Invariant measures of randomized quantum trajectories

INV ARIANT MEASURES OF RANDOMIZED QUANTUM TRAJECTORIES TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL Abstra ct. Quan tum tra jectories are Mark ov c hains mo deling quantum systems sub jected to re- p eated indirect measurements. Their stationary regime dep ends on what observ ables are measured on the prob es used to indirectly measure the system. In this article w e explore the prop erties of quan tum tra jectories when the c hoice of probe observ able is randomized. The randomization induces some regularization of the quantum tra jectories. W e show that non-singular randomiza- tion ensures that quantum tra jectories purify and therefore accept a unique inv ariant probability measure. W e furthermore study the regularity of that in v ariant measure. In that endeav our, w e in tro duce a new notion of ergodicity for quan tum c hannels, whic h w e call m ultiplicativ e primitivit y . It is a priory stronger than primitivity but w eak er than positivity impro ving. Finally , we compute some inv ariant measures for canonical quan tum channels and explore the limits of our assumptions with sev eral examples. Contents 1. In tro duction 1 2. Setup and assumptions 3 3. Main results 5 4. On multiplicativ e primitivity 7 5. Pro of of the main results 8 6. Examples 13 App endix A. On the diﬀerent notions of irreducibilit y 17 App endix B. Space of op erators induced by instruments 18 App endix C. GAP Measures 20 App endix D. SageMath 10 co de listing 22 References 25 1. Intr oduction Quan tum systems sub jected to rep eated indirect measurements ev olv e randomly according to sto c hastic pro cesses called quan tum tra jectories. They typically mo del exp erimen ts in quan tum optics – see for example [ WM10 ; HR06 ]. In [ MK06 ], Maassen and Kümmerer prov ed that the state of the system has a tendency to purify along the quantum tra jectory , meaning that, if it starts in a mixed state, it has a tendency to get closer to the set of pure states, reaching it asymptotically under the righ t conditions. That property was leveraged in [ Ben+19 ] to pro v e uniqueness of the in v ariant measure for quantum tra jectories. The pro of of that fundamen tal result required a new approac h since standard tec hniques used for Marko v chains are not eﬃcien t for quan tum tra jectories. Indeed, they are neither φ -irreducible nor contracting in general – see [ BFP23 , Section 8]. While inspired b y w orks on random products of matrices – esp ecially [ GLP16 ] – new techniques were developed Date : March 31, 2026. 2020 Mathematics Subje ct Classiﬁc ation. 60J05, 81P15. Key wor ds and phr ases. Quantum tra jectories, Marko v chains, In v ariant measures, GAP measure, Quantum mea- suremen t theory . 2 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL to deal with non-in v ertible and non strongly irreducible matrices. F ollo wing [ Ben+19 ], several ﬁner limit theorems ha v e b een pro v ed in [ BFP23 ; BHP25 ]. The pro ofs of [ BHP25 ] were obtained through the establishmen t of a sp ectral gap for the Mark o v k ernel. In [ BPS24 ], the set of inv arian t measures has b een classiﬁed when puriﬁcation do es not hold. The a v erage evolution of a quan tum tra jectory results from the repeated application of a quan- tum c hannel Φ ∗ 1 to the initial densit y matrix state of the system. A ﬁrst condition required for the uniqueness of the inv arian t measure is that Φ ∗ accepts a unique inv arian t state. As already men- tioned, the second assumption used to prov e uniqueness in [ Ben+19 ] is that the quan tum tra jectory puriﬁes. That prop ert y not only depends on Φ ∗ , but also on a Kraus decomp osition that is c hosen for Φ ∗ : b y the Kraus decomposition theorem, w e may alw a ys write Φ ∗ ( ρ ) = k X i =1 v i ρv ∗ i , for k ∈ N large enough and ( v i ) k i =1 some indexed family of matrices. This c hoice of matrices is not unique. It corresp onds to a choice of observ able, or more precisely orthonormal basis, measured on the prob e used to indirectly measure the system – see [ Bus+16 , Theorem 7.14]. Dep ending on the c hoice of ( v i ) i , for a ﬁxed Φ ∗ , puriﬁcation ma y or may not happ en. In the present article, we explore the prop erties of quan tum tra jectories when the c hoice of decom- p osition is randomized at eac h time step. In particular, w e are in terested in the agnostic p osition c haracterized b y c hoosing uniformly the orthonormal basis along which the prob e is measured. This represen ts a canonical example of a quan tum tra jectory that gains regularit y from the randomiza- tion. All our results assume the quantum channel Φ ∗ is irreducible, so it accepts a unique inv ariant state of full rank. Our ﬁrst main result concerns conditions for puriﬁcation. In Theorem 3.3 , we pro v e that if the randomization is not singular, then puriﬁcation holds. Thanks to [ Ben+19 , Theorem 1.1], this implies uniqueness of the inv ariant measure. A ctually , in Lemma 5.2 , w e pro v e more generally that an y informationally complete instrumen t leads to puriﬁcation. Our second main result is that, con trary to the general case as prov ed in [ BFP23 , Section 8], suﬃcien tly randomized quantum tra jectories are φ -irreducible with resp ect to the uniform measure on the set of states – see Theorem 3.5 . If moreov er a matrix v i is in v ertible, then the in v arian t measure is equiv alent to the uniform one. T o pro v e these results we introduce a new notion of ergo dicit y for quan tum channels w e call multiplicativ e primitivit y – see Deﬁnition 2.4 . It is a priory stronger than primitivit y ( i.e. , irreducible and ap eriodic) but w eak er than positivity impro ving ( Φ ∗ ( ρ ) > 0 for any ρ ≥ 0 ) – see Prop ositions 4.1 and 4.2 . In dimension 2 , multiplicativ e primitivity and primitivity are equiv alen t – see Proposition 6.1 . Our third main result concerns the sp ecial case of a uniformly randomized orthonormal basis measured on the prob e. Here, symmetries of th e c hannel Φ ∗ translate in to symmetries of the in v ariant measure of the quan tum tra jectory – see Theorem 3.11 . The proof relies on the expression of the Mark o v c hain k ernel as a GAP measure as introduced in [ JR W94 ] (see also [ Gol+06 ; Gol+16 ]). W e illustrate our results with some examples. First, w e provide trivial examples of in v ariant measures: One is the GAP measure for some state ρ and the other is the uniform measure. Second, in dimension 2 , w e explicitly state the integral equation satisﬁed by the in v arian t probability density when the c hoice of measured probe’s orthonormal basis is uniform. The equation is a consequence of the explicit densit y expression for GAP measures from [ Gol+06 , Equation (18)], whose v alidity requires a nontrivial pro of that in dimension 2 the matrices v i are almost alwa ys inv ertible. W e 1 In this article, Φ ∗ denotes the completely positive, trace-preserving map acting on density matrices ρ , while its dual Φ is a unit-preserving map acting on observ ables. W e call b oth maps “quan tum c hannels”. UNIFORM QUANTUM TRAJECTORIES 3 then pro vide t w o examples of m ultiplicativ ely primitive c hannels in dimension 3 , one with in v ertible matrices and the other with only non-in v ertible matrices. These results raise sev eral questions. First: is the new notion of multiplicativ e primitivity nec- essary to prov e φ -irreducibility? Second: can one ﬁnd primitive coun ter-examples to m ultiplicativ e primitivit y in dimension 3 or higher? Finally , can an explicit expression of the inv ariant measure densit y b e deriv ed, at least for some simple nontrivial c hannels? More generally the present w ork motiv ates further study of quan tum tra jectories deﬁned b y randomized measurements. The article is structured as follows. In Section 2 , we pro vide the mathematical setup and state our main assumptions. In Section 3 , we state our main results. In Section 4 , we discuss the new m ultiplicativ e primitivity assumption. In Section 5 we gather all the pro ofs of our main results. In Section 6 , w e provide diﬀerent examples of c hannels for whic h we can compute the inv ariant measure. W e also discuss the speciﬁc case of dimension 2 and w e provide some mathematically relev ant examples in dimension 3 . In App endix A , w e recall some equiv alent c haracterizations of diﬀeren t notions of ergo dicit y for quan tum channels. In Appendix B , we derive the space of op erators generated b y informationally complete instrumen ts. In App endix C , w e summarize some prop erties of GAP measures useful to us. Finally , in App endix D , w e pro vide the SageMath 10 code w e used to pro v e that our examples in dimension 3 are multiplicativ ely primitiv e. 2. Setup and assumptions Consider the set M d ( C ) of d × d complex matrices equipp ed with its Borel σ -algebra. Let Φ : M d ( C ) → M d ( C ) be a completely p ositiv e unit preserving map, i.e. , a quantum channel. By Stinespring’s dilation theorem, or the Kraus decomposition theorem, there exists a ﬁnite tuple ( v i ) k i =1 ∈ M d ( C ) k suc h that Φ( X ) = k X i =1 v ∗ i X v i . The c hoice of k and ( v i ) k i =1 is not unique. Fixing k , following Stinespring’s theorem, any k -tuple of matrices ( w i ) k i =1 also satisﬁes Φ( X ) = k X i =1 w ∗ i X w i , if and only if there exist a u ∈ U ( k ) , with U ( k ) denoting the Lie group of k × k unitary matrices, suc h that w i = k X j =1 u ij v j . A set of suc h matrices { w i } i is called a Kraus decomp osition of Φ . The freedom in the c hoice of Kraus decomp osition implies that, for any probability measure λ on U ( k ) equipp ed with its Borel σ -algebra, denoting v i ( u ) = P k j =1 u ij v j ∈ M d ( C ) , Φ( X ) = k X i =1 Z U ( k ) v i ( u ) ∗ X v i ( u ) d λ ( u ) . The sum ov er i can b e absorb ed into the probabilit y measure: for i ∈ { 1 , . . . , k } , let u ( i ) ∈ U ( k ) with ﬁrst ro w being u ( i ) 1 l = δ l,i , and let λ i b e the image measure of λ b y the map u 7→ u ( i ) u . Then, with µ := k X i =1 λ i , (1) 4 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL w e may write Φ( X ) = Z U ( k ) v ( u ) ∗ X v ( u ) d µ ( u ) , where v : U ( k ) → M d ( C ) is deﬁned by v ( u ) = v 1 ( u ) . Note that µ is such that µ ( U ( k )) = k . It ensures that, for any ﬁxed x ∈ C d with ∥ x ∥ = 1 , ∥ v ( u ) x ∥ 2 2 d µ ( u ) deﬁnes a probabilit y measure on U ( k ) . Seeing Φ as the result of a unitary interaction b et w een a system and a prob e – see [ Bus+16 , Theorem 7.14] – the measure λ can b e interpreted as a random choice of observ able measured on the prob e. Then, conditioning on the measurement result and the detector adjustmen t, b oth of them summarized in u ∈ U ( k ) , the ev olution of the state v ector also b ecomes random. W e describ e state vectors ˆ x as elements of the pro jective space P( C d ) , which consists of equiv- alence classes ˆ x = { z x : z ∈ C } ∈ P( C d ) for any x ∈ C d \ { 0 } . Giv en ˆ x ∈ P( C d ) , we will denote b y x an arbitrary norm 1 representativ e of ˆ x . W e equip P( C d ) with its Borel σ -algebra and ν unif denotes the uniform probabilit y measure on P( C d ) . F urther, for w ∈ M d ( C ) and ˆ x ∈ P( C d ) such that w x  = 0 , we denote w · ˆ x = c w x . If w x = 0 w e set w · ˆ x arbitrarily . Then ν unif is the unique probabilit y measure on P( C d ) such that u · ˆ x has the same law as ˆ x for an y unitary matrix u . W e also often rely on Dirac’s notation, where a v ector x in a Hilb ert space is denoted | x ⟩ and its dual by ⟨ x | . So, | x ⟩ is not necessarily normalized and ⟨ x | | x ′ ⟩ = ⟨ x, x ′ ⟩ is the scalar pro duct of the Hilb ert space. It follows that, ⟨ wx || w x ⟩ = ⟨ x | w ∗ w | x ⟩ = ∥ w x ∥ 2 . In this notation, P( C d ) is in bijection with the set of rank-one orthogonal pro jectors on C d (pure state densit y matrices) through the map ˆ x 7→ | x ⟩⟨ x | . Giv en an initial datum ˆ x 0 ∈ P( C d ) and the abov e-described measure µ on U ( k ) , the random ev olution of the state vector is then described b y a Marko v c hain ( ˆ x n ) n ∈ N 0 2 with ˆ x n +1 = v ( U n ) · ˆ x n with U n ∼ ∥ v ( u ) x n ∥ 2 2 d µ ( u ) . This pro cess is what is called a quan tum tra jectory . The Mark o v k ernel of this chain is given, for an y b ounded measurable function f : P( C d ) → C , b y Π f ( ˆ x ) = E ( f ( ˆ x 1 ) | ˆ x 0 = ˆ x ) = Z U ( k ) f ( v ( u ) · ˆ x ) ∥ v ( u ) x ∥ 2 2 d µ ( u ) . F or probability measures ν o v er P( C d ) , ν Π denotes the probability measure deﬁned by E ν Π ( f ) = E ν (Π f ) for any b ounded measurable function f . In terms of ensembles, Π( ˆ x, A ) = Π 1 A ( ˆ x ) for any measurable subset A of P( C d ) , where 1 A is the c haracteristic function of A . If λ = δ u 0 for some u 0 ∈ U ( k ) , then the quan tum tra jectory corresponds to the evolution of the quantum system when a ﬁxed observ able is measured on the prob es. As already men tioned, b y con trast, taking λ to not be an extreme p oin t in the set of probability measures corresp onds to a randomization of the observ able measured on the prob es. In this article w e focus on suc h randomizations, and especially on those ones that ha v e some regularity with respect to a Haar measure on U ( k ) . Deﬁnition 2.1 (Non-singular µ ) . L et µ unif b e a Haar me asur e over U ( k ) . W e say that µ is non- singular if µ unif (supp µ ) > 0 . The choice µ = µ unif corresp onds to a random uniform c hoice of the basis measured on the probe at each time step. It corresp onds to an agnostic p osition with resp ect to the prob e measurement observ able. The only other assumptions we use concern the ergo dic prop erties of the quan tum c hannel Φ . 2 W e use the con ven tion N = { 1 , 2 . . . } and N 0 = { 0 } ∪ N . UNIFORM QUANTUM TRAJECTORIES 5 Deﬁnition 2.2 (Irreducible channels) . A quantum channel Φ is c al le d irr e ducible if ther e exists n ∈ N such that for any p ositive semi-deﬁnite X ∈ M d ( C ) , (Id M d ( C ) + Φ) n ( X ) is p ositive deﬁnite. This is the standard assumption leading to the P erron–F rob enius theorem for positive maps on ﬁnite-dimensional C ∗ -algebras – see [ EHK78 ]. A stronger assumption requires that Φ n maps p ositiv e semi-deﬁnite matrices to p ositiv e deﬁnite ones. It is called primitivit y and implies irreducibilit y . Deﬁnition 2.3 (Primitiv e c hannels) . A quantum channel Φ is c al le d primitive if ther e exists n ∈ N such that for any p ositive semi-deﬁnite X ∈ M d ( C ) , Φ n ( X ) is p ositive deﬁnite. There exists a notion of p erio d for irreducible quan tum c hannels – see [ EHK78 ]. Primitiv e maps are actually the ap eriodic irreducible maps – see the same reference. F or some of our results, we assume a new, stronger notion of primitivity , whic h we call m ulti- plicativ e primitivity . Giv en a quantum c hannel Φ , for p ∈ N w e deﬁne (2) V p := linspan { v i 1 · · · v i p : ( i 1 , . . . , i p ) ∈ { 1 , . . . , k } p } ⊂ M d ( C ) where the matrices ( v i ) k i =1 are a Kraus decomp osition of Φ . Note that, by linearit y , V p do es not dep end on the particular choice of Kraus decomposition. F rom Theorem A.2 , Φ is primitiv e if and only if for an y non-zero x ∈ C d , we get V p x = C d for some p ∈ N . Our stronger form of primitivity is the follo wing. Deﬁnition 2.4 (Multiplicative primitivity) . A quantum channel Φ is c al le d multiplic atively primi- tive if for any ˆ x ∈ P( C d ) , ther e exists p ∈ N such that V p 1 x := { a p · · · a 1 x : a 1 , . . . , a p ∈ V 1 } satisﬁes V p 1 x = C d . Since V 1 do es not dep end on the c hoice of Kraus decomposition, m ultiplicativ e primitivity is indeed a property of Φ . Since V p 1 ⊆ V p , Theorem A.2 yields that m ultiplicativ e primitivit y im- plies primitivity . The v alidity of the reverse implication in d ≥ 3 is, as far as we know, an op en question, while for d = 2 , we prov e this reverse implication in Prop osition 4.3 . W e further discuss m ultiplicativ e primitivity in Section 4 . W e are no w equipp ed to state our main results. 3. Main resul ts 3.1. Uniqueness of the inv ariant measure. Our ﬁrst result is the uniqueness of the inv arian t measure. T o formulate it we introduce the 1 -W asserstein distance W 1 ( ν 1 , ν 2 ) = inf P Z d ( ˆ x, ˆ y ) d P ( ˆ x, ˆ y ) , where d ( ˆ x, ˆ y ) = p 1 − |⟨ x, y ⟩| 2 is a metric o v er P( C d ) and the inﬁm um is taken o v er all probability measures P on P( C d ) × P( C d ) such that P ( · , P( C d )) = ν 1 and P (P( C d ) , · ) = ν 2 . Theorem 3.1. Assume Φ is irr e ducible and µ is non-singular, se e Deﬁnitions 2.1 and 2.2 . Then, Π ac c epts a unique invariant me asur e ν inv on P( C d ) , and ther e exist m ∈ { 1 , . . . , d } and two c onstants C > 0 and γ ∈ [0 , 1) such that for any pr ob ability me asur e ν on P( C d ) and any n ∈ N , W 1 1 m m − 1 X l =0 ν Π mn + l , ν inv ! ≤ C γ n . This theorem is a corollary of Theorem 3.3 and [ Ben+19 , Theorem 1.1]. Indeed, as discussed on page 310 of [ Ben+19 ], the irreducibility assumption of Deﬁnition 2.2 is equiv alen t to assumption ( ϕ -erg) in [ Ben+19 ] with E = C d . W e provide a detailed pro of of this equiv alence in Theorem A.1 . The only hypothesis missing to apply [ Ben+19 , Theorem 1.1] is puriﬁcation. It is a central notion in the study of quan tum tra jectories. W e form ulate the usual equiv alent assumption for a general 6 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL measure space, as Lemma 5.2 in the proof uses it to pro vide an original result of independent in terest. Deﬁnition 3.2 (Puriﬁcation) . L et (Ω , κ ) b e a me asur e sp ac e. L et v : Ω → M d ( C ) b e a me asur able map such that R Ω v ∗ ( ω ) v ( ω ) d κ ( ω ) = Id C d . Then, the me asur e κ is said to purify if al l the ortho gonal pr oje ctors π satisfying that for any p ∈ N and κ ⊗ p -almost any ( ω 1 , . . . , ω p ) , (3) π v ( ω 1 ) ∗ · · · v ( ω p ) ∗ v ( ω p ) · · · v ( ω 1 ) π = ∥ v ( ω p ) · · · v ( ω 1 ) π ∥ 2 π , ar e of r ank at most 1 . In [ Ben+19 ], puriﬁcation is formulated sligh tly diﬀerently: Equation ( 3 ) is required to hold for all matrices v 1 , . . . , v p in the supp ort of the image measure of κ b y v . Ho w ev er, b y con tin uit y , Equa- tion ( 3 ) holds for this larger set of matrices if and only if it holds for κ ⊗ p -almost every ( ω 1 , . . . , ω p ) . Hence, the presen t deﬁnition of puriﬁcation is equiv alen t to the one of [ Ben+19 ]. The next theorem sho ws that for non-singular µ , puriﬁcation holds. Theorem 3.3. Assume Φ is irr e ducible and µ is non-singular, se e Deﬁnitions 2.1 and 2.2 . Then µ satisﬁes the puriﬁc ation assumption of Deﬁnition 3.2 . This theorem is pro v ed in Section 5.1 . 3.2. ν unif -irreducibilit y. The notion of φ -irreducibilit y is standard for Mark o v c hains. It is a generalization of the irreducibilit y criterion for stochastic matrices to an arbitrary space and can b e understo od as a notion of ergo dicity . W e refer the reader to [ MT09 ] for a comprehensive presen tation of the sub ject. Deﬁnition 3.4 (Prop osition 4.2.1 in [ MT09 ]) . L et X b e a me asur able sp ac e. A Markov chain with kernel P over X is φ -irr e ducible for a me asur e φ if for any x ∈ X and any me asur able A ⊂ X with φ ( A ) > 0 , ther e exists n ∈ N such that P n ( x , A ) > 0 . In general, quantum tra jectories are not φ -irreducible (see [ BFP23 , Section 8]). Ho w ev er, the follo wing result gives a suﬃcient criterion for φ -irreducibility of a quan tum tra jectory . W e recall that µ ≫ µ unif means that µ unif is absolutely con tin uous with resp ect to µ . Theorem 3.5. Assume Φ is multiplic atively primitive, se e Deﬁnition 2.4 , and µ ≫ µ unif . Then, the Markov chain ( ˆ x n ) n on P( C d ) is φ -irr e ducible with φ = ν unif . Mor e over, ν inv ≫ ν unif . The condition µ ≫ µ unif ensures that an y ray in V 1 is p otentially selected through the measure- men t. The φ -irreducibilit y of the quantum tra jectory has n umerous consequences – see [ MT09 ]. In particular, since P( C d ) is compact, the chain ( ˆ x n ) is positive Harris. It follo ws that the W asserstein distance in Theorem 3.1 can be switc hed to the total v ariation one – see [ MT09 , Theorem 13.3.4]. Moreo v er, the law of large n um bers holds for any L 1 ( ν inv ) function – see [ MT09 , Theorem 17.0.1]. This contrasts with [ BFP23 , Prop osition 8.2] which sho ws that the la w of large num bers may fail for bounded discon tin uous functions for general quan tum tra jectories. W e will not list all the prop erties implied b y φ -irreducibilit y here, as w e fo cus on the symmetries and regularit y of the in v ariant measure ν inv . W e refer the reader interested in to other consequences to [ MT09 ]. An imp ortan t result that is partly a consequence of the previous theorem is that for µ ≫ µ unif and v ( u ) in v ertible, the inv arian t measure is equiv alent to the uniform one. W e recall that ν 1 ∼ ν 2 means that ν 1 ≪ ν 2 and ν 1 ≫ ν 2 . UNIFORM QUANTUM TRAJECTORIES 7 Theorem 3.6. Assume Φ is multiplic atively primitive, se e Deﬁnition 2.4 , µ ≫ µ unif and v ( u ) is invertible for µ -almost every u . Then, ν inv ∼ ν unif . Remark 3.7. The assumption of invertibility is crucial to the pr o of. In Se ction 6.1 we pr ovide a c ounter example to The or em 3.6 when µ ( { u : det v ( u ) = 0 } ) > 0 . Remark 3.8. When µ ∼ µ unif , the assumption of invertibility of v ( u ) c an b e simpliﬁe d: In L emma 5.4 we show that when µ ≪ µ unif , it is suﬃcient to assume invertibility of v ( u ) for one single u ∈ U ( k ) . Remark 3.9. In Se ction 6.4.2 we pr ovide an example of Φ multiplic atively primitive such that det v ( u ) = 0 for any u ∈ U ( k ) . Mor e gener al ly, ther e exist numer ous irr e ducible line ar sp ac es of non-invertible matric es. F or example, the set of anti-symmetric matric es in o dd dimension, or the adjoint r epr esentation of a semisimple c omplex Lie algebr a. Se e, for example, [ L ov89 ; Dr a06 ] for these and other examples. The theorems of this section are pro v ed in Section 5.2 . 3.3. Uniformly randomized measuremen t and symmetries. W e ﬁnally fo cus on the case µ = µ unif . Lev eraging Haar measure inv ariance, we sho w that the symmetries of Φ are recov ered in the inv arian t measure. Deﬁnition 3.10. A quantum channel Φ is U -c ovariant for some U ∈ U ( d ) if for any X ∈ M d ( C ) , Φ( U ∗ X U ) = U ∗ Φ( X ) U. The gr oup gener ate d by al l the U ∈ U ( d ) such that Φ is U -c ovariant is c al le d the symmetry gr oup of Φ and is denote d by G Φ . Theorem 3.11. Assume Φ is irr e ducible, se e Deﬁnition 2.2 , and µ = µ unif . Then, for any U ∈ G Φ , ν inv is invariant under the image ˆ x 7→ U · ˆ x . F rom this theorem, we can deduce the inv ariant measure when the symmetry group is the full unitary group. Corollary 3.12. Assume Φ is irr e ducible, se e Deﬁnition 2.2 , and µ = µ unif . Assume mor e over that G Φ = U ( d ) . Then ν inv = ν unif . Theorem 3.11 is pro v ed in Section 5.3 . 4. On mul tiplica tive primitivity As it is an uncommon notion of ergo dicit y for quantum channels, we discuss m ultiplicativ e prim- itivit y . F or future reference, w e explicitly state in Prop osition 4.1 that multiplicativ ely primitive quan tum c hannels are also primitive. Moreo v er, w e sho w that a p ositivit y impro ving quan tum c hannel is m ultiplicativ ely primitiv e – see Proposition 4.2 . So p ositivit y impro ving = ⇒ m ultiplicativ ely primitive = ⇒ primitiv e . The question whether multiplicativ e primitivity is equiv alent to primitivity is op en in dimension d ≥ 3 . In Prop osition 4.3 , we sho w equiv alence in d = 2 . W e ha v e not found an y coun terexample y et when d ≥ 3 . Before we pro v e these results, we highligh t the diﬀerence betw een primitivit y and its multiplica- tiv e version using the p oin t of view of purely generated ﬁnitely correlated states as introduced in [ FNW92 ], also kno wn as matrix pro duct states (MPS). F or an y n ∈ N , let V n : C d → C d ⊗ ( C k ) ⊗ n b e deﬁned by V n = P k i 1 ,...,i n =1 v i n · · · v i 1 ⊗ | e i 1 ⟩ ⊗ · · · ⊗ | e i n ⟩ where ( v i ) is a Kraus decomp osition of Φ and ( e i ) is an orthonormal basis of C k . Then, for an y ˆ x ∈ P( C d ) , | V n x ⟩ is an element of S ( C d ⊗ ( C k ) ⊗ n ) , 8 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL the unit sphere 3 of C d ⊗ ( C k ) ⊗ n . Therefore, primitivit y is equiv alen t to the exist ence, for any ˆ x ∈ P( C d ) , of an n ∈ N such that  (Id C d ⊗ ⟨ Ψ n | ) | V n x ⟩ ∥ (Id C d ⊗ ⟨ Ψ n | ) | V n x ⟩∥ : Ψ n ∈ S (( C k ) ⊗ n ) , ∥ (Id C d ⊗ ⟨ Ψ n | ) | V n x ⟩∥ > 0  = S ( C d ) , while multiplicativ e primitivity is equiv alen t to the existence, for any ˆ x ∈ P( C d ) , of an n ∈ N such that  (Id C d ⊗ ⟨ ψ 1 | ⊗ · · · ⊗ ⟨ ψ n | ) | V n x ⟩ ∥ (Id C d ⊗ ⟨ ψ 1 | ⊗ · · · ⊗ ⟨ ψ n | ) | V n x ⟩∥ : ( ψ i ) n i =1 ∈ S ( C k ) n , ∥ (Id C d ⊗ ⟨ ψ 1 | ⊗ · · · ⊗ ⟨ ψ n | ) | V n x ⟩∥ > 0  = S ( C d ) . Hence, primitivity requires that any state in S ( C d ) can b e obtained b y pro jecting | V n x ⟩ along any , p ossibly en tangled, state | Ψ n ⟩ in S (( C k ) ⊗ n ) , whereas its multiplicativ e v ersion requires that the same state | Ψ n ⟩ is a pro duct state. That suggests the implication stated in the next prop osition. Prop osition 4.1. If Φ is multiplic atively primitive as in Deﬁnition 2.4 , then it is primitive, se e Deﬁnition 2.3 . Pr o of. This is a direct consequence of linspan V p 1 = V p (compare Equation ( 2 )), using Theorem A.2 . □ Prop osition 4.2. Assume Φ is p ositivity impr oving, that is, Φ( X ) > 0 for any non-zer o X ≥ 0 . Then, Φ is multiplic atively primitive as in Deﬁnition 2.4 with p = 1 for any ˆ x ∈ P( C d ) . Pr o of. Let ˆ x ∈ P( C d ) . Assume V 1 x  = C d . Then, since V 1 x is a subspace of C d , there exists some ˆ y ∈ P( C d ) suc h that V 1 x ⊥ C y . That implies ⟨ y , Φ( | x ⟩⟨ x | ) y ⟩ = 0 with | x ⟩⟨ x | the orthogonal pro jector onto C x . So V 1 x  = C d con tradicts the fact that Φ is positivity improving. Therefore, V 1 x = C d . □ Prop osition 4.3. F or d = 2 , a quantum channel on M 2 ( C ) is multiplic atively primitive if and only if it is primitive. Pr o of. Proposition 4.1 prov es the forw ard implication. It remains to pro v e that primitivity implies m ultiplicativ e primitivity . Assume there exists ˆ x ∈ P( C 2 ) such that for an y p ∈ N , V p 1 x  = C 2 . Then, there exists a sequence ( ˆ y p ) p ∈ N ⊂ P( C 2 ) such that V p 1 x = C y p . That implies in turn that for an y p ∈ N , Φ p ( | x ⟩⟨ x | ) = | y p ⟩⟨ y p | . This con tradicts Φ n ( | x ⟩⟨ x | ) > 0 for some n ∈ N . Therefore, Φ is m ultiplicativ ely primitive. □ This pro of relies on the fact that in dimension 2 , either V 1 x = C 2 , or it is one-dimensional. In b oth cases, V 2 1 x is a v ector space. Already in dimension 3 , w e lose this prop ert y as V 1 x can b e of dimension 2 and V 2 1 x may not b e a v ector space, but a con tin uous union of v ector spaces. W e could not ﬁnd any examples of primitive quan tum channels on M 3 ( C ) that are not multi- plicativ ely primitive. Thus, the equiv alence b et w een primitivity and multiplicativ e primitivity in arbitrary dimension remains an op en question of in terest. 5. Pr oof of the main resul ts 5.1. Puriﬁcation: Pro of of Theorem 3.3 . The proof relies on the notion of informationally complete instrumen ts. In the following M k, ≥ ( C ) denotes the set of p ositiv e semi-deﬁnite k × k complex matrices. 3 F or an y Hilb ert space H w e denote b y S ( H ) its unit sphere. UNIFORM QUANTUM TRAJECTORIES 9 Deﬁnition 5.1 (Informationally complete instruments) . L et (Ω , F ) b e a me asur able sp ac e and let J : F → L ( M d ( C )) b e an instrument, i.e. , J ( A ) is c ompletely p ositive for every A ∈ F , J is σ -additive and J (Ω) is a quantum channel. W e say that J is informational ly c omplete if ther e exist V : C d → C d ⊗ C k , µ a me asur e over (Ω , F ) and M : Ω → M k, ≥ ( C ) an element of L 1 ( µ ) such that  Z Ω f ( ω ) M ( ω ) d µ ( ω ) : f : Ω → C b ounded and measurable  = M k ( C ) (4) and J ( A ) : X 7→ Z A V ∗ ( X ⊗ M ( ω )) V d µ ( ω ) (5) for al l A ∈ F . With a smal l abuse of notation we denote J ( ω ) : X 7→ V ∗ ( X ⊗ M ( ω )) V . Without assuming Equation ( 4 ), [ Bus+16 , Theorem 7.11] yields that a dilation as in Equation ( 5 ) exists for an y instrumen t J . This is a consequence of Stinespring’s dilation theorem. Then, Equation ( 4 ) is actually the condition for M b eing an informationally complete p ositiv e op erator-v alued measure (IC-PO VM). Equiv alently , IC-POVMs are POVMs suc h that the map ρ 7→ tr( ρM ( ω )) d µ ( ω ) from the set of densit y matrices to probability measures ov er Ω is injective. Hence, kno wing the distribution of outcomes of the measurement mo deled by the POVM allo ws for the iden tiﬁcation of the system state. So, an IC-POVM can b e used to perform a complete tomograph y of the system state. Considering that instruments can alwa ys b e thought of as resulting from an indirect measure- men t via prob es, see [ Bus+16 , Theorem 7.14], informationally complete instrumen ts corresp ond to indirect measurements allowing for a complete prob e tomography . Lemma B.1 sho ws that for informationally complete instrumen ts, in contrast to the prob e PO VM M , the induced PO VM on the system, M p ( ω 1 , . . . , ω p ) := J ( ω 1 ) ◦ · · · ◦ J ( ω p )(Id C d ) with arbitrarily large p , is not alwa ys informationally complete. It is so when Φ = J (Ω) is primitive and p is large enough, since m in Lemma B.1 is actually the perio d of Φ and m = 1 for primitiv e maps. The notion of informationally complete instrumen ts has been used in [ Ben+25 ] to prov e the equiv alence b et w een the v anishing of en trop y pro duction of repeated quan tum measurements and a notion of quan tum detailed balance. Lemma 5.2. Assume Φ is irr e ducible and J is an informational ly c omplete instrument, se e Deﬁ- nition 5.1 , such that J ( A )( X ) = R A v ( ω ) ∗ X v ( ω ) d µ ( ω ) for some me asur able v : Ω → M d ( C ) . Then, µ puriﬁes in the sense of Deﬁnition 3.2 . Pr o of. F ollowing Lemma B.1 , there exist an orthogonal partition { E i } m i =1 of C d and p ∈ N suc h that  Z Ω p f ( ω 1 , . . . , ω p ) J ( ω 1 ) ◦ · · · ◦ J ( ω p )(Id C d ) d µ ⊗ p ( ω 1 , . . . , ω p ) : f ∈ L ∞ ( µ ⊗ p )  = m M i =1 L ( E i ) where for an y ﬁnite dimensional linear space E , L ( E ) denotes the set of endomorphisms of E . Then, puriﬁcation is equiv alen t to: An y orthogonal pro jector π suc h that (6) π m M i =1 L ( E i ) ! π = C π is of rank at most 1 . So let π to be a nonzero orthogonal pro jector satisfying Equation ( 6 ). Let { p 1 , . . . , p d } be a resolution of the identit y by rank- 1 orthogonal pro jectors such that p j ∈ L m i =1 L ( E i ) for any j ∈ { 1 , . . . , d } . Since the p j are p ositive semi-deﬁnite, and sum up to the iden tit y , there exists j ∈ { 1 , . . . , d } suc h that π p j π  = 0 . The rank submultiplicativit y implies that π p j π has rank 1 . Then, 10 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL from Equation ( 6 ), π p j π = ∥ p j π ∥ 2 π , so also π has rank 1 . Hence, puriﬁcation holds, and the lemma is prov ed. □ Pr o of of The or em 3.3 . W e will apply Lemma 5.2 , which readily establishes the claimed puriﬁcation assumption. F or this, set Ω = U ( k ) and F = B ( U ( k )) , i.e. , F is the Borel σ -algebra ov er the unitary group U ( k ) . Recall that Φ : X 7→ P k i =1 v ∗ i X v i for some ( v i ) k i =1 ∈ M d ( C ) k and consider as an instrumen t J ( u ) : X 7→ v ( u ) ∗ X v ( u ) , where we recall v ( u ) = P k j =1 u 1 j v j . By Deﬁnition 5.1 , we ma y apply Lemma 5.2 and therefore immediately conclude the pro of, if we can ﬁnd some ( M , V ) suc h that Equations ( 4 ) and ( 5 ) hold. Let ( e i ) k i =1 b e the canonical orthonormal basis of C k and let V := P k i =1 v i ⊗ e i . Then, setting M : U ( k ) → M d ( C ) , M ( u ) := u ∗ | e 1 ⟩⟨ e 1 | u , we can write J ( u ) : X 7→ V ∗ ( X ⊗ M ( u )) V , and J ( A ) = R A J ( u ) d µ ( u ) implies Equation ( 5 ). So it remains to show that the PO VM M veriﬁes Equation ( 4 ) whenev er µ is non-singular. Let B := { R U ( k ) f ( u ) M ( u ) d µ ( u ) : f ∈ L ∞ ( µ ) } . By deﬁnition and con tin uit y of M in u , B = linspan { u ∗ | e 1 ⟩⟨ e 1 | u : u ∈ supp µ } . Assume B  = M k ( C ) . Then, there exists a nonzero X ∈ M k ( C ) suc h that tr( u ∗ | e 1 ⟩⟨ e 1 | uX ) = 0 for an y u ∈ supp µ . Then, [ Mit20 , Prop osition 1] applied to the p olynomial u 7→ tr( u ∗ | e 1 ⟩⟨ e 1 | uX ) implies µ unif (supp µ ) = 0 . That implies µ do es not verify the non-singularity assumption of Deﬁni- tion 2.1 . This contradiction yields the theorem. □ 5.2. Regularit y: Pro of of Theorems 3.5 and 3.6 . W e start with the pro of of Theorem 3.5 . Recalling Deﬁnition 3.4 and Deﬁnition 2.4 , we must pro v e that for an y ˆ x ∈ P( C d ) and A ⊆ P( C d ) measurable, such that ν unif ( A ) > 0 , we ha v e Π p ( ˆ x, A ) > 0 , (7) with p b eing the ˆ x -dep enden t integer of Deﬁnition 2.4 . Since ν unif ( A ) > 0 only if A has non-empt y in terior, it is suﬃcient to pro v e Equation ( 7 ) for A b eing a ball B ( ˆ y , ε ) , with resp ect to the metric d ( ˆ x, ˆ y ) deﬁned ab ov e Theorem 3.1 , centered in an arbitrary ˆ y ∈ P( C d ) and with arbitrary radius ε > 0 . A ctually , Equation ( 7 ) is the only statemen t that needs to b e pro v ed, since then ν inv ≫ ν unif is implied by the following prop osition from [ MT09 ]. Prop osition 5.3 ([ MT09 , Prop osition 10.1.2], Item (ii)) . L et ( ˆ x n ) n ∈ N b e a φ -irr e ducible Markov chain with φ = ν unif on P( C d ) . Then ν inv ≫ ν unif . So let us now pro v e Equation ( 7 ). Fix ˆ x, ˆ y ∈ P( C d ) and ε > 0 . Let p ∈ N b e the ˆ x -dep enden t in teger of Deﬁnition 2.4 . Let { v i } k i =1 b e a set of matrices such that Φ : X 7→ P k i =1 v ∗ i X v i . The assumption of m ultiplicativ e primitivit y ( i.e. Deﬁnition 2.4 ) implies that there exist λ > 0 and ( ϕ 1 , . . . , ϕ p ) ∈ S ( C k ) p suc h that y = λ k X i 1 ,...,i p =1 ( ⟨ e i 1 , ϕ 1 ⟩ v i 1 ) · · · ( ⟨ e i p , ϕ p ⟩ v i p ) x. Cho osing ( u (1) , . . . , u ( p )) ∈ U ( k ) p suc h that u ( l ) 1 i = ⟨ e i , ϕ l ⟩ for an y l ∈ { 1 , . . . , p } and i ∈ { 1 , . . . , k } , ˆ y = v ( u ( p )) · · · v ( u (1)) · ˆ x. UNIFORM QUANTUM TRAJECTORIES 11 Then, Π p ( ˆ x, B ( ˆ y , ε )) = Z U ( k ) p χ ( d ( v ( w p ) · · · v ( w 1 ) · ˆ x, ˆ y ) < ε ) ∥ v ( w p ) · · · v ( w 1 ) x ∥ 2 d µ ⊗ p ( w 1 , . . . , w p ) = Z U ( k ) p χ ( d ( v ( w p ) · · · v ( w 1 ) · ˆ x, v ( u ( p )) · · · v ( u (1)) · ˆ x ) < ε ) ∥ v ( w p ) · · · v ( w 1 ) x ∥ 2 d µ ⊗ p ( w 1 , . . . , w p ) , with χ ( S ) = 1 if the statement S is true, and χ ( S ) = 0 otherwise. Since v ( u ( p )) · · · v ( u (1)) x  = 0 , ( w 1 , . . . , w p ) 7→ v ( w p ) · · · v ( w 1 ) · ˆ x is con tin uous in an op en neigh- b orhoo d of ( u (1) , . . . , u ( p )) . Hence, there exists η > 0 such that ( w 1 , . . . , w p ) ∈ B ( u (1) , η ) × · · · × B ( u ( p ) , η ) implies d ( v ( w p ) · · · v ( w 1 ) · ˆ x, v ( u ( p )) · · · v ( u (1)) · ˆ x ) < ε . So, Π p ( ˆ x, B ( ˆ y , ε )) ≥ Z B ( u (1) ,η ) ×···× B ( u ( p ) ,η ) ∥ v ( w p ) · · · v ( w 1 ) x ∥ 2 d µ ⊗ p ( w 1 , . . . , w p ) . Again, since ( w 1 , . . . , w p ) 7→ ∥ v ( w p ) · · · v ( w 1 ) x ∥ 2 is contin uous, for η > 0 small enough, there exists δ > 0 such that inf ( w 1 ,...,w p ) ∈ B ( u (1) ,η ) ×···× B ( u ( p ) ,η ) ∥ v ( w p ) · · · v ( w 1 ) x ∥ 2 ≥ δ. It follows that Π p ( ˆ x, B ( ˆ y , ε )) ≥ δ p Y l =1 µ ( B ( u ( l ) , η )) . Since µ ≫ µ unif , for an y l ∈ { 1 , . . . , p } , µ ( B ( u ( l ) , η )) > 0 and Theorem 3.5 is pro v ed. □ W e turn to the proof of Theorem 3.6 . The pro of relies on the fact that when µ -almost all of the matrices v ( u ) are in v ertible, ν inv is either pure point, absolutely con tin uous with resp ect to ν unif , or singular con tin uous. It cannot b e a mixture of the three. A similar result for pro ducts of independent and identically distributed matrices can b e found in [ BL85 , Prop osition 4.4, Section VI.4]. W e follow essen tially the proof in that reference. But, b efore, w e pro v e the lemma justifying Remark 3.8 . It sho ws that when µ ≪ µ unif , it is suﬃcien t to c hec k that v ( u ) is in v ertible for one u ∈ U ( k ) . Lemma 5.4. Assume µ ≪ µ unif and ther e exists u ∈ U ( k ) such that v ( u ) is invertible. Then, v ( u ) is invertible for µ -almost every u . Pr o of. Assume there exists a measurable subset A of U ( k ) such that µ ( A ) > 0 and v ( u ) is non- in v ertible for an y u ∈ A . Assuming v ( u ) is non-in v ertible for any u ∈ A means that for an y u ∈ A , det v ( u ) = 0 . Since there exists u ∈ U ( k ) such that det v ( u )  = 0 , [ Mit20 , Prop osition 1] implies µ unif ( A ) = 0 and therefore µ ( A ) = 0 since µ ≪ µ unif . This con tradiction yields the theorem. □ W e no w turn to the pro of of our v ersion of [ BL85 , Proposition 4.4, Section VI.4]. Lemma 5.5. Assume that for µ -almost every u , v ( u ) is invertible. Assume Π ac c epts a unique invariant pr ob ability me asur e ν inv . Then, one of the thr e e fol lowing alternatives o c cur: • ν inv is pur e p oint, • ν inv ≪ ν unif , • ν inv ⊥ ν unif and has no pur e p oint c omp onent. Pr o of. Let ν ac b e the absolutely contin uous part of ν inv with resp ect to ν unif , ν sc its singular con- tin uous part and ν pp its pure p oin t part in Leb esgue’s decomp osition of Borel measures. Denote furthermore ν c = ν ac + ν sc . First, there is no ˆ y ∈ P( C d ) such that R P( C d ) Π( ˆ x, { ˆ y } ) d ν c ( ˆ x ) > 0 . Indeed, if it w as the case, there would exist a measurable subset A of U ( k ) × P( C d ) suc h that ( µ ⊗ ν c )( A ) > 0 and for any ( u, ˆ x ) ∈ A , v ( u ) is in v ertible and v ( u ) · ˆ x = ˆ y . That is equiv alen t to ˆ x = v ( u ) − 1 · ˆ y for an y ( u, ˆ x ) ∈ A . Hence, ( µ ⊗ ν c )( A ) ≤ R U ( k ) ν c ( { ˆ x : ˆ x = v ( u ) − 1 · ˆ y } ) d µ ( u ) = 0 since 12 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL { ˆ x : ˆ x = v ( u ) − 1 · ˆ y } is at most a singleton for µ -almost every u . That con tradicts the assumption ( µ ⊗ ν c )( A ) > 0 . Thus, ν c Π has no pure p oin t comp onen t. Then, with ν inv = ν pp + ν c = ν pp Π + ν c Π w e conclude ν c ≥ ν c Π . Since Π 1 ( ˆ x ) = 1 , ν c Π(P( C d )) = ν c (P( C d )) and it follo ws that ν c = ν c Π . By uniqueness of the in v ariant measure, either ν c = 0 and therefore ν inv = ν pp , or ν c (P( C d )) = 1 and ν inv = ν c . It remains to sho w that for ν c (P( C d )) = 1 , either ν inv = ν ac or ν inv = ν sc . F or any v in v ertible, the measure deﬁned by ν v ( A ) = R P( C d ) 1 A ( v · ˆ x ) ∥ v x ∥ 2 d ν unif ( ˆ x ) is equiv alen t to ν unif . Indeed, for an y measurable subset A of P( C d ) , ∥ v − 1 ∥ − 2 Z P( C d ) 1 A ( v · ˆ x ) d ν unif ( ˆ x ) ≤ ν v ( A ) ≤ ∥ v ∥ 2 Z P( C d ) 1 A ( v · ˆ x ) d ν unif ( ˆ x ) . Since the image measure of ν unif b y ˆ x 7→ v · ˆ x is equiv alen t to ν unif (see [ BL85 , Exercise 5.4, Section I.5] for example), ν v is equiv alent to ν unif . Since ν unif Π = R U ( k ) ν v ( u ) d µ ( u ) and v ( u ) is inv ertible for µ -almost every u , ν unif Π is equiv alent to ν unif . Hence, if ν unif ( A ) = 0 , then ν unif Π( A ) = 0 and ν ac ( A ) = 0 . Assume ν ac Π( A ) > 0 . Then there exists a measurable B ⊂ P( C d ) such that ν ac ( B ) > 0 and ∀ ˆ x ∈ B , Π( ˆ x, A ) > 0 . The absolute con tin uit y ν ac ≪ ν unif implies ν unif ( B ) > 0 . Then, Π( ˆ x, A ) > 0 for an y ˆ x ∈ B implies ν unif Π( A ) > 0 whic h contradicts ν unif Π( A ) = ν unif ( A ) = 0 . Therefore, ν ac Π ≪ ν unif , and ν c = ν ac + ν sc = ν ac Π + ν sc Π implies ν ac ≥ ν ac Π . Thus, we infer as before that either ν inv = ν ac or ν inv = ν sc . This concludes the pro of of the lemma. □ Pr o of of The or em 3.6 . Since v ( u ) is inv ertible for µ -almost an y u , Lemma 5.5 implies that either ν inv is pure p oin t, absolutely contin uous or singular con tin uous with resp ect to ν unif . Since µ ≫ µ unif , Theorem 3.5 implies ν inv ≫ ν unif , therefore ν inv cannot be pure p oin t or singular con tin uous. Hence, ν inv ∼ ν unif . □ 5.3. Symmetries: Pro of of Theorem 3.11 . The pro of of this theorem and the related corol- lary rely on the observ ation that when µ is uniform, then δ ˆ x Π is the GAP measure asso ciated to Φ ∗ ( | x ⟩⟨ x | ) . W e refer the reader to App endix C and [ JR W94 ; Gol+06 ; Gol+16 ; T um20 ] for an in tro duction to GAP measures. In the app endix and those references, the GAP measure is deﬁned on S ( C d ) and not P( C d ) . Giv en a GAP measure on S ( C d ) , one can alwa ys deﬁne the respective GAP measure on P( C d ) as a pullback under the pro jection S ( C d ) → P( C d ) . Con v ersely , by Item (c) in Prop osition C.1 , GAP measures (on S ( C d ) ) are inv arian t under multiplication by a phase, so given a GAP measure on P( C d ) , one can uniquely reconstruct the resp ectiv e GAP measure on S ( C d ) . Thus, the notions of GAP measures on S ( C d ) and P( C d ) are equiv alen t. Lemma 5.6. Assume µ = µ unif . Then δ ˆ x Π is the GAP me asur e asso ciate d to Φ ∗ ( | x ⟩⟨ x | ) . Pr o of. This lemma is a consequence of [ Gol+16 , Lemma 1]: Let v 1 , . . . , v k b e Kraus op erators of Φ . W e deﬁne V : C d → C d ⊗ C k b y | V x ⟩ = X j v j | x ⟩ ⊗ | e j ⟩ , where ( e j ) is the canonical basis of C k . Then, Φ ∗ ( | x ⟩⟨ x | ) = tr C k | V x ⟩⟨ V x | , where tr C k is the partial trace o v er C