of the Hilb ert sp ac es H and J , r esp e ctively, and let E W V : D ( V ⊗ W ) → [1 , + ∞ ] b e the entanglement function on D ( V ⊗ W ) . Then, for every D ∈ D ( V ⊗ W ) , E W V ( D ) = E ( D ) , (422) wher e, with a slight abuse, we have identifie d the density op er ator D on V ⊗ W with its image via the natur al (isometric) emb e dding of B 1 ( V ⊗ W ) into B 1 ( H ⊗ J ) . Pr o of. Let π ∈ B ( H ), ϖ ∈ B ( J ) b e the orthogonal pro jections onto V and W . The Banac h space B ( V ⊗ W ) can b e identified with the closed subspace  L ∈ B ( H ⊗ J ) : L = ( π ⊗ ϖ ) L ( π ⊗ ϖ )  of B ( H ⊗ J ), i.e., the range of the con tinuous pro jection B ( H ⊗ J ) ∋ L 7→ ( π ⊗ ϖ ) L ( π ⊗ ϖ ) ∈ B ( H ⊗ J ) (whic h is a norm-closed b y a well-kno wn result; see, e.g., Corollary 3.2.10 of [46]). Let us denote b y ∥ · ∥ Γ the injective norm of B ( V ⊗ W ) (defined as in (345)); namely , for ev ery L ∈ B ( V ⊗ W ), ∥ L ∥ Γ : = sup  | tr( C L ) | : C ∈ B 1 ( V ) b ⊗ B 1 ( W ) , W V ∥ C ∥ b ⊗ 1 = 1  , where W V ∥ · ∥ b ⊗ 1 denotes the norm of the pro jective tensor product B 1 ( V ) b ⊗ B 1 ( W ) (recall that, b y Prop osition 3.13, W V ∥ C ∥ b ⊗ 1 = ∥ C ∥ b ⊗ 1 ). Observ e that ∥ · ∥ Γ coincides with the restriction to B ( V ⊗ W ) of the injective norm of B ( H ⊗ J ) (whic h justifies our use of a unique sym b ol for denoting b oth norms). This fact can b e easily 91 c heck ed, e.g., recalling relation (346), and observing that, for ev ery L ∈ B ( V ⊗ W ) — i.e., for ev ery L ∈ B ( H ⊗ J ) suc h that L = ( π ⊗ ϖ ) L ( π ⊗ ϖ ) — we ha v e: ∥ L ∥ Γ = sup {|⟨ ϕ ⊗ ψ , L ( η ⊗ χ ) ⟩| : ϕ, η ∈ V , ψ , χ ∈ W , ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } = sup {|⟨ ϕ ⊗ ψ , L ( η ⊗ χ ) ⟩| : ϕ, η ∈ H , ψ , χ ∈ J , ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1 } . (423) By the previous facts, it follows that, for every D ∈ D ( V ⊗ W ), E W V ( D ) = sup {| tr( DL ) | : L ∈ B ( H ⊗ J ) , L = ( π ⊗ ϖ ) L ( π ⊗ ϖ ) , ∥ L ∥ Γ = 1 } ≤ E ( D ) = sup {| tr(( π ⊗ ϖ ) D ( π ⊗ ϖ ) L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } = sup {| tr( D ( π ⊗ ϖ ) L ( π ⊗ ϖ )) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } ≤ sup {| tr( D ( π ⊗ ϖ ) L ( π ⊗ ϖ )) | : L ∈ B ( H ⊗ J ) , ∥ ( π ⊗ ϖ ) L ( π ⊗ ϖ ) ∥ Γ = 1 } = sup {| tr( D L ) | : L ∈ B ( H ⊗ J ) , L = ( π ⊗ ϖ ) L ( π ⊗ ϖ ) , ∥ L ∥ Γ = 1 } = E W V ( D ) . (424) Here, the first (and the last) equality , and the first inequality , hold by the fact that the injectiv e norm of B ( V ⊗ W ) coincides with the restriction to B ( V ⊗ W ) of the injective norm of B ( H ⊗ J ), while, for the second inequalit y , we ha ve used the fact that ∥ ( π ⊗ ϖ ) L ( π ⊗ ϖ ) ∥ Γ = sup {|⟨ ϕ ⊗ ψ , L ( η ⊗ χ ) ⟩| : ϕ, η ∈ V , ψ , χ ∈ W , ∥ ϕ ∥ = · · · = ∥ χ ∥ = 1 } ≤ ∥ L ∥ Γ , ∀ ∈ B ( H ⊗ J ) . (425) Therefore, in conclusion, for ev ery densit y op erator D ∈ D ( V ⊗ W ), E W V ( D ) = E ( D ). The entanglemen t function E is well-behav ed wrt the action of suitable ‘lo cal quantum maps’; i.e., the action of such a map do es not incr e ase the entanglemen t of a cross state. T o illustrate this fact, let us recall that a an y p ositive, tr ac e-pr eserving line ar map E : B 1 ( H ) → B 1 ( H ) ∈ L 1 ( H ) — in short, a PTP map — is b ounded and mild ly c ontr active , i.e., ∥ E ∥ [1] = 1 (see Proposition 1 of [59]). The following result that further characterizes the entanglemen t function E holds: Prop osition 6.2. L et E : B 1 ( H ) → B 1 ( H ) , G : B 1 ( J ) → B 1 ( J ) b e p ositive, tr ac e-pr eserving line ar maps. Then, ther e is a unique b ounde d line ar map E b ⊗ G : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ) b ⊗ B 1 ( J ) such that  E b ⊗ G  ( S ⊗ T ) = E ( S ) ⊗ G ( T ) , for al l S ∈ B 1 ( H ) and T ∈ B 1 ( J ) . The map E b ⊗ G is tr ac e pr eserving, i.e., tr  E b ⊗ G  ( C )  = tr( C ) , for al l C ∈ B 1 ( H ) b ⊗ B 1 ( J ) . If it is also p ositive — namely, if C ≥ 0 = ⇒  E b ⊗ G  ( C ) ≥ 0 — then we have: •  E b ⊗ G  ( D ( H ⊗ J ) b ) ⊂ D ( H ⊗ J ) b . • F or every D ∈ D ( H ⊗ J ) b , E  E b ⊗ G  ( D )  ≤ E ( D ) . Pr o of. The linear maps E : B 1 ( H ) → B 1 ( H ) and G : B 1 ( J ) → B 1 ( J ), b eing p ositiv e and trace- preserving, are b ounded, and ∥ E ∥ [1] = ∥ G ∥ [1] = 1. By Corollary 3.12, there is a unique b ounded linear map E b ⊗ G : B 1 ( H ) b ⊗ B 1 ( J ) → B 1 ( H ) b ⊗ B 1 ( J ) such that  E b ⊗ G  ( S ⊗ T ) = E ( S ) ⊗ G ( T ), for all S ∈ B 1 ( H ) and T ∈ B 1 ( J ), and, moreov er,   E b ⊗ G   B = ∥ E ∥ [1] ∥ G ∥ [1] = 1. Let us pro ve that this map is trace-preserving. In fact, for every C ∈ B 1 ( H ) b ⊗ B 1 ( J ), there is a decomp osition of the form C = ∥ · ∥ b ⊗ 1 – P k S k ⊗ T k , with T k ⊗ S k ∈ θ ( B 1 ( H ) , B 1 ( J )) (Corollary 3.5). Then, w e ha v e that tr  E b ⊗ G  ( C )  = tr  ∥ · ∥ b ⊗ 1 – P k  E b ⊗ G  ( S k ⊗ T k )  = P k tr( E ( S k )) tr( G ( T k )) = tr( C ) , (426) where, for the last equalit y , we hav e used the fact that E and G are trace-preserving. Therefore, if E b ⊗ G is also p ositive, then  E b ⊗ G  ( D ( H ⊗ J ) b ) ⊂ D ( H ⊗ J ) b . Moreov er, since   E b ⊗ G   B = 1, then for ev ery D ∈ D ( H ⊗ J ) b , E  E b ⊗ G  ( D )  =    E b ⊗ G  ( D )   b ⊗ 1 ≤ ∥ D ∥ b ⊗ 1 = E ( D ). 92 W e stress that the hypothesis that the map E b ⊗ G in Prop osition 6.2 b e p ositive is nontrivial, and is realized, e.g., in the case where b oth the Hilb ert spaces H , J are finite-dimensional, and, moreo ver, the PTP maps E and G therein are, in particular, c ompletely p ositive [69, 70]. By Theorem 6.1, we will now prov e that the entanglemen t of any given state D ∈ D ( H ⊗ J ) can b e detected b y a sp ecial t yp e of quan tum observ ables, a so-called entanglement witness (for the en tangled state D ) [26, 27]. Precisely , for ev ery entangle d state D ∈ D ( H ⊗ J ), there is a quantum observ able E on H ⊗ J , suc h that the exp ectation v alue of E is strictly larger than 1 — and, in particular, by a suitable c hoice of E , arbitrarily close to E ( D ) > 1, if E ( D ) < + ∞ (i.e., if D is a cross state) — when measuring the observ able on D , whereas it is not larger than 1, not even in mo dulus, when measuring the observ able on any sep ar able state ς ∈ D ( H ⊗ J ) se . T o mak e a long story short, a state is entangled iff it admits a witness observ able. Corollary 6.2 (Existence of suitable en tanglement witnesses) . L et D b e an entangle d state on the bip artite Hilb ert sp ac e H ⊗ J ; i.e., let D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se . Then, ther e exists a b ounde d selfadjoint op er ator E on H ⊗ J (an entanglement witness for D ) satisfying the c onditions E ∈ F ( H ) R ˘ ⊗ F ( J ) R , ∥ E ∥ Γ = 1 (427) — wher e F ( H ) R is the line ar sp ac e of finite-r ank selfadjoint op er ators on H — and such that tr( D E ) > 1 , wher e as | tr( ς E ) | ≤ 1 , ∀ ς ∈ D ( H ⊗ J ) se . (428) Mor e over, if D ∈ D ( H ⊗ J ) b \ D ( H ⊗ J ) se — i.e., if D b e an entangle d cr oss state — then, for every ϵ > 0 , the finite-r ank selfadjoint op er ator E ∈ F ( H ) R ˘ ⊗ F ( J ) R c an b e chosen in such a way that ∥ D ∥ b ⊗ 1 − ϵ < tr( D E ) ≤ ∥ D ∥ b ⊗ 1 = E ( D ) . (429) If, inste ad, D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b — i.e., if D is not a cr oss state, and henc e E ( D ) = + ∞ — then, for every n ∈ N , the finite-r ank selfadjoint op er ator E ∈ F ( H ) R ˘ ⊗ F ( J ) R c an b e chosen in such a way that tr( D E ) > n . (430) Pr o of. Let