The quantum moment problem and bounds on entangled multi-prover games

The quan tum momen t problem and b ounds on en tangle d m ulti-pro v er games Andrew C. Dohert y ∗ Y eong-Cherng Liang ∗ Ben T oner † Stephanie W ehner ‡ Octob er 29, 2018 Abstract W e study the quantum moment pr oblem : Given a conditional probability distribution to- gether with some p olynomia l constraints, do es ther e exist a quantum state ρ and a collec tion of measurement op erator s such that (i) the pro bability of o bta ining a par ticular outcome when a par ticular measur ement is p erfor med on ρ is sp eciﬁed by the conditio nal pro bability distribu- tion, and (ii) the measur ement op erato rs sa tisfy the constraints. F or example, the cons traints might sp ecify tha t some measur ement op era tors must commute. W e s how that if an instance of the quantum moment pr o blem is unsatis ﬁa ble, then there exists a cer tiﬁcate of a particula r form proving this. Our pro of is base d on a recent result in algebraic geometry , the noncommutativ e Positivstellensa tz of Helton and McCullough [ T r ans. Amer. Math. So c. , 3 56(9):37 21, 200 4]. A sp ecial cas e of the quantum moment problem is to co mpute the v alue of one-r ound multi- prov er g ames with entangled prov er s. Under the conjecture that the provers need only s hare states in ﬁnite-dimensional Hilbert s paces, we prove that a hierarch y o f semideﬁnite programs similar to the one g iven by Nav ascu ´ es, Pironio and Ac ´ ın [ Phys. R ev. L ett . , 98:0 1 0401 , 20 07] conv erges to the entangled v alue of the game. It follo ws that the class of la nguages r ecognized by a multi-prov er interactive pro of system where the pro vers share e ntanglement is recurs ive. ∗ School of Physical Sciences, The Universit y of Queen sland, Qu eensland 4072, Au stralia † Cen tru m voor Wiskun de en Informatica, Kru islaan 413, 1098 SJ Amsterdam, The N etherlands ‡ Institute for Quantum Information, California Institute of T echnology , Pa sadena, CA 91125, USA 1 In tro duc tion The stud y of multi -pr o v er games h as led to man y exciting results in cla ssical complexit y theory . A one-round multi-pro v er c o op er at ive game of inc omp lete information is a game p la y ed by a ve riﬁ er against tw o pro v ers, Alice and Bob. The strategy of the v eriﬁer is ﬁ xed. He randomly chooses t wo questions according to some ﬁxed prob ab ility distribu tion and send s one question to eac h pro ve r. Alice and Bob then eac h return an ans wer to the ve riﬁ er . Th e v eriﬁ er decides whether to acce pt these answers on the basis of some p r e-deﬁned rules of the game that s p ecify whether the give n answ ers are winning answ ers for the questions s ent. T o win the game, Alice and Bob ma y th ereb y agree on an y strategy b eforehand, bu t they ma y no longer communicate once the game h as started. The maximum p robabilit y with whic h Alice and Bob can cause the ve riﬁ er to accept is known as the value of the game. A simp le example is the w ell-kno wn CHSH game [14, 15]. In this case, the questions and answ ers are bits. The veriﬁer chooses qu estions s ∈ { 0 , 1 } and t ∈ { 0 , 1 } uniformly at random and sen d s s to Alice and t to Bob. I n order to win the game, Alice and Bob m ust r eply with bits a, b ∈ { 0 , 1 } such that s ∧ t = a ⊕ b , i.e., the logica l AND of s and t sh ould b e equal to the X OR of a and b . It is straigh tforwa rd to ve rify that the CHSH game has v alue 3 / 4. In teractiv e pro of syste ms ha ve receiv ed considerable atten tion since their in tro du ction by Babai [2] and Goldwasser, Micali and Rac k oﬀ [20] in 1985. Of sp ecial inte rest to us are pr o of systems with multiple pro vers [3, 5, 10, 17, 18, 29] as introdu ced b y Ben-Or, Goldw asser, Kilian and Widger- son [5], whic h can b e describ ed in terms of multi- pr o v er games b et w een a veriﬁer, and t w o or more pro vers. Whereas the pro vers are computationally unb ounded, the veriﬁer is limited to prob ab ilistic p olynomial time. Both the p ro ve rs and the ve riﬁ er ha ve access to a common input string x . The goal of the pro v ers is to con vince th e v eriﬁer that x b elongs to a pre-sp eciﬁed language L . The v eriﬁer’s aim, on the other hand, is to determine wh ether the prov ers’ claim is ind eed v alid. In eac h round, the v eriﬁer sends a p oly( | x | ) size qu ery to the pro vers, wh o return a p olynomial size answ er. A t the end of the p roto col, the v eriﬁer either accepts, meaning that he concludes x ∈ L , or rejects, based on the messages exc h an ged and his o wn pr iv ate r andomness. A language L has a m ulti-prov er in teractiv e pro of system if there is a p roto col suc h that, if x ∈ L , there exist answ ers the p ro v ers can give w h ic h will cause the v eriﬁer to accept with high probability . Ho w eve r, if x 6∈ L , then there exists n o s trategy f or the pro ve rs that will only cause the ve riﬁ er to accept, except with v ery lo w probabilit y . Here, x and L lead to p articular game. Let MIP denote the class of languages having a m ulti-prov er interactiv e pro of system. It has b een sh o wn that classical t wo -pr o v er in teractiv e pr o of systems are just as p ow erful as pro of systems inv olving m ore than tw o pro vers. Indeed, Babai, F ortno w and Lund [3], and F eige and Lo v´ asz [18] ha ve sh o wn that a language is in NEXP if and only if it has a two -pr o v er one-round pro of system, i.e., MIP = NEXP . 