k in C d ⊗ C k . Let Ψ := V x ∈ S ( C d ⊗ C k ) and { u 1 , . . . , u k } b e an orthonormal basis of C k . Let U be the unitary matrix whose j th ro w is the en try-wise complex UNIFORM QUANTUM TRAJECTORIES 13 conjugate of u j for j in { 1 , . . . , k } . Then, the conditional w a v e function ψ U is the random C d -v ector deﬁned as ψ U = (Id C d ⊗ ⟨ u J | ) | V x ⟩ ∥ (Id C d ⊗ ⟨ u J | ) | V x ⟩∥ , where J is random with distribution P ( J = j ) = ∥ (Id C d ⊗ ⟨ u J | ) | V x ⟩∥ 2 . Direct computation leads to (Id C d ⊗ ⟨ u j | ) | V x ⟩ = v j ( U ) x and therefore ψ U = v J ( U ) x ∥ v J ( U ) x ∥ . Let µ Ψ ,U 1 denote the distribution of the conditional wa v e function ψ U . Then, for any measurable set A ⊂ S ( C d ) , we ﬁnd that µ Ψ ,U 1 ( A ) = X j ∥ v j ( U ) x ∥ 2 1 A ( v j ( U ) · ˆ x ) . No w let U b e distributed according to the Haar probabilit y measure λ o v er U ( k ) . Then, the random v ariables ( v j ( U )) k j =1 are identically distributed and E λ ( µ Ψ ,U 1 ( A )) = k E λ ( ∥ v ( U ) x ∥ 2 1 A ( v ( U ) · ˆ x )) = δ ˆ x Π( A ) , where we used that E λ ( v ( U ) ∗ X v ( U )) = 1 k Φ( X ) . F rom [ Gol+16 , Lemma 1] it follo ws that E λ µ Ψ ,U 1 = GAP tr C k | Ψ ⟩⟨ Ψ | = GAP Φ ∗ ( | x ⟩⟨ x | ) and the claim is prov ed. □ The second and last lemma we require sho w the impact of the symmetry on the la w of the Mark o v c hain itself. Lemma 5.7. Assume µ = µ unif . Assume U ∈ U ( d ) is such that for any X ∈ M d ( C ) , Φ( U ∗ X U ) = U ∗ Φ( X ) U . Then, for any ˆ x ∈ P( C d ) , δ U · ˆ x Π is the image me asur e of δ ˆ x Π by the map ˆ y 7→ U · ˆ y . Pr o of. Property 2 of GAP measures in [ Gol+06 ] implies that GAP of U Φ ∗ ( | x ⟩⟨ x | ) U ∗ is equal to the image measure of GAP of Φ ∗ ( | x ⟩⟨ x | ) by the map ˆ y 7→ U · ˆ y . Then the symmetry Φ ∗ ( U | x ⟩⟨ x | U ∗ ) = U Φ ∗ ( | x ⟩⟨ x | ) U ∗ and Lemma 5.6 yield the lemma. □ Pr o of of The or em 3.11 . Since µ = µ unif , it is trivially non-singular and Theorem 3.1 yields that Π accepts a unique inv arian t measure ν inv . Let ν U b e the image measure of ν inv b y ˆ x 7→ U · ˆ x . Then, ν U Π = R P( C d ) δ U · ˆ x Π d ν inv ( ˆ x ) . Then, Lemma 5.7 implies that ν U Π is the image measure of ν inv Π by ˆ x 7→ U · ˆ x . Since ν inv Π = ν inv , w e deduce that ν U is Π -inv ariant and the uniqueness of the inv arian t measure yields the theorem. □ 6. Examples 6.1. In v arian t measure non-equiv alent to uniform. Let us pro vide a coun terexample to The- orem 3.6 where Φ is primitiv e, µ ≫ µ unif , but µ ( { u : det v ( u ) = 0 } ) > 0 . W e sho w that for this example, ν inv ≫ ν unif as implied b y Theorem 3.5 but there exists a measurable subset A of P( C d ) suc h that ν inv ( A ) > 0 but ν unif ( A ) = 0 . Let d = k = 2 , v 1 = | e 2 ⟩⟨ e 1 | and v 2 = | e + ⟩⟨ e 2 | with { e 1 , e 2 } the canonical basis of C 2 and ˆ e + the equiv alenc e class of e 1 + e 2 . Now Φ 3 ( X ) = 1 4 X 22 (Id C 2 + | e 1 ⟩⟨ e 1 | ) + X ++ ( 1 2 Id C 2 + 1 4 | e 2 ⟩⟨ e 2 | ) with X 22 = ⟨ e 2 , X e 2 ⟩ and X ++ = ⟨ e + , X e + ⟩ . Since | e 2 ⟩⟨ e 2 | + | e + ⟩⟨ e + | is p ositiv e deﬁnite, w e conclude Φ 3 ( X ) > 0 for an y X ≥ 0 and b y Prop osition 4.3 , Φ is multiplicativ ely primitive. Let 4 λ = 1 2 Haar U (2) + 1 2 δ Id C 2 and deﬁne µ as in Equation ( 1 ). Then, µ ≫ µ unif and for an y probabilit y measure ν o v er P( C 2 ) , ν Π( { ˆ e 2 , ˆ e + } ) ≥ Z P( C 2 ) 1 2 ( |⟨ e 1 , x ⟩| 2 + |⟨ e 2 , x ⟩| 2 ) d ν ( ˆ x ) = 1 2 > 0 . Th us, ν inv ( { ˆ e 2 , ˆ e + } ) > 0 , but ν unif ( { ˆ e 2 , ˆ e + } ) = 0 . 4 A t this p oin t, w e normalize Haar U (2) ( U (2)) = 1 . 14 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL 6.2. Uniformly randomized quan tum tra jectories with explicit ν inv . In this section, w e alw a ys consider µ = µ unif . As a simple example with an explicit ν inv , consider the pro jection c hannel Φ( X ) = Id C d tr( ρX ) for some p ositiv e deﬁnite density matrix ρ . So Φ ∗ ( ϱ ) = ρ for any densit y matrix ϱ . Then Φ is trivially p ositivit y improving and therefore multiplicativ ely primitive thanks to Prop osition 4.2 . Moreo v er, Lemma 5.6 implies ν inv = GAP ρ where GAP ρ is the GAP measure for the densit y matrix ρ . Another example with an explicit ν inv is the dep olarizing c hannel Φ( X ) = (1 − p ) X + p Id C d 1 d tr( X ) for p ∈ (0 , 1] . Here, the symmetry group of Φ is the full unitary group and Corollary 3.12 implies ν inv = ν unif . Ho w ev er, for a generic multiplicativ ely primitive Φ , and for µ = µ unif , computing a closed form for in v ariant measures do es not seem to b e an easy task. F or instance, as so on as the maximally mixed state Id C d /d in the dep olarizing channel is replaced b y an arbitrary p ositiv e deﬁnite density matrix, establishing a closed form for ν inv is nontrivial, as illustrated in the next section. 6.3. Uniformly randomized in dimension 2 . Let us no w concen trate on d = 2 . In that case our diﬀeren t assumptions reduce to standard ones and the density of ν inv with resp ect to ν unif satisﬁes an integral k ernel equation. Prop osition 6.1. Assume d = 2 and Φ is primitive, se e Deﬁnition 2.3 . L et µ = µ unif . Then, ν inv ∼ ν unif and the density f inv ( ˆ x ) = d ν inv d ν unif ( ˆ x ) is a solution to f inv ( ˆ x ) = Z P( C 2 ) 2 det Φ ∗ ( | x ′ ⟩⟨ x ′ | ) ⟨ x, Φ ∗ ( | x ′ ⟩⟨ x ′ | ) − 1 x ⟩ − 3 f inv ( ˆ x ′ ) d ν unif ( ˆ x ′ ) . (8) This result is an application of Lemma 5.6 and [ Gol+06 , Equation (18)] (see Equation ( 10 )). It also requires ν inv ∼ ν unif whic h is assured by primitivit y , Prop osition 4.3 , Theorem 3.6 and Lemma 6.4 . Remark 6.2. If Φ is irr e ducible but not primitive ( i.e. its p erio d is 2 ), then ther e exists an or- thonormal b asis { x 1 , x 2 } of C 2 such that Φ ∗ ( | x 1 ⟩⟨ x 1 | ) = | x 2 ⟩⟨ x 2 | and Φ ∗ ( | x 2 ⟩⟨ x 2 | ) = | x 1 ⟩⟨ x 1 | . Thus, if Π ac c epts a unique invariant pr ob ability me asur e, it is e qual to 1 2 ( δ ˆ x 1 + δ ˆ x 2 ) . Remark 6.3. With minor mo diﬁc ations to the pr o of of Pr op osition 6.1 , Equation ( 8 ) c an b e gener- alize d to arbitr ary dimension d using [ Gol+06 , Equation (18)], pr ovide d det v ( u )  = 0 for almost al l u ∈ U ( k ) , so we c an apply The or em 3.6 . Thanks to L emma 5.4 , it even suﬃc es to have det v ( u )  = 0 for only one single u ∈ U ( k ) . In d = 2 , the required inv ertibilit y of v ( u ) is actually automatically true for irreducible Φ . Lemma 6.4. Assume d = 2 and Φ is irr e ducible, se e Deﬁnition 2.2 . Then, det v ( u )  = 0 for µ unif -almost every u ∈ U ( k ) . Pr o of. Assume det v ( u ) = 0 within some non-null set with resp ect to µ unif . Then, there is some op en set U ⊂ U ( k ) with det v ( u ) = 0 ∀ u ∈ U . Now, recall that det v ( u ) is a p olynomial in the matrix entries of u , so det v ( u ) = 0 ∀ u ∈ U ( k ) . Fix u 1 , u 2 ∈ U ( k ) suc h that v ( u 1 ) ∝ v ( u 2 ) and v ( u 1 ) and v ( u 2 ) are non zero. Consider ﬁrst the case where v ( u 1 ) is diagonalizable. Then, for any w ∈ C , there exists u w ∈ U ( k ) suc h that v ( u w ) ∝ wv ( u 1 ) + v ( u 2 ) . W orking in an eigenbasis of v ( u 1 ) , w e can write any v ( u w ) , as a scalar m ultiple of X w := w v ( u 1 ) + v ( u 2 ) =  x + αw y z t  where v ( u 2 ) =  x y z t  ∈ M 2 ( C ) where α is the non zero eigenv alue of v ( u 1 ) . Since v ( u w ) is not in v ertible for an y w ∈ C , 0 = det X w = ( xt − y z ) + w αt for all w ∈ C , so t = y z = 0 . Since u 2 w as arbitrary , for an y u ∈ U ( k ) , UNIFORM QUANTUM TRAJECTORIES 15 w e hav e the dichotom y: either v ( u ) =  a ( u ) b ( u ) 0 0  or v ( u ) =  a ( u ) 0 c ( u ) 0  , for some a, b, c : U ( k ) → C . Moreo v er, either b ≡ 0 or c ≡ 0 , since if we had u 3 , u 4 ∈ U ( k ) with b ( u 3 ) c ( u 4 )  = 0 , then there would b e some u ∈ U ( k ) with v ( u ) = 1 √ 2 ( v ( u 3 ) + v ( u 4 )) not b eing in one of the abov e forms. No w, let f b e an non-zero element of ker v 1 . If b ≡ 0 , then v ( u ) f = 0 for any u ∈ U ( k ) , th us R U ( k ) ∥ v ( u ) f ∥ 2 d µ ( u ) = 0 which is incompatible with the condition R U ( k ) v ( u ) ∗ v ( u ) d µ ( u ) = Id C 2 . Th us, b ≡ 0 and c ≡ 0 . Then, the image of v ( u 1 ) is a one-dimensional inv arian t subspace of v ( u ) for any u ∈ U ( k ) . That contradicts the irreducibilit y of Φ . Therefore v ( u ) is inv ertible for µ -almost ev ery u . Finally , if v ( u 1 ) is not diagonalizable, working in a basis of its Jordan form, X w = w v ( u 1 ) + v ( u 2 ) =  x y + w z t  , and det X w = xt − y z − z w . Hence, det X w = 0 for an y w ∈ C if and only if z = xt = 0 . It follows that v ( u ) is not inv ertible for any u ∈ U ( k ) only if, for any u ∈ U ( k ) , v ( u ) =  a ( u ) b ( u ) 0 c ( u )  . That implies that for any u ∈ U ( k ) , the image of v ( u 1 ) is an in v ariant subspace of v ( u ) for all u ∈ U ( k ) , which con tradicts again the irreducibilit y of Φ . □ Pr o of of Pr op osition 6.1 . Since d = 2 and Φ is primitive, Prop osition 4.3 implies that Φ is multi- plicativ ely primitiv e. Then, Lemma 6.4 and Theorem 3.6 imply ν inv ∼ ν unif . Now assume Φ ∗ ( | y ⟩⟨ y | ) is inv ertible for ν unif -almost every ˆ y . Then Lemma 5.6 and ( 10 ) yield the prop osition. It remains to pro v e that Φ ∗ ( | y ⟩⟨ y | ) is inv ertible for ν unif -almost every ˆ y . If Φ ∗ ( | y ⟩⟨ y | ) is inv ertible for one ˆ y ∈ P( C 2 ) , then det Φ ∗ ( | y ⟩⟨ y | )  = 0 and [ Mit20 , Prop osition 1] imply that Φ ∗ ( | y ⟩⟨ y | ) is in v ertible for ν unif -almost every ˆ y . So we m ust exclude that Φ ∗ ( | y ⟩⟨ y | ) is non-inv ertible for all ˆ y ∈ P( C 2 ) . If this was the case, then for any ˆ y ∈ P( C 2 ) , there w ould exist a sequence ( ˆ y n ) n ∈ N in P( C 2 ) such that Φ ∗ n ( | y ⟩⟨ y | ) = | y n ⟩⟨ y n | for any n ∈ N . That con tradicts the assumption that Φ is primitiv e. □ Remark 6.5. A lr e ady for the simple channel Φ( X ) = (1 − p ) X + p Id C 2 tr( ρX ) with p ∈ (0 , 1) and ρ a p ositive deﬁnite density matrix distinct fr om Id C 2 / 2 , solving Equation ( 8 ) for f inv is non-trivial. During the pr o duction of this article a r esult showing stability of the GAP me asur e under me asur e- ment app e ar e d [ T um26 ]. However, this stability c onc erns the state c onditione d on the me asur ement outc ome ( i.e. for a ﬁxe d pr e determine d outc ome) and as mentione d in the same r efer enc e that do es not imply invarianc e of the GAP me asur e under indir e ct me asur ement. 