D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se . By prop erties (E5) and (E6) in Theorem 6.1, and taking in to accoun t Remark 6.2, w e ha ve: 1 < E ( D ) = sup { tr( D F ) : F ∈ F ( H ) R ˘ ⊗ F ( J ) R , ∥ F ∥ Γ = 1 } . (431) Th us, if D ∈ D ( H ⊗ J ) b \ D ( H ⊗ J ) se , then by prop ert y (E2) in Theorem 6.1, E ( D ) = ∥ D ∥ b ⊗ 1 , and, for ev ery ϵ > 0, there is some E ∈ F ( H ) R ˘ ⊗ F ( J ) R — with ∥ E ∥ Γ = 1 — suc h that ∥ D ∥ b ⊗ 1 − ϵ < tr( D E ) ≤ ∥ D ∥ b ⊗ 1 = E ( D ). Assume that, in particular, 0 < ϵ < ∥ D ∥ b ⊗ 1 − 1, and hence 1 < ∥ D ∥ b ⊗ 1 − ϵ < tr( D E ) ≤ ∥ D ∥ b ⊗ 1 = E ( D ) . (432) Therefore, w e can and do c ho ose E in suc h a wa y that tr( D E ) > 1 and condition (429) is satisfied. If, instead, D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b — i.e., if D is not a cross state — then E ( D ) = + ∞ , and hence, by (431), for ev ery n ∈ N , there is some E ∈ F ( H ) R ˘ ⊗ F ( J ) R — with ∥ E ∥ Γ = 1 — suc h that tr( D E ) > n ≥ 1 (thus, also condition (430) is satisfied). In b oth cases, tr( D E ) > 1, and, by Theorem 5.4 (or b y prop erty (E5) in Theorem 6.1), for ev ery separable state ς ∈ D ( H ⊗ J ) se , we ha ve that E ( ς ) = ∥ ς ∥ b ⊗ 1 = 1; hence, taking in to account that ∥ E ∥ Γ = 1, w e ha ve: | tr( ς E ) | ≤ E ( ς ) = ∥ ς ∥ b ⊗ 1 = 1, for all ς ∈ D ( H ⊗ J ) se . Thus, the b ounded op erator E satisfies all the required conditions, and the pro of is complete. 93 R emark 6.6 . T o conform to the con ven tion adopted in the literature [26, 27], w e should redefine our en tanglement witness by considering the b ounded selfadjoin t op erator W = I − E ∈ A ( H ⊗ J ), where E ∈ F ( H ) R ˘ ⊗ F ( J ) R is as in Corollary 6.2, so that the expectation v alue of W is strictly smaller than zero when measuring the witness on a given en tangled state D (whose entanglemen t w e wish to detect), whereas it is not smaller than zero when measuring the observ able on any separable state. The standard pro of relies on a suitable geometric version of the Hahn-Banach theorem [1]. The adv antage of our presen t result (and definition of the entanglemen t witness) is t wofold. On the one hand, w e pro ve that a witness can alw ays be c hosen, for ev ery dimension of the lo cal Hilb ert spaces H and J , in the linear space F ( H ) R ˘ ⊗ F ( J ) R of all finite linear combinations of finite-rank lo c al observables . On the other hand, w e show that this witness can be chosen in suc h a w a y as to appro ximate (by measuring it), the v alue E ( D ) > 1 of the en tanglement function on the en tangled state D we are in terested to detect, pro vided that D is a cross state, i.e., E ( D ) < + ∞ . Definition 6.1 (Entanglemen t witnesses) . W e call a bounded selfadjoin t op erator E ∈ B ( H ⊗ J ) R an entanglement witness if | tr( DE ) | > 1, for some entangled state D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se , whereas, for every separable state ς ∈ D ( H ⊗ J ) se , | tr( ς E ) | ≤ 1; in suc h a case, w e s a y that E is an entanglement witness for D . Moreov er, w e say that an en tangled state D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se is dete cte d by a certain entanglemen t witness E ∈ B ( H ⊗ J ) R if | tr( D E ) | > 1. Prop osition 6.3. If E ∈ B ( H ⊗ J ) R is an entanglement witness, then ∥ E ∥ ⊗ ∞ > 1 . A b ounde d selfadjoint op er ator E ∈ B ( H ⊗ J ) R is an entanglement witness if one of the fol lowing two e quivalent c onditions is satisfie d: (W1) ∥ E ∥ Γ = 1 and | tr( D E ) | > 1 , for some state D ∈ D ( H ⊗ J ) (which in then entangle d). (W2) ∥ E ∥ ⊗ ∞ > ∥ E ∥ Γ = 1 . Every entangle d state D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se is dete cte d by some entanglement