6.4. Examples in dimension 3 satisfying m ultiplicativ e primitivit y. Recall for a quantum c hannel Φ that “positivity impro ving” implies “m ultiplicativ ely primitive”, whic h implies “primitiv e”. W e provide tw o non-trivial examples for d = 3 suc h that m ultiplicativ e primitivit y holds, but Φ is not p ositivity-impro ving. The method of pro of diﬀers betw een the t w o examples, but alw a ys relies on algebraic indep endence. In the ﬁrst example, v ( u ) is µ -almost alwa ys inv ertible, whereas, in the second one, v ( u ) is not inv ertible for any u . 6.4.1. A n example with almost sur e invertible v ( u ) . W e provide a nontrivial example where all our assumptions are v eriﬁed. Let, v 1 = √ 2 2   0 1 0 1 0 1 0 0 0   and v 2 = √ 2 2   0 0 0 0 − 1 0 1 0 − 1   . 16 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL Prop osition 6.6. The channel Φ( X ) := v ∗ 1 X v 1 + v ∗ 2 X v 2 is not p ositivity-impr oving, but multiplic a- tively primitive, se e Deﬁnition 2.4 , and for µ unif -almost any u ∈ U (2) , det v ( u )  = 0 . Pr o of. One easily c hec ks Φ( | e 1 ⟩⟨ e 1 | ) = 1 2 | e 2 ⟩⟨ e 2 | , whic h is not positive deﬁnite, so Φ is not p ositivit y- impro ving. W e will next show that multiplicativ e primitivity holds with p = 8 . Consider the 3 × 3 matrix of p olynomials deﬁned b y P ( z 1 , . . . , z 16 ) = Q 7 i =0 ( z 2 i +1 v 1 + z 2 i +2 v 2 ) . Let J ( z 1 , . . . , z 16 ) b e the 9 × 16 Jacobian matrix of the entries of this matrix with resp ect to ( z 1 , . . . , z 16 ) . Then, J (1 , 2 , 3 , 1 , 2 , 3 , 1 , 2 , 3 , 1 , 2 , 3 , 1 , 2 , 3 , 1) has rank 9 . This computation can be made in SageMath 10 using the co de av ailable in Appendix D . Then, it follows from [ ER93 , Theorem 2.3], that the p olynomials deﬁned b y the en tries of the matrix P are algebraically indep enden t. Hence, P ( C 16 ) = M 3 ( C ) , whic h in particular implies m ultiplicativ e primitivit y . It remains to prov e that det v ( u )  = 0 for almost every u ∈ U ( k ) . This follows from det( av 1 + bv 2 )  = 0 for an y a, b ∈ C suc h that ab  = 0 . Indeed, av 1 + bv 2 = √ 2 2   0 a 0 a − b a b 0 − b   , so det( av 1 + bv 2 ) = √ 2 2 a 2 b and the prop osition is prov ed. □ 6.4.2. A n example with non-invertible v ( u ) . W e conclude our list of examples with one that is mul- tiplicativ ely primitiv e but no Kraus decomposition contains an inv ertible matrix. So Theorem 3.6 cannot b e applied. F or this, we consider the matrices w 1 =   1 1 0 − 1 1 0 0 0 0   and w 2 =   0 0 1 0 0 1 1 0 0   , and turn them in to Kraus operators v 1 , v 2 b y a similarit y transformation. T o do so, w e ﬁrst sho w that T : X 7→ w ∗ 1 X w 1 + w ∗ 2 X w 2 is irreducible. Let us assume T is not irreducible, so from Theorem A.1 there exists a prop er subspace E ⊂ C 3 that is both w 1 - and w 2 -in v ariant. If e 3 ⊥ E , then w 1 w 2 E = C e 1 . Then, w 1 e 1 = e 1 − e 2 and w 2 e 1 = e 3 imply E = C 3 and E is not a proper subspace. Hence, e 3 ⊥ E and E ⊂ C e 1 + C e 2 . So in that case, w 2 E = C e 3 or w 2 E = 0 . In the ﬁrst case, w 2 E ⊂ E . In the second case, E ⊂ k er w 2 = C e 2 . It follows that E = C e 2 , but w 1 e 2 = e 1 + e 2 . Th us, there do es not exist a prop er subspace E of C 3 suc h that w 1 E ⊂ E and w 2 E ⊂ E . W e conclude that T is irreducible and [ EHK78 , Theorem 2.3] implies that there exist r > 0 and C ∈ M 3 ( C ) , C > 0 suc h that T ( C ) = λC . Then, setting v 1 = r − 1 / 2 C 1 / 2 w 1 C − 1 / 2 and v 2 = r − 1 / 2 C 1 / 2 w 2 C − 1 / 2 , the map Φ : M 3 ( C ) → M 3 ( C ) deﬁned b y Φ( X ) = v ∗ 1 X v 1 + v ∗ 2 X v 2 is a quan tum c hannel. Prop osition 6.7. The channel Φ just deﬁne d is not p ositivity-impr oving, but multiplic atively prim- itive, se e Deﬁnition 2.4 , and det v ( u ) = 0 for any u ∈ U (2) . Pr o of. By direct computation, T ( | e 3 ⟩⟨ e 3 | ) = | e 1 ⟩⟨ e 1 | . W e conclude that also Φ( | C − 1 / 2 e 3 ⟩⟨ C − 1 / 2 e 3 | ) is of rank 1, so Φ is not positivity-impro ving. UNIFORM QUANTUM TRAJECTORIES 17 Next, we verify Φ is m ultiplicativ ely primitive with p = 4 . Fix x ∈ C 3 \ { 0 } . Consider the C 3 -v ector of polynomials in ( z 1 , . . . , z 8 ) deﬁned b y P x ( z 1 , . . . , z 8 ) = 3 Y i =0 ( z 2 i +1 w 1 + z 2 i +2 w 2 ) ! x. T o establish m ultiplicativ e primitivit y , we must show that P x : C 8 → C 3 is surjectiv e. Let J x ( z 1 , . . . , z 8 ) ∈ M 3 ( C ) b e the Jacobian of P x . Then, for ( z 1 , . . . , z 8 ) ∈ C 8 ﬁxed, the determinant D x ( z 1 , . . . , z 8 ) of ( J x ( z 1 , . . . , z 8 ) ij ) 3 i,j =1 , is 0 if and only if x is an element of one of the following three sets  a, b, − a z 2 z 6 z 2 z 5 + z 1 z 6 + z 7 z 8 ( a − b ) z 2 z 5 − ( a + b ) z 1 z 6 z 2 z 5 + z 1 z 6  : a, b ∈ C  ,  a, b, − 2 a z 3 z 6 2 z 3 z 5 + z 4 z 6 + z 7 z 8 2( a − b ) z 3 z 5 − ( a + b ) z 4 z 6 2 z 3 z 5 + z 4 z 6  : a, b ∈ C  , or  a, b, − a 2 z 6 z 5 − b z 7 z 8  : a, b ∈ C  . These sets can b e computed using the SageMath 10 code a v ailable in Appendix D . It follows that for any x ∈ C 3 \ { 0 } , there exist ( z 1 , . . . , z 8 ) ∈ C 8 suc h that D x ( z 1 , . . . , z 8 )  = 0 . Then [ ER93 , Theorem 2.3] implies that the 3 en tries of P x ( z 1 , . . . , z 8 ) are algebraically indep enden t as p olynomials in ( z 1 , . . . , z 8 ) . Hence, for any ˆ x ∈ P( C 3 ) , P x ( C 8 ) = C 3 . Since  Q 3 i =0 ( z 2 i +1 v 1 + z 2 i +2 v 2 )  x = r − 2 C 1 / 2 P (( z 1 , . . . , z 8 ) × C − 1 / 2 x ) and C 1 / 2 > 0 , Φ is multi- plicativ ely primitive with p = 4 in Deﬁnition 2.4 . It remains to pro v e det v ( u ) = 0 for an y u ∈ U (2) . By deﬁnition, v ( u ) = r − 1 / 2 C 1 / 2 ( u 11 w 1 + u 12 w 2 ) C − 1 / 2 . So, it is equiv alent to prov e det( aw 1 + bw 2 ) = 0 for an y a, b ∈ C . Then, det( aw 1 + bw 2 ) = b det  a b a b  = 0 yields the proposition. □ Remark 6.8. Sinc e det v ( u ) = 0 for any u ∈ U (2) , for any p ∈ N , V p 1 is a subset of the singular matric es. So, the set of p olynomials in ( z i ) 2 p i =1 deﬁne d by the entries of Q p − 1 ℓ =0 ( z 2 ℓ +1 v 1 + z 2 ℓ +2 v 2 ) c annot b e algebr aic al ly indep endent. Inde e d, det  Q p − 1 ℓ =0 ( z 2 ℓ +1 v 1 + z 2 ℓ +2 v 2 )  = 0 . It fol lows that the metho d of pr o of use d for Pr op osition 6.6 is ineﬃcient for this example and multiplic ative primitivity, as deﬁne d in Deﬁnition 2.4 , is not e quivalent to V p 1 = M d ( C ) for some p ∈ N . Appendix A. On the different notions of irreducibility In this section w e gather some equiv alent formulations of irreducibilit y as deﬁned in Deﬁnition 2.2 and primitivity as deﬁned in Deﬁnition 2.3 . These results are not new, but w e did not ﬁnd an appropriate published reference. Theorem A.1. L et Φ : M d ( C ) → M d ( C ) b e c ompletely p ositive with Kr aus r ank 5 r . L et k ∈ N b e lar ger or e qual to r . L et µ b e a me asur e over U ( k ) such that Φ : X 7→ R U ( k ) v ( u ) ∗ X v ( u ) d µ ( u ) . Then, Φ is irr e ducible, as deﬁne d in Deﬁnition 2.2 or [ EHK78 , L emma 2.1], if and only if any subsp ac e E of C d such that v ( u ) E ⊆ E for µ -almost every u is either { 0 } or C d . 5 W e recall that the Kraus rank is the minimal k ∈ N suc h that Φ can be written as Φ( X ) = P k i =1 v ∗ i X v i with { v i } k i =1 a set of d × d matrices. 18 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL Pr o of. While this result can b e derived as a consequence of [ EHK78 , Lemma 2.1], w e provide a direct pro of. Assume that any non-zero subspace E of C d with v ( u ) E ⊆ E for µ -almost ev ery u is suc h that E = C d . Fix ˆ x ∈ P( C d ) and let E x = linspan ( { x } ∪ { v ( u n ) · · · v ( u 1 ) x : n ∈ N , ( u 1 , . . . , u n ) ∈ supp µ n } ) . By deﬁnition E x is non trivial and in v arian t under µ -almost every v ( u ) . Hence, E x = C d . Then, a dimensional argument leads to E x = linspan ( { x } ∪ { v ( u n ) · · · v ( u 1 ) x : n ∈ { 1 , . . . , d − 1 } , ( u 1 , . . . , u n ) ∈ supp µ n } ) = C d . It follo ws that, for any ˆ y ∈ P( C d ) , there exist u 1 , . . . , u p ∈ U ( k ) with p ≤ d − 1 such that either ⟨ y , x ⟩  = 0 or ⟨ y , v ( u p ) · · · v ( u 1 ) x ⟩  = 0 . Then, b y p ositivit y and linearity , ⟨ x, (Φ + Id M d ( C ) ) d − 1 ( | y ⟩⟨ y | ) x ⟩ > 0 . Since ˆ x and ˆ y are arbitrary and X 7→ (Φ + Id M d ( C ) ) d − 1 ( X ) is linear, it follows that Φ is irreducible. F or the rev erse implication, assume Φ is irreducible and let E b e a non trivial subspace of C d suc h that v ( u ) E ⊆ E for µ -almost ev ery u , but E  = C d . Then, there exists ˆ y ∈ P( C d ) such that y ⊥ E . Since E is v ( u ) -inv arian t, and b y con tin uit y of u 7→ v ( u ) , for an y u 1 , . . . , u p ∈ supp µ with p ≤ d − 1 , y ⊥ v ( u p ) · · · v ( u 1 ) E = 0 . It follows that ⟨ x, (Φ + Id M d ( C ) ) d − 1 ( | y ⟩⟨ y | ) x ⟩ = 0 for an y x ∈ E , which con tradicts the irreducibilit y of Φ . □ Theorem A.2. L et Φ b e a quantum channel. L et µ b e a me asur e over U ( k ) such that Φ : X 7→ R U ( k ) v ( u ) ∗ X v ( u ) d µ ( u ) . Then, Φ is primitive, as in Deﬁnition 2.3 , if and only if for any ˆ x ∈ P( C d ) ther e exists n ∈ N such that linspan { v ( u n ) · · · v ( u 1 ) x : ( u 1 , . . . , u n ) ∈ supp µ n } = C d . Pr o of. F or ˆ x ∈ P( C d ) and n ∈ N , let E x,n = linspan { v ( u n ) · · · v ( u 1 ) x : ( u 1 , . . . , u n ) ∈ supp µ n } . Assume Φ is primitiv e and let n b e the integer of Deﬁnition 2.3 . Assume E x,n  = C d , so there exists ˆ y ∈ P( C d ) such that y ⊥ E x,n . It follows for an y ( u 1 , . . . , u n ) ∈ supp µ n , that ⟨ y , v ( u n ) · · · v ( u 1 ) x ⟩ = 0 . Hence, ⟨ x, Φ n ( | y ⟩⟨ y | ) x ⟩ = 0 , whic h contradicts the assumption that Φ is primitiv e. Therefore, E x,n = C d . Con v ersely , if E x,n = C d then, for an y ˆ y ∈ P( C d ) , there exist ( u 1 , . . . , u n ) ∈ supp µ n suc h that ⟨ y , v ( u n ) · · · v ( u 1 ) x ⟩  = 0 . Thus, ⟨ x, Φ n ( | y ⟩⟨ y | ) x ⟩ > 0 . The linearity of Φ n and the fact that ˆ x and ˆ y are arbitrary yield that Φ is primitiv e. □ Appendix B. Sp a ce of opera tors induced by instr uments Lemma B.1. L et J b e an informational ly c omplete instrument, se e Deﬁnition 5.1 , over the me a- sur able sp ac e (Ω , F ) and assume Φ = J (Ω) is irr e ducible with p erio d m . Then ther e exist n ∈ N and a de c omp osition of C d into ortho gonal subsp ac es { E i } m i =1 , such that for any p ≥ n ,  Z Ω p f ( ω 1 , . . . , ω p ) J ( ω 1 ) ◦ · · · ◦ J ( ω p )(Id C d ) d µ ⊗ p ( ω 1 , . . . , ω p ) : f ∈ L ∞ ( µ ⊗ p )  = m M i =1 L ( E i ) , wher e we r e c al l that L ( E ) is the set of endomorphisms of the ﬁnite dimensional line ar sp ac e E . Pr o of. Let B p J :=  Z Ω p f ( ω 1 , . . . , ω p ) J ( ω 1 ) ◦ · · · ◦ J ( ω p )(Id C d ) d µ ⊗ p ( ω 1 , . . . , ω p ) : f ∈ L ∞ ( µ ⊗ p )  . F rom [ EHK78 , Theorem 4.2], there exists an orthogonal resolution of the identit y { P i } m i =1 suc h that Φ( P i ) = P i − 1 with P 0 = P m . Since Φ = J (Ω) and J is σ -additive, J ( A )( X ) ≤ Φ( X ) UNIFORM QUANTUM TRAJECTORIES 19 for an y A ∈ F and X ≥ 0 . Then, J ( A )( P i ) ≤ P i − 1 for an y i ∈ { 1 , . . . , m } and A ∈ F . It follo ws that setting E i = P i C d for i ∈ { 1 , . . . , m } implies that for any A ∈ F and i ∈ { 1 , . . . , m } , J ( A )(Id C d ) ∈ L m ℓ =1 L ( E ℓ ) . Hence, for any p ∈ N , B p J ⊂ m M i =1 L ( E i ) . F or the reverse inclusion, ﬁrst note that J (Ω)(Id C d ) = Id C d implies B p J ⊂ B p +1 J . Hence, if the rev erse inclusion is prov ed for some p = p 0 , it is true for an y p larger than p 0 . Since Φ is irreducible, so is its trace dual Φ ∗ – see [ EHK78 , p. 2]. Therefore, Φ ∗ accepts a unique ﬁxed p oin t ρ > 0 , tr( ρ ) = 1 – see [ EHK78 , Theorem 2.4]. Let ( e ℓ ) d ℓ =1 b e the canonical basis of C d and ( F k ) d 2 k =1 b e the canonical basis of M d ( C ) . Let ψ b e the element of C d ⊗ C d deﬁned by ψ = e 1 ⊗ e 1 + · · · + e d ⊗ e d . Then | ψ ⟩⟨ ψ | = P d 2 k =1 F k ⊗ F k . It follo ws from [ EHK78 , Theorem 4.2] that lim n →∞ (Φ mn ⊗ Id M d ( C ) )( | ψ ⟩⟨ ψ | ) = d 2 X k =1 m X i =1 tr( F ∗ k ρ i ) P i ⊗ F k = m X i =1 P i ⊗ ρ i , where ρ 1 , . . . , ρ m are p ositiv e semi-deﬁnite trace one ﬁxed p oin ts of Φ ∗ m suc h that supp ρ i = E i . Hence, there exist n ∈ N and a constan t c > 0 such that (Φ mn ⊗ Id M d ( C ) )( | ψ ⟩⟨ ψ | ) ≥ c m X i =1 P i ⊗ P i . That implies range  m X i =1 P i ⊗ P i  ⊆ range  (Φ mn ⊗ Id M d ( C ) )( | ψ ⟩⟨ ψ | )  . W e in troduce Kraus op erators ( v i ) k i =1 b y taking V : C d → C d ⊗ C k and M : Ω → M k, ≥ ( C ) as in Deﬁnition 5.1 , ﬁxing an orthonormal basis ( f i ) k i =1 of C k and deﬁning v i suc h that ⟨ x ⊗ f i , V y ⟩ = ⟨ x, v i y ⟩ for an y x, y ∈ C d . Then, with the index notation i := ( i 1 , . . . , i mn ) ∈ { 1 , . . . , k } mn , v i := v i 1 . . . v i mn , w e hav e m M i =1 E i ⊗ E i = range( m X i =1 P i ⊗ P i ) ⊆ range  (Φ mn ⊗ Id M d ( C ) )( | ψ ⟩⟨ ψ | )  = range X i ∈{ 1 ,...,k } mn ( v i ⊗ Id C d ) ∗ | ψ ⟩⟨ ψ | ( v i ⊗ Id C d ) ! ⊆ linspan { ( v i ⊗ Id C d ) ∗ ψ : i ∈ { 1 , . . . , k } mn } . No w, ⟨ ϕ 1 ⊗ ϕ 2 , ( a ⊗ Id C d ) ψ ⟩ = ⟨ ϕ 1 , aϕ 2 ⟩ for any a ∈ M d ( C ) , ϕ 1 , ϕ 2 ∈ C d where ϕ 2 is the complex conjugate of ϕ 2 in the canonical basis. Th us, m M i =1 L ( E i ) ⊆ linspan { v ∗ i : i ∈ { 1 , . . . , k } mn } . Finally , we take the step from Kraus op erators v ∗ i to operators in B mn J : If for an y X ∈ L m i =1 L ( E i ) , w e can sho w that there exists some g X ∈ L ∞ ( µ ⊗ mn ) such that (9) X = Z Ω mn g X ( ω 1 , . . . , ω mn ) J ( ω 1 ) ◦ · · · ◦ J ( ω mn )(Id C d ) d µ ⊗ mn ( ω 1 , . . . , ω mn ) , 20 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL then L m i =1 L ( E i ) ⊂ B mn J , and the pro of is done. T o construct g X , w e write X = Id ∗ C d X and deﬁne the co eﬃcien t v ector ϕ X ∈ ( C k ) ⊗ mn (so ϕ X ( i ) ∈ C for i ∈ { 1 , . . . , k } mn ) suc h that X = P i ϕ X ( i ) v i and ϕ Id ∈ ( C k ) ⊗ mn suc h that Id C d = P i ϕ Id ( i ) v i . No w, note that the POVM M b eing informationally complete ( i.e. satisfying Equation ( 4 )) implies that for an y ℓ ∈ N ,  Z Ω ℓ g ( ω 1 , . . . , ω ℓ ) M ( ω 1 ) ⊗ · · · ⊗ M ( ω ℓ ) d µ ⊗ ℓ ( ω 1 , . . . , ω ℓ ) : g ∈ L ∞ ( µ ⊗ ℓ )  = M k ( C ) ⊗ ℓ . W e can therefore ﬁnd a function g X ∈ L ∞ ( µ ⊗ mn ) such that | ϕ Id ⟩⟨ ϕ X | = Z Ω mn g X ( ω 1 , . . . , ω mn ) M ( ω 1 ) ⊗ · · · ⊗ M ( ω mn ) d µ ⊗ mn ( ω 1 , . . . , ω mn ) . Then, using that for any Y ∈ M d ( C ) , J ( ω 1 ) ◦ · · · ◦ J ( ω mn )( Y ) = V ∗ mn ( Y ⊗ M ( ω 1 ) ⊗ · · · ⊗ M ( ω mn )) V mn , with V mn : C d → C d ⊗ ( C k ) ⊗ mn deﬁned by V mn x = X i ( v i x ) ⊗ f i , where f i := f i 1 ⊗ · · · ⊗ f i mn , the righ t-hand side of Equation ( 9 ) amounts to Z Ω mn g X ( ω 1 , . . . , ω mn ) J ( ω 1 ) ◦ · · · ◦ J ( ω mn )(Id C d ) d µ ⊗ mn ( ω 1 , . . . , ω mn ) = V ∗ mn (Id C d ⊗ | ϕ Id ⟩⟨ ϕ X | ) V mn = X i,j ϕ Id ( j ) ϕ X ( i ) v ∗ j v i = Id ∗ C d X = X, as claimed. So indeed, L m i =1 L ( E i ) ⊂ B p 0 J with p 0 = mn . □ Appendix C. GAP Measures In this app endix we brieﬂy discuss Gaussian adjuste d pr oje cte d (GAP) me asur es , an imp ortan t class of probabilit y distributions on the sphere of a separable Hilb ert space H . Roughly sp eaking, for an y density matrix ρ on H , in terms of information gained by a measuremen t, GAP ρ is the most spread out distribution o v er the sphere S ( H ) that has density matrix ρ . W e motiv ate the use of GAP measures, giv e an explicit construction and alternative deﬁnitions (in ﬁnite dimensions) and state some crucial prop erties. F or further details we refer the reader to [ Gol+06 ; Gol+16 ] and [ V og25 ]. C.1. Motiv ation. GAP measures were ﬁrst in troduced by Jozsa, Robb, and W o otters [ JR W94 ] in an information theoretic con text. They considered the “accessible information” of an ensemble, a quan tiﬁer for the maximal amoun t of classical information that can b e extracted from it, and sho w ed that under the constrain t that its densit y matrix is given by ρ , this information is minimized b y GAP ρ . More precisely , let ν b e a probabilit y measure on P( C d ) and M a POVM on the space Ω . The join t law of the system state and measuremen t outcome is d p ν,M ( ˆ x, ω ) = ⟨ x, M ( d ω ) x ⟩ d ν ( ˆ x ) . Its marginal on P( C d ) is ν and, on Ω , it is Q ρ ν ( d ω ) = tr( M ( d ω ) ρ ν ) with ρ ν = E ν ( | x ⟩⟨ x | ) . Then the m utual information betw een ( ˆ x, ω ) 7→ ˆ x and ( ˆ x, ω ) 7→ ω with resp ect to p ν,M is the Kullbac k-Leibler div ergence (or relativ e en trop y) I ( p ν,M ) = D K L ( p ν,M | ν ⊗ Q ρ ν ) . In [ JR W94 ], the authors show that GAP ρ is the measure ν minimizing sup M I ( p ν,M ) UNIFORM QUANTUM TRAJECTORIES 21 o v er the set of probability measures ν such that ρ ν = ρ . Namely , GAP ( ρ ) = argmin ν : ρ ν = ρ sup M I ( p ν,M ) . They named the measure Scr o o ge me asur e , referring to Eb enezer Scro oge, the v ery stingy protagonist of Charles Dic k en’s nov ella A Christmas Car ol (1843), as GAP ρ is “particularly stingy with its information”. A couple of years later, Goldstein, Lebowitz, Mastro donato, T um ulk a and Zanghì [ Gol+06 ; Gol+16 ] realized that GAP measures also naturally o ccur in quantum statistical mec hanics: They sho w ed that if ρ is a canonical density matrix, i.e. , if it is of the form ρ can = 1 Z e − β H , where H is the Hamiltonian, β the in v erse temp erature and Z a normalization constant, then GAP ρ describ es the thermal equilibrium distribution of the w a v e function and can b e seen as a quan tum analogue of the canonical ensem ble of classical statistical mechanics. More precisely , they sho w that for bipartite systems, t ypically the w a v e function of the subsystem (in the sense of the c onditional wave function [ DGZ92 ; GN99 ; Gol+06 ]) is GAP ρ can -distributed where ρ can is a canonical density matrix for the subsystem. W e remark that similar considerations can also be made for grand- canonical density matrices [ ITV26 ]. C.2. Construction. W e now giv e the construction of GAP measures to which its acronym refers and for simplicity we restrict ourselv es to ﬁnite-dimensional Hilbert spaces H . Note that a similar construction is also p ossible for separable Hilbert spaces, see [ T um20 ]. The starting point for the construction is a Gaussian measure, which we then adjust and ﬁnally pro ject on to the sphere. Let ρ b e a density matrix on H , and Ψ G a Gaussian complex random vector of zero mean and co v ariance ρ . Let G ρ denote the probabilit y measure on H describing the distribution of Ψ G . As w e require that GAP ρ has exp ected density matrix ρ , i.e. , that ρ GAP ρ := Z S ( H ) | ψ ⟩⟨ ψ | dGAP ρ ( ψ ) = ρ, w e cannot simply pro ject G ρ on to S ( H ) ; this would result in the wrong densit y matrix. Thus, some adjustmen t is necessary , and w e deﬁne the Gaussian adjuste d me asur e GA ρ b y dGA ρ ( ψ ) = ∥ ψ ∥ 2 d G ρ ( ψ ) , i.e., w e adjust the density with the factor ∥ ψ ∥ 2 . Note that E ∥ Ψ G ∥ 2 = tr( ρ ) = 1 ensures that GA ρ is also a probabilit y measure on H . In the last step, w e pro ject the measure GA ρ to the sphere S ( H ) . Let Ψ GA b e a GA ρ -distributed v ector. W e then deﬁne GAP ρ as the distribution of the random v ector Ψ GAP := Ψ GA ∥ Ψ GA ∥ . W e remark that as Ψ GA is contin uously distributed, ∥ Ψ GA ∥  = 0 with probability 1 and thus Ψ GAP is well-deﬁned. Moreov er, b y deﬁnition GAP ρ satisﬁes ρ GAP ρ = ρ . C.3. Alternativ e deﬁnitions. In ﬁnite dimension, there are diﬀeren t w a ys to deﬁne GAP mea- sures. W e discuss tw o that are relev an t to us. The ﬁrst one is deﬁning GAP ρ b y its density with resp ect to the uniform measure. More precisely , supp ose that ρ has ﬁnite rank r = dim supp( ρ ) . 22 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL Then the densit y of GAP ρ relativ e to the uniform probability measure u on S (supp( ρ )) is given b y [ Gol+06 , Equation (18)]: d GAP ρ d u ( ψ ) = r det ρ + ⟨ ψ | ρ − 1 + | ψ ⟩ − r − 1 , (10) where ρ + denotes the restriction of ρ to its supp ort supp( ρ ) . F or the second alternativ e deﬁnition let K b e a copy of H and let Φ ∈ S ( H ⊗K ) with tr K | Φ ⟩⟨ Φ | = ρ , where tr K denotes the partial trace o v er K . Moreov er, let u K b e the uniform probability measure on S ( K ) and let Ψ K ∼ µ K , where µ K is the measure with density d µ K d u K ( ψ K ) = dim( K ) ∥ (Id H ⊗ ⟨ ψ K | ) | Φ ⟩∥ 2 . Then, GAP ρ is the distribution of the random v ector Ψ := (Id H ⊗ ⟨ Ψ K | ) | Φ ⟩ ∥ (Id H ⊗ ⟨ Ψ K | ) | Φ ⟩∥ , see again [ Gol+16 , Section 1.4]. As the authors there point out, the measure µ K can b e though t of as the distribution of a quantum measurement of Id H ⊗ M K on a system in the pure state Φ , where M K is the PO VM deﬁned by d M K ( ψ ) = dim( K ) | ψ ⟩⟨ ψ | d u K ( ψ ) . C.4. Prop erties. W e collect a couple of useful prop erties of GAP measures in the following propo- sition. F or the pro ofs we refer to [ Gol+06 ; Gol+16 ]; see also Section 1.5.3 in [ V og25 ] for a longer collection of properties of GAP measures. Prop osition C.1. L et ρ b e a density matrix on a sep ar able Hilb ert sp ac e H . Then, (a) GAP ρ has density matrix ρ , i.e. , ρ GAP ρ = ρ . (b) The map ρ 7→ GAP ρ is c ovariant: F or any unitary op er ator U on H and any me asur able subset A ⊂ S ( H ) , GAP ρ ( { U ∗ | ψ ⟩ : | ψ ⟩ ∈ A } ) = GAP U ρU ∗ ( A ) . (c) GAP ρ is invariant under glob al phase changes, i.e. , for any θ ∈ R and any me asur able subset A ⊂ S ( H ) , GAP ρ ( e iθ A ) = GAP ρ ( A ) . Appendix D. SageMa th 10 code listing Listing of the SageMath 10 co de used to chec k for algebraic indep endence in the examples of Section 6.4 . It is also av ailable in the Git rep ository plmlab.math.cnrs.fr/tb enoist/m ultiplicativ e- primitivit y . 1 # ! / u s r / b i n / e n v s a g e 2 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 3 # M u l t P r i m . s a g e : C h e c k i n g f o r m u l t i p l i c a t i v e p r i m i t i v i t y 4 # E x a c t c o m p u t a t i o n s m a d e o n t h e a l g e b r a i c f i e l d ( Q Q b a r ) b y d e f a u l t . 5 # L a n g u a g e : S a g e M a t h 1 0 6 # D a t e : 2 0 2 6 - 0 2 - 0 7 7 # A u t h o r s : T r i s t a n B e n o i s t , S a s c h a L i l l , C o r n e l i a V o g e l 8 # V e r s i o n : 0 . 2 9 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 10 11 # C o m p u t a t i o n o f B p 12 d e f c o m p u t e _ B p ( K r a u s , p , b a s e _ r i n g = Q Q b a r ) : 13 " " " 14 R e t u r n s t h e m a t r i x o f p o l y n o m i a l s B _ p = V _ 1 ^ p g i v e n a n i n t e g e r p UNIFORM QUANTUM TRAJECTORIES 23 15 a n d a l i s t o f K r a u s s q u a r e m a t r i c e s . 16 " " " 17 d i m _ K r a u s = K r a u s [ 0 ] . n c o l s ( ) # D i m e n s i o n o f t h e K r a u s m a t r i c e s 18 n K r a u s = l e n ( K r a u s ) # N u m b e r o f K r a u s o p e r a t o r s 19 20 R = P o l y n o m i a l R i n g ( b a s e _ r i n g , ’ z ’ , n K r a u s * p ) 21 22 B p = m a t r i x ( R , d i m _ K r a u s , d i m _ K r a u s , s p a r s e = F a l s e ) 23 B p = i d e n t i t y _ m a t r i x ( R , d i m _ K r a u s ) 24 f o r k i n r a n g e ( p ) : 25 V = z e r o _ m a t r i x ( R , d i m _ K r a u s ) 26 f o r i i n r a n g e ( n K r a u s ) : 27 V = V + R . g e n ( n K r a u s * k + i ) * K r a u s [ i ] 28 B p = B p * V 29 r e t u r n B p 30 31 # A r e t h e e n t r i e s o f B p a l g e b r a i c a l l y i n d e p e n d e n t ? 32 d e f i s _ a l g e b r a i c a l l y _ i n d e p ( l i s t _ p o l y , c o o r d s = N o n e ) : 33 " " " 34 I f T r u e i s r e t u r n e d , t h e n t h e p o l y n o m i a l s i n t h e l i s t l i s t _ p o l y 35 a r e a l g e b r a i c a l l y i n d e p e n d e n t . 36 R e t u r n s a l s o , a s a l i s t , t h e c o o r d i n a t e s w h e r e t h e J a c o b i a n 37 h a s b e e n e v a l u a t e d . 38 " " " 39 p o l y n o m i a l _ r i n g = l i s t _ p o l y [ 0 ] . p a r e n t ( ) 40 v a r i a b l e s = p o l y n o m i a l _ r i n g . g e n s ( ) 41 r i n g = p o l y n o m i a l _ r i n g . b a s e _ r i n g ( ) 42 43 i f l e n ( c o o r d s ) ! = p o l y n o m i a l _ r i n g . n g e n s ( ) : 44 r a i s e V a l u e E r r o r ( " M i s m a t c h n u m b e r o f c o o r d i n a t e s f o r e v a l u a t i o n " ) 45 46 i f c o o r d s ! = N o n e : 47 m a t r i x _ j a c o b i = j a c o b i a n ( l i s t _ p o l y , v a r i a b l e s ) ( c o o r d s ) 48 e l s e : # I f n o c o o r d i n a t e s a r e g i v e n , g e n e r a t e r a n d o m c o o r d i n a t e s . 49 c o o r d s = [ r i n g ( c h o i c e ( r a n g e ( 1 , 1 1 ) ) ) f o r i i n r a n g e ( l e n ( v a r i a b l e s ) ) ] 50 m a t r i x _ j a c o b i = j a c o b i a n ( l i s t _ p o l y , v a r i a b l e s ) ( c o o r d s ) 51 52 i f m a t r i x _ j a c o b i . r a n k ( ) < l e n ( l i s t _ p o l y ) : 53 r e t u r n F a l s e , c o o r d s 54 e l s e : 55 r e t u r n T r u e , c o o r d s 56 57 # I s m u l t i p l i c a t i v e p r i m i t i v i t y t r u e ? 58 d e f i s _ m P r i m _ r a n k _ j a c o b i _ B p ( K r a u s , p , c o o r d s = N o n e , b a s e _ r i n g = Q Q b a r ) : 59 " " " 60 R e t u r n s t h e c o o r d i n a t e s w h e r e t h e J a c o b i a n h a s b e e n e v a l u a t e d a s a l i s t 61 a n d T r u e i f B p = V _ 1 ^ p i s t h e f u l l m a t r i x s p a c e . 62 p i s a n i n t e g e r a n d K r a u s a l i s t o f s q u a r e m a t r i c e s . 63 " " " 64 65 B p = c o m p u t e _ B p ( K r a u s , p , b a s e _ r i n g ) 66 r e t u r n i s _ a l g e b r a i c a l l y _ i n d e p ( B p . l i s t ( ) , c o o r d s ) 67 68 d e f r o o t s _ o f _ j a c o b i a n _ d e t _ B p x ( K r a u s , p , i _ m i n o r = 0 , b a s e _ r i n g = Q Q b a r ) : 24 TRIST AN BENOIST, SASCHA LILL, AND CORNELIA V OGEL 69 " " " 70 R e t u r n s t h e z e r o s ( a s a l i s t ) o f t h e J a c o b i a n d e t e r m i n a n t o f B p * x 71 w i t h t h e e n t r i e s o f t h e x a s t h e v a r i a b l e s . 72 i _ m i n o r i s t h e f i r s t i n d e x o f t h e m i n o r o f t h e J a c o b i a n m a t r i x 73 w h o s e d e t e r m i n a n t i s c o m p u t e d . 74 " " " 75 76 B p = c o m p u t e _ B p ( K r a u s , p , b a s e _ r i n g ) 77 n K r a u s = l e n ( K r a u s ) 78 d i m = K r a u s [ 0 ] . n c o l s ( ) 79 80 l i s t _ v a r = v a r ( [ ’ x ’ + s t r ( i ) f o r i i n r a n g e ( d i m ) ] ) 81 82 R _ B x = P o l y n o m i a l R i n g ( b a s e _ r i n g , ’ z ’ , n K r a u s * p + d i m ) 83 B x = B p * v e c t o r ( [ R _ B x . g e n ( n K r a u s * p + i ) f o r i i n r a n g e ( d i m ) ] ) 84 85 l i s t _ p o l y = B x . l i s t ( ) 86 v a r i a b l e s = l i s t _ p o l y [ 0 ] . p a r e n t ( ) . g e n s ( ) [ : - d i m ] 87 88 m a t r i x _ j a c o b i = j a c o b i a n ( l i s t _ p o l y , v a r i a b l e s ) 89 90 d d e t = d e t ( m a t r i x _ j a c o b i [ : , i _ m i n o r : i _ m i n o r + d i m ] ) 91 92 r e t u r n s o l v e ( [ d d e t ( l i s t ( v a r i a b l e s ) + l i s t ( l i s t _ v a r ) ) = = 0 ] , l i s t _ v a r ) 93 94 95 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 96 # I f t h e s c r i p t i s n o t i m p o r t e d b u t d i r e c t l y c a l l e d 97 i f _ _ n a m e _ _ = = ’ _ _ m a i n _ _ ’ : 98 # L i s t o f K r a u s o p e r a t o r s ( l i s t o f s q u a r e m a t r i c e s o f t h e s a m e s i z e ) 99 K r a u s _ x p l 1 = [ m a t r i x ( [ [ 0 , 1 , 0 ] , [ 1 , 0 , 1 ] , [ 0 , 0 , 0 ] ] ) , 100 m a t r i x ( [ [ 0 , 0 , 0 ] , [ 0 , - 1 , 0 ] , [ 1 , 0 , - 1 ] ] ) ] 101 K r a u s _ x p l 2 = [ m a t r i x ( [ [ 1 , 1 , 0 ] , [ - 1 , 1 , 0 ] , [ 0 , 0 , 0 ] ] ) , 102 m a t r i x ( [ [ 0 , 0 , 1 ] , [ 0 , 0 , 1 ] , [ 1 , 0 , 0 ] ] ) ] 103 104 # E x a m p l e 1 105 p = 8 106 m P r i m , c o o r d s = i s _ m P r i m _ r a n k _ j a c o b i _ B p ( K r a u s _ x p l 1 , p , 107 [ i % 108 i f m P r i m : 109 p r i n t ( " I n E x a m p l e 1 , m u l t i p l i c a t i v e p r i m i t i v i t y i s v e r i f i e d . " ) 110 p r i n t ( " J a c o b i a n e v a l u a t e d i n : { } \ n " . f o r m a t ( v e c t o r ( c o o r d s ) ) ) 111 e l s e : 112 p r i n t ( " E x a m p l e 1 p o s s i b l y n o t m u l t i p l i c a t i v e l y p r i m i t i v e . \ n " ) 113 114 # E x a m p l e 2 115 p = 4 116 r o o t s = r o o t s _ o f _ j a c o b i a n _ d e t _ B p x ( K r a u s _ x p l 2 , p ) 117 p r i n t ( " I n E x a m p l e 2 , t h e r o o t s o f t h e J a c o b i a n d e t e r m i n a n t o f a 3 x 3 m i n o r 118 o f B p * x a r e : " ) 119 s h o w ( [ v e c t o r ( [ x . r h s ( ) f o r x i n r o o t ] ) f o r r o o t i n r o o t s ] ) 120 121 # E O F The co de output is: REFERENCES 25 In Example 1, multiplicative primitivity is verified. Jacobian evaluated in: (1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1) In Example 2, the roots of the Jacobian determinant of a 3x3 minor of Bp*x are: [(r1, r2, -(r1*z1*z5*z7 - ((r1 - r2)*z1*z4 - (r1 + r2)*z0*z5)*z6)/((z1*z4 + z0*z5)*z7)), (r3, r4, -(2*r3*z2*z5*z7 - (2*(r3 - r4)*z2*z4 - (r3 + r4)*z3*z5)*z6)/((2*z2*z4 + z3*z5)*z7)), (r5, r6, -1/2*(2*r6*z4*z6 + r5*z5*z7)/(z4*z7))] A c kno wledgemen ts. The researc h of TB w as partly funded by ANR pro ject DYNA CQUS, gran t n um ber ANR-24-CE40-5714. SL ackno wledges ﬁnancial supp ort by the Europ ean Union (ER C FermiMa th nr. 101040991 and ER C Ma thBEC nr. 101095820). CV was supp orted b y the Deutsc he F orsch ungsgemeinschaft (DFG, German Researc h F oundation) – TRR 352 – Pro ject-ID 470903074. Moreov er, C.V. ac kno wledges ﬁnancial supp ort by the ER C Starting Grant “F ermiMath" No. 101040991 and the ERC Consolidator Gran t “RAMBAS” No. 10104424, funded by the Europ ean Union. Views and opinions expressed are those of the authors and do not necessarily reﬂect those of the Europ ean Union or the European Research Council Executiv e Agency . Neither the European Union nor the gran ting authorit y can be held responsible for them. Conﬂicts of interest. The authors declare no conﬂict of in terest with resp ect to the present article. References [Ben+19] T. Benoist, M. F raas, Y. P autrat, and C. P ellegrini. “ In v ariant Measure for Quan tum T ra jectories”. Pr ob ability The ory and R elate d Fields 174 (2019), pp. 307 –334. [Ben+25] T. Benoist, N. Cuneo, V. Jakšić, and C.-A. Pillet. “ On en trop y production of repeated quan tum measurements I II. Quan tum detailed balance”. arXiv preprint 2025. [BFP23] T. Benoist, J.-L. F atras, and C. Pellegrini. “ Limit theorems for quantum tra jectories”. Sto chastic Pr o c esses and their Applic ations 164 (2023), pp. 288–310. [BHP25] T. Benoist, A. Hauteco eur, and C. P ellegrini. “ Quantum tra jectories. 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Cam bridge Univ ersit y Press, 2010. Institut de Ma théma tiques de Toulouse, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France Email addr ess : tristan.benoist@math.univ-toulouse.fr Dep ar tment of Ma thema tical Sciences, Universitetsp arken 5, DK-2100 Copenhagen, Denmark Email addr ess : sali@math.ku.dk Dep ar tment of Ma thema tics, LMU Munich, Theresienstr. 39, 80333 Munich, Germany Email addr ess : cornelia.vogel@math.lmu.de

Invariant measures of randomized quantum trajectories

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