witness; in p articular, it c an b e dete cte d by a finite-r ank entanglement witness E such that ∥ E ∥ Γ = 1 , and, whenever D is a cr oss state, one c an further r e quir e that the exp e ctation value tr( D E ) of the observable E when me asuring D is arbitr arily close to E ( D ) , with 1 < tr( D E ) ≤ E ( D ) < + ∞ . Pr o of. If E ∈ B ( H ⊗ J ) R is an en tanglement witness, then | tr( D E ) | > 1, for some entangled state D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) se , so that, by F act 2.6, ∥ E ∥ ⊗ ∞ > 1. Let us now pro ve the second assertion. In fact, if ∥ E ∥ Γ = 1, then | tr( ς E ) | ≤ 1, for every separable state ς ∈ D ( H ⊗ J ) se , b ecause sup {| tr( ς E ) | : ς ∈ D ( H ⊗ J ) se } ≤ sup  | tr( C E ) | : C ∈ B 1 ( H ) b ⊗ B 1 ( J ) , ∥ C ∥ b ⊗ 1 = 1  = : ∥ E ∥ Γ = 1, where the inequality holds by the fact that ς ∈ D ( H ⊗ J ) se = ⇒ ∥ ς ∥ b ⊗ 1 = 1. Thus, if, in addition, | tr( D E ) | > 1, for some state D ∈ D ( H ⊗ J ), then this state must b e en tangled, and hence E is an entanglemen t witness; moreov er, b y F act 2.6, ∥ E ∥ ⊗ ∞ > 1 = ∥ E ∥ Γ , and hence condition (W1) implies (W2) . Now, if E ∈ B ( H ⊗ J ) R is suc h that ∥ E ∥ ⊗ ∞ > 1, then, again by F act 2.6, | tr( D E ) | > 1, for some state D ∈ D ( H ⊗ J ); hence, condition (W2) implies (W1) , and b oth conditions imply that E is an entanglemen t witness. This prov es the second assertion. The third assertion is essen tially a rephrasing of Corollary 6.2. Let us now study the pro jective norm ∥ · ∥ b ⊗ 1 and the entanglemen t function on the pur e states P ( H ⊗ J ) (the rank-one pro jections on H ⊗ J ). W e will denote b y P ( H ⊗ J ) b : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) b = P ( H ⊗ J ) ∩ B 1 ( H ) b ⊗ B 1 ( J ) (433) the set of all pure states on H ⊗ J that are cr oss states. 94 Example 6.1. Let a ∈ H ⊗ J b e a normalized nonzero vector — ∥ a ∥ = 1 — and let a = s X l =1 a l ( ϕ l ⊗ ψ l ) , 1 ≤ s = srank ( a ) ≤ min { dim( H ) , dim( J ) } , { a l } s l =1 = scfs ( a ) ⊂ R + ∗ , (434) b e the Sch midt decomposition of a (recall F act 3.1). As usual, w e set b a ≡ c aa : = | a ⟩ ⟨ a | ∈ P ( H ⊗ J ). Then, we ha ve the expansion b a = s X k, l =1 a k a l  [ ϕ k ϕ l ⊗ [ ψ k ψ l  , [ ϕ k ϕ l ⊗ [ ψ k ψ l ≡ | ϕ k ⟩ ⟨ ϕ l | ⊗ | ψ k ⟩ ⟨ ψ l | = | ϕ k ⊗ ψ k ⟩ ⟨ ϕ l ⊗ ψ l | , (435) whic h con verges absolutely wrt the pro jective norm ∥ · ∥ b ⊗ 1 if P s k, l =1 a k a l = P s k =1 a k P s l =1 a l < ∞ — i.e., if P s l =1 a l < ∞ — and, otherwise, conv ergence wrt the trace norm ∥ · ∥ ⊗ 1 should b e understo o d, namely , b a = ∥ · ∥ ⊗ 1 – P s k, l =1 a k a l | ϕ k ⊗ ψ k ⟩ ⟨ ϕ l ⊗ ψ l | . Precisely , ∥ · ∥ ⊗ 1 -con vergence ` a la Pringsheim holds; i.e., for every ϵ > 0, there is some N ϵ ∈ N such that ∥ b a − P m k =1 P n l =1 a k a l ( | ϕ k ⊗ ψ k ⟩ ⟨ ϕ l ⊗ ψ l | ) ∥ ⊗ 1 < ϵ , for all m, n > N ϵ . Let us also consider the related vector c N ≡ c N ( a ) = P N l =1 ϕ l ⊗ ψ l ∈ H ⊗ J , — where 1 ≤ N ≤ s , if s = srank ( a ) < ∞ , and N ∈ N , otherwise — and the asso ciated rank-one p ositiv e op erator E N ≡ E N ( b a ) : = | c N ⟩ ⟨ c N | = N X k, l =1 [ ϕ k ϕ l ⊗ [ ψ k ψ l ∈  F ( H ) ˘ ⊗ F ( J )  ∩  R + ∗ P ( H ⊗ J )  . (436) Recalling relation (346), w e easily conclude that ∥ E N ∥ Γ = 1, b ecause ⟨ ϕ 1 ⊗ ψ 1 , E N ( ϕ 1 ⊗ ψ 1 ) ⟩ = 1 and, moreov er, for every ϕ, η ∈ H , ψ , χ ∈ J , with ∥ ϕ ∥ = ∥ η ∥ = ∥ ψ ∥ = ∥ χ ∥ = 1, we hav e 1 ≤ |⟨ ϕ ⊗ ψ , E N ( η ⊗ χ ) ⟩| = |⟨ ϕ ⊗ ψ , c N ⟩⟨ c N , η ⊗ χ ⟩| =   P N k, l =1 ⟨ ϕ ⊗ ψ , ϕ k ⊗ ψ k ⟩⟨ ϕ l ⊗ ψ l , η ⊗ χ ⟩   =   P N k, l =1 ⟨ ϕ, ϕ k ⟩⟨ ψ , ψ k ⟩⟨ ϕ l , η ⟩⟨ ψ l , χ ⟩   =   P N k =1 α k β k     P N l =1 γ l δ l   ≤ 1 . (437) Here, α k ≡ ⟨ ϕ k , ϕ ⟩ , β k ≡ ⟨ ψ , ψ k ⟩ , γ l ≡ ⟨ η , ϕ l ⟩ and δ l ≡ ⟨ ψ l , χ ⟩ — with P N k =1 | α k | 2 ≤ ∥ ϕ ∥ 2 = 1, P N k =1 | β k | 2 ≤ ∥ ψ ∥ 2 = 1, P N l =1 | γ l | 2 ≤ ∥ η ∥ 2 = 1 and P N l =1 | δ l | 2 ≤ ∥ χ ∥ 2 = 1 — so that the final inequalit y follo ws directly from the Cauch y-Sc hw arz inequality . Notice that ∥ E N ∥ ⊗ ∞ = ∥ c N ∥ 2 = N ; hence, for every N > 1, ∥ E N ∥ ⊗ ∞ > ∥ E N ∥ Γ = 1, so that E N is an entanglemen t witness (recall the sufficient condition (W2) in Prop osition 6.3). Let us assume, at first, that a ∈ H ⊗ J is suc h that s = srank ( a ) < ∞ ; hence, b a ∈ P ( H ⊗ J ) b , b ecause, by (435), b a ∈ F ( H ) ˘ ⊗ F ( J ). Exploiting the fact that, by Theorem 5.2 (see rela- tion (353)), ∥ b a ∥ b ⊗ 1 = sup  | tr( b a F ) | = |⟨ a , F a ⟩| : F ∈ F ( H ) ˘ ⊗ F ( J ) , ∥ F ∥ Γ = 1  (438) and the triangle inequality , one concludes that, for ev ery N ≤ s ,  P N l =1 a l  2 = ⟨ a , c N ⟩ 2 = |⟨ a , c N ⟩| 2 = tr( b a E N ) ≤ ∥ b a ∥ b ⊗ 1 ≤ P s k, l =1 a k a l   [ ϕ k ϕ l ⊗ [ ψ k ψ l   b ⊗ 1 =  P s l =1 a l  2 , (439) 95 where E N ∈  F ( H ) ˘ ⊗ F ( J )  ∩  R + ∗ P ( H ⊗ J )  , with ∥ E N ∥ Γ = 1, is the rank-one positive op erator defined b y (436). Then, the estimate (439), for N = s = srank ( a ), provides a simple expression of ∥ b a ∥ b ⊗ 1 in terms of the Sc hmidt co efficien ts of a , i.e., ∥ b a ∥ b ⊗ 1 =  P s l =1 a l  2 = 1 + P k  = l a k a l , (440) where, for obtaining the second equality , w e hav e used the fact that P s k =1 a 2 k = 1. It is clear that, if min { dim( H ) , dim( J ) } < ∞ (hence, s = srank ( a ) < ∞ ), this is actually the general expression of the pro jectiv e norm of a pure state in P ( H ⊗ J ) = P ( H ⊗ J ) b . Let us now fo cus on the genuinely infinite-dimensional case where dim( H ) = dim( J ) = ∞ . W e claim that, in this case, P ( H ⊗ J ) ⊋ P ( H ⊗ J ) b ; sp ecifically , given a normalized nonzero vector a ∈ H ⊗ J — with s = srank ( a ) ≤ ∞ — we ha ve that b a ∈ P ( H ⊗ J ) b ⇐ ⇒ s X l =1 a l < ∞ (441) and, for every b a ∈ P ( H ⊗ J ) b , ∥ b a ∥ b ⊗ 1 =  P s l =1 a l  2 . Otherwise stated, b a ∈ P ( H ⊗ J ) b iff either srank ( a ) < ∞ , or srank ( a ) = ∞ and the sequence its Schmidt co efficien ts { a l } l ∈ N is contained in ℓ 1 ⊊ ℓ 2 (i.e., { a l } l ∈ N is, up to normalization, a probability distribution). As a consequence, we ha ve: P ( H ⊗ J ) \ P ( H ⊗ J ) b =  b a ∈ P ( H ⊗ J ) : srank ( a ) = ∞ and { a l } l ∈ N ∈ ℓ 2 \ ℓ 1   = ∅ . (442) Moreo ver, relation (440) actually provides the most general expression of the pro jective norm of a pure state b a in P ( H ⊗ J ) b . Let us prov e relation (441). W e first consider the “if part” of the claim. Then, let a ∈ H ⊗ J b e a normalized nonzero vector, with a Schmidt decomp osition of the form (434), and suc h that P s l =1 a l < ∞ . In this case, as previously noted, the expansion (435) is absolutely conv ergen t wrt the pro jectiv e norm ∥ · ∥ b ⊗ 1 if P s k, l =1 a k a l = P s k =1 a k P s l =1 a l < ∞ , i.e., if P s l =1 a l < ∞ ; hence, b a ∈ P ( H ⊗ J ) b . Consider, now, the “only if part” of the claim. It is sufficien t to supp ose that b a ∈ P ( H ⊗ J ) b , s = srank ( a ) = ∞ and P ∞ l =1 a l = ∞ , and show that this assumption leads to a con tradiction. T o this end, let us put, for every n ∈ N , b a n : = | a n ⟩ ⟨ a n | = n X k, l =1 a k a l  [ ϕ k ϕ l ⊗ [ ψ k ψ l  ∈ B 1 ( H ) ˘ ⊗ B 1 ( J ) , where a n : = n X l =1 a l ( ϕ l ⊗ ψ l ) . (443) As previously noted, b a = ∥ · ∥ ⊗ 1 – lim m,n | a m ⟩ ⟨ a n | = ∥ · ∥ ⊗ 1 – lim n b a n , whereas — since, b y relation (440) (that, of course, admits a direct generalization to an y rank-one p ositiv e op erator), ∥ b a n ∥ b ⊗ 1 =  P n l =1 a l  2 = ⇒ lim n ∥ b a n ∥ b ⊗ 1 = ∞ (444) — the sequence  b a n  n ∈ N cannot conv erge to b a wrt the pro jective norm. This fact, how ever, do es not automatic al ly imply that b a ∈ P ( H ⊗ J ) b . Let us prov e that this is indeed the case; i.e., that w e find a con tradiction wrt our initial assumption. Consider the orthogonal pro jection op erator P n : =  P n k =1 [ ϕ k ϕ k  ⊗  P n l =1 [ ψ l ψ l  = n X k, l =1 [ ϕ k ϕ k ⊗ [ ψ l ψ l (445) on to the finite-dimensional subspace span { ϕ k ⊗ ψ l : k , l = 1 , . . . , n } of H ⊗ J . Note that a n = P n a and b a n = P n b a P n , (446) 96 and, moreov er, P n = π n ⊗ ϖ n , where π n = P n k =1 [ ϕ k ϕ k , ϖ n = P n l =1 [ ψ l ψ l are orthogonal pro jections on to span { ϕ k : k = 1 , . . . , n } ⊂ H and span { ψ l : l = 1 , . . . , n } ⊂ J , resp ectiv ely . Hence, by Prop osition 3.15 (see relation (167)), ∥ b a n ∥ b ⊗ 1 ≤ ∥ b a ∥ b ⊗ 1 , ∀ n ∈ N . (447) Therefore, b y (444) and (447), we must conclude that if srank ( a ) = ∞ and P ∞ l =1 a l = ∞ , then b a = ∥ · ∥ ⊗ 1 – lim n b a n ∈ P ( H ⊗ J ) b . Ha ving pro ved our first claim (441), let us next pro ve that, for every pure state b a ∈ P ( H ⊗ J ) b , relation (440) holds true. By what previously prov ed, w e kno w that P ∞ l =1 a l < ∞ . A t this p oin t, defining the rank-one p ositiv e op erator b a n as in (443), in this case we ha v e that b a = ∥ · ∥ b ⊗ 1 – lim n b a n and ∥ b a n ∥ b ⊗ 1 =  P n l =1 a l  2 = ⇒ ∥ b a ∥ b ⊗ 1 = lim n ∥ b a n ∥ b ⊗ 1 =  P ∞ l =1 a l  2 . (448) Summarizing, w e ha v e no w sho wn — with no assumption on the dimension of the Hilbert spaces H and J — that a pure state b a ∈ P ( H ⊗ J ) belongs to P ( H ⊗ J ) b : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) b iff P s l =1 a l < ∞ , where { a l } s l =1 is the set of the Sc hmidt co efficients of a , and, moreo ver, for ev ery b a ∈ P ( H ⊗ J ) b , ∥ b a ∥ b ⊗ 1 =  P s l =1 a l  2 = 1 + P k  = l a k a l . It is w orth observing that the expansion (435) of b a ∈ P ( H ⊗ J ) b , which conv erges absolutely wrt the pro jective norm ∥ · ∥ b ⊗ 1 , is an optimal standar d de c omp osition — recall Definition 3.8 — b ecause   [ ϕ k ϕ l ⊗ [ ψ k ψ l   b ⊗ 1 =   [ ϕ k ϕ l   1   [ ψ k ψ l   1 = 1 (449) and ∥ b a ∥ b ⊗ 1 = P s k, l =1 a k a l . By the previous results, and by Theorem 6.1, the v alue of the en tanglement function (409) on a pure state is no w completely determined, i.e., E ( b a ) = ( ∥ b a ∥ b ⊗ 1 =  P s l =1 a l  2 if b a ∈ P ( H ⊗ J ) b ⇐ ⇒ P s l =1 a l < ∞ + ∞ if b a ∈ P ( H ⊗ J ) \ P ( H ⊗ J ) b ⇐ ⇒ P s l =1 a l = ∞ . (450) Note that, in the case where M min { dim( H ) , dim( J ) } < ∞ (hence, s = srank ( a ) ≤ M < ∞ ), b a is maximal ly entangle d iff s = M and, moreov er, a 1 = · · · = a s = 1 / √ s , and, in suc h a case, E ( b a ) = ∥ b a ∥ b ⊗ 1 = s = M . Ev en if the fact that the entanglemen t function is not finite on those pure states that are not cross states is a consequence of Theorem 6.1, it is in teresting to see a direct pro of of this fact. F or every b a ∈ P ( H ⊗ J ), we ha ve: E ( b a ) : = sup {| tr( b a L ) | : L ∈ B ( H ⊗ J ) , ∥ L ∥ Γ = 1 } . (451) Assume that dim( H ) = dim( J ) = ∞ and b a ∈ P ( H ⊗ J ) \ P ( H ⊗ J ) b (hence, s = srank ( a ) = ∞ ). W e claim that, in this case, E ( b a ) = + ∞ . In fact, by relation (451), and b y the fact that E N , N ∈ N , is a p ositiv e op erator in F ( H ) ˘ ⊗ F ( J ) such that ∥ E N ∥ Γ = 1, we obtain the estimate E ( b a ) ≥ tr( b a E N ) = |⟨ a , c N ⟩| 2 = ⟨ a , c N ⟩ 2 =  P N l =1 a l  2 , ∀ N ∈ N . (452) A t this p oin t, since b a ∈ P ( H ⊗ J ) \ P ( H ⊗ J ) b = ⇒ lim N P N l =1 a l = P ∞ l =1 a l = ∞ , we conclude that E ( b a ) = + ∞ , coheren tly with the prop ert y (E3) in Theorem 6.1. Finally , observ e that, if the pure state b a ∈ P ( H ⊗ J ) is en tangled — i.e., if E ( b a ) > 1 (or, equiv alently , if s = srank ( a ) > 1) — then, by the estimates (439) and (452), for N > 1 sufficiently large, the rank-one p ositiv e op erator E N ∈  F ( H ) ˘ ⊗ F ( J )  ∩  R + ∗ P ( H ⊗ J )  , with ∥ E N ∥ Γ = 1, defined b y (436), is an entanglement witness for b a , because tr( b a E N ) > 1 (recall the suffic ien t condition (W1) in Prop osition 6.3). 97 In Example 6.1, w e ha ve actually pro ved the following result, which completely c haracterizes the pro jectiv e norm and the entanglemen t of pure states on H ⊗ J : Prop osition 6.4. L et a ∈ H ⊗ J b e a nonzer o ve ctor — with ∥ a ∥ = 1 — and let a = s X l =1 a l ( ϕ l ⊗ ψ l ) , 1 ≤ s = srank ( a ) ≤ min { dim( H ) , dim( J ) } , { a l } s l =1 = scfs ( a ) ⊂ R + ∗ , (453) b e the Schmidt de c omp osition of a , wher e s = srank ( a ) denotes the Schmidt r ank of a . L et us set b a ≡ c aa : = | a ⟩ ⟨ a | . Then, b a ∈ P ( H ⊗ J ) b : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) b iff P s l =1 a l < ∞ . In the c ase wher e M ≡ min { dim( H ) , dim( J ) } < ∞ , the pr oje ctive norm and the entanglement of the pur e state b a ∈ P ( H ⊗ J ) = P ( H ⊗ J ) b ar e pr ovide d by the expr ession E ( b a ) = ∥ b a ∥ b ⊗ 1 =  P s l =1 a l  2 = 1 + P k  = l a k a l . (454) Henc e, in this c ase, the pr oje ctive norm and the entanglement of any pur e state b a ∈ P ( H ⊗ J ) satisfy the r elations 1 ≤ E ( b a ) = ∥ b a ∥ b ⊗ 1 ≤ s = srank ( a ) ≤ M , (455) wher e al l ine qualities c an b e satur ate d. In p articular, b a is sep ar able iff E ( b a ) = ∥ b a ∥ b ⊗ 1 = s = 1 , wher e as it is entangle d iff 1 < E ( b a ) = ∥ b a ∥ b ⊗ 1 ≤ s = srank ( a ) ≤ M . Supp ose now that dim( H ) = dim( J ) = ∞ . Then, the set P ( H ⊗ J ) b ⊊ P ( H ⊗ J ) (456) of those pur e states on H ⊗ J that ar e cr oss states c onsists of those states of the form b a : = | a ⟩ ⟨ a | , wher e a ∈ H ⊗ J — ∥ a ∥ = 1 — is such that either srank ( a ) < ∞ , or srank ( a ) = ∞ and the se quenc e of its Schmidt c o efficients { a l } l ∈ N is c ontaine d in ℓ 1 ⊊ ℓ 2 (i.e., P ∞ l =1 a l < ∞ and P ∞ l =1 a 2 l = 1) . Mor e over, for every b a ∈ P ( H ⊗ J ) b , and given the Schmidt de c omp osition (453) , we have that 1 ≤ E ( b a ) = ∥ b a ∥ b ⊗ 1 =  P s l =1 a l  2 < ∞ , (457) wher e as, for every b a ∈ P ( H ⊗ J ) \ P ( H ⊗ J ) b  = ∅ , E ( b a ) = + ∞ . In this c ase, b a is sep ar able iff E ( b a ) = ∥ b a ∥ b ⊗ 1 = s = 1 , wher e as it is entangle d iff 1 < E ( b a ) ≤ s = srank ( a ) ≤ ∞ . Final ly, every element of P ( H ⊗ J ) b , for any dimension of the Hilb ert sp ac es H and J , is ∥ · ∥ b ⊗ 1 -optimal ly de c omp osable. The preceding results allow us to prov e the following important facts: Lemma 6.1. Supp osing that dim( H ) = dim( J ) = ∞ , let P ∈ P ( H ⊗ J ) b e such that E ( P ) = + ∞ ; i.e., let P ∈ P ( H ⊗ J ) \ P ( H ⊗ J ) b . Then, for every D 0 ∈ D ( H ⊗ J ) and every p ∈ (0 , 1] , the density op er ator D = p P + (1 − p ) D 0 is such that E ( D ) = + ∞ ; i.e., D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . Pr o of. Suppose that dim( H ) = dim( J ) = ∞ and the rank-one pro jection P = b a ≡ | a ⟩ ⟨ a | — for some normalized nonzero vector a ∈ H ⊗ J — do es not b elong to P ( H ⊗ J ) b , and let a = s X l =1 a l ( ϕ l ⊗ ψ l ) , s = srank ( a ) , { a l } s l =1 = scfs ( a ) ⊂ R + ∗ , (458) b e the Sc hmidt decomp osition of a . Then, b y Prop osition 6.4, w e m ust ha v e that s = srank ( a ) = ∞ and, moreo v er, b a ∈ P ( H ⊗ J ) b = ⇒ P ∞ l =1 a l = ∞ . No w, b y prop ert y (E6) in Theorem 6.1, w e ha ve: E ( D ) = sup { tr( D F ) : F ∈ F ( H ) ˘ ⊗ F ( J ) , F = F ∗ , ∥ F ∥ Γ = 1 } . (459) 98 T o exploit this useful expression of ∥ D ∥ b ⊗ 1 , let us consider, as in Example 6.1, the vector c N = P N l =1 ϕ l ⊗ ψ l ∈ H ⊗ J , N ∈ N , and the asso ciated rank-one p ositiv e op erator E N : = | c N ⟩ ⟨ c N | = N X k, l =1 [ ϕ k ϕ l ⊗ [ ψ k ψ l ∈  F ( H ) ˘ ⊗ F ( J )  ∩  R + ∗ P ( H ⊗ J )  . (460) As argued in Example 6.1 (see (437)), ∥ E N ∥ Γ = 1. Therefore, b y relation (459), and b y the fact that E N is a p ositiv e op erator in F ( H ) ˘ ⊗ F ( J ) suc h that ∥ E N ∥ Γ = 1, for the density op erator D = p P + (1 − p ) D 0 — with D 0 ∈ D ( H ⊗ J ) and p ∈ (0 , 1] — we obtain the estimate E ( D ) ≥ tr( D E N ) = p tr( P E N ) + (1 − p ) tr( D 0 E N ) ≥ p tr( P E N ) = p |⟨ a , c N ⟩| 2 = p ⟨ a , c N ⟩ 2 = p  P N l =1 a l  2 , ∀ N ∈ N . (461) A t this p oin t, since lim N P N l =1 a l = P ∞ l =1 a l = ∞ and p > 0, b y the previous estimate we see that E ( D ) = + ∞ ; i.e., by property (E3) in Theorem 6.1, D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . Theorem 6.2. L et D = P j p j P j — with p j > 0 , P j p j = 1 and P j ∈ P ( H ⊗ J ) — b e the sp e ctr al de c omp osition of a cr oss st ate D ∈ D ( H ⊗ J ) b , expr esse d in terms of minimal pr oje ctions (and c onver ging wrt the tr ac e norm ∥ · ∥ ⊗ 1 ). Then, we have that { P j } ⊂ P ( H ⊗ J ) b : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) b . (462) Pr o of. Our first claim is ob vious in the case where min { dim( H ) , dim( J ) } < ∞ , b ecause, in this case, D ( H ⊗ J ) = D ( H ⊗ J ) b and P ( H ⊗ J ) = P ( H ⊗ J ) b . Let us then supp ose that dim( H ) = dim( J ) = ∞ , and let D = P j p j P j ∈ D ( H ⊗ J ) b ⊊ D ( H ⊗ J ), where the set { P j } consists of minimal — i.e., rank-one — sp ectral pro jections of D . Reasoning by con tradiction, assume that there is some rank-one pro jection P m = b a ≡ | a ⟩ ⟨ a | ∈ { P j } (for some normalized nonzero v ector a ∈ H ⊗ J ), that do es not b elong to P ( H ⊗ J ) b . Then, we ha ve that D = p m P m + (1 − p m ) D 0 , where D 0 = P j  = m p j 1 − p m P j ∈ D ( H ⊗ J ). By Lemma 6.1, it follo ws that E ( D ) = + ∞ ; i.e., D ∈ D ( H ⊗ J ) \ D ( H ⊗ J ) b . Th us, we are led to a contradiction wrt our initial assumption that D ∈ D ( H ⊗ J ) b and there exists some element P m of the set of rank-one pro jections { P j } such that P m ∈ P ( H ⊗ J ) b . Thus, actually , { P j } ⊂ P ( H ⊗ J ) b . W e are now able to characterize the extreme p oin ts of the conv ex set D ( H ⊗ J ) b . Corollary 6.3. P ( H ⊗ J ) b : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) b = ext( D ( H ⊗ J ) b ) . Pr o of. First note that, since D ( H ⊗ J ) b ⊂ D ( H ⊗ J ), then, by F act 2.17, w e ha ve: P ( H ⊗ J ) b : = P ( H ⊗ J ) ∩ D ( H ⊗ J ) b = ext( D ( H ⊗ J )) ∩ D ( H ⊗ J ) b ⊂ ext( D ( H ⊗ J ) b ). Then, to prov e our claim, w e need to pro v e the rev erse con tainment relation, i.e., that P ( H ⊗ J ) b ⊃ ext( D ( H ⊗ J ) b ), as w ell. In fact, let D ∈ D ( H ⊗ J ) b \ P ( H ⊗ J ) b , and let D = P j p j P j — with p j > 0, P j p j = 1 and P j ∈ P ( H ⊗ J ) — b e the sp ectral decomp osition of the cross state D , expressed in terms of rank-one pro jections. Note that, here, the set { P j } must contain at least t wo elements, b ecause, otherwise, w e w ould ha ve that D = P 1 ∈ P ( H ⊗ J ) b , which would contradict our preceding assumption. Thus, we can write D = p 1 P 1 + (1 − p 1 ) D 1 , where D 1 = P j > 1 p j 1 − p 1 P j ∈ D ( H ⊗ J ). But actually , since D is a cross state, then, by Theorem 6.2, { P j } ⊂ P ( H ⊗ J ) b . Thus, in particular, P 1 ∈ D ( H ⊗ J ) b , and hence D 1 = 1 1 − p 1 D + p 1 p 1 − 1 P 1 ∈ D ( H ⊗ J ) b = D ( H ⊗ J ) ∩ B 1 ( H ) b ⊗ B 1 ( J ) , (463) as well. 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