1.1 Multi-pro v er games with en tanglemen t In this p ap er, we stu dy multi-pro v er games in a quant um s etting. In particular, we allo w Alice and Bob to share an en tangled quan tum s tate as part of their strategy . After receiving th eir questions, the prov ers may p erform an y lo cal measurement on their part of the en tangled state, and decide on an answer based on the outcome of their measurement. All communicat ion b et ween the ve riﬁ er and the pr ov ers r emains classical. It turns out that sharing enta n glement can increase the p robabilit y that the prov ers can cause the v eriﬁer to accept, an eﬀect k n o wn as qu an tum nonlo c ality . F or example, if the pro v ers share a maximally en tangled state of t w o qubits they can win the C HSH game (cause the v eriﬁer to accept) with probabilit y p ∗ CHSH ≈ 85% > 3 / 4. 1 W e write MIP ∗ for the set of languages that ha ve inte ractiv e pr o ofs with en tangled prov ers. V ery little is kno wn ab out MIP ∗ . Most imp ortan tly , it was not kno wn b efore our w ork whether there exists an algorithm of an y complexit y for deciding mem b ership in MIP ∗ , except f or extremely restricted classes of games. In particular, if we restrict to ga mes where Alice and Bob eac h answ er a single bit a, b ∈ { 0 , 1 } , and the v eriﬁer only looks at the X OR of these t w o b its, then the (en tangled) v alue of the game can b e computed in time p olynomial in the num b er of questions [12, 15]. Let ⊕ MIP[2] d enote the restricted class where the veriﬁer’s outpu t is a function of the X OR of tw o binary answers. Then ⊕ MIP ∗ [2] ⊆ EXP [15], while it is known that ⊕ MIP = NEXP, for certain completeness and soun dness p arameters [22], i.e., the r esulting pro of system is signiﬁcan tly w eak ened if the pro vers are allo w ed to share en tanglemen t. In fact, suc h a pro of sy s tem can ev en b e sim ulated usin g just a single quantum pro v er, i.e., ⊕ MIP ∗ [2] ⊆ QIP(2) ⊆ EXP [48, 27]. Unfortunately , ve ry little is kno wn for more general games wh ere we allo w shared ent anglement b et ween the pr o v ers, and w here Alice and Bob giv e m uch longer answe rs, or wh en w e ha ve m ore than tw o pr o v ers. Unlik e in the classical case, general m ulti-prov er games are not equiv alent to t w o-prov er games w hen the pro v ers sh are en tanglemen t [25]. F or t wo -pr ov er u n ique ga mes (where for eac h pair of questions there exists exactly one p air of winnin g answers), it is kno wn that w e can appro ximate the v alue of a t wo-pro ver game to within a certain accuracy in p olynomial time [26]. Masanes [33] has sho wn ho w to compute the v alue of m ulti-prov er games where the questions to, and the answers f rom eac h p ro ve r are bits. But ev en f or v ery small games with a v ery limited n umb er of questions, the entangled v alue is t ypically unk n o wn [9]. Assuming that th e pro vers share quant u m en tanglemen t is a reasonable mo del b ecause it cap- tures the prop erties of a m ulti-prov er game that a v eriﬁer can enforce ph ysically: w h ile the v eriﬁ er can en f orce the condition th at the pr o v ers cannot comm unicate by ensurin g that they are spacelik e- separated, he has no wa y to ensu re th at pr o v ers in a quan tum universe do not sh are entangle ment. Multi-pro v er games with en tangled pr o v ers are also kno wn as non-lo c al games with ent anglement . The entangle d value of a game is the maximum probabilit y with which Alice and Bob can win using en tanglemen t. Here, we are concerned with the follo wing question: Ho w can we compute the en tangled v alue of non-lo cal games with multiple pro vers? And, h ow can w e decide mem b ersh ip of MIP ∗ ? 1.2 Results Quan tum momen t problem. T o r eac h our goal, w e ﬁrst introdu ce the quantum mo ment pr ob- lem that is a generalization of our problem. Informally , the quan tum moment problems a sks whether giv en a cond itional probability distributions and some p olynomial constrain ts on ob s erv ables, we can ﬁ nd a quantum state and quan tum measuremen ts satisfying the constraints, that pr o vide us with the requir ed pr obabilities. W e ma y use the constraints to imp ose certain restrictions on the form of our quan tum measurements. F or example, w e ma y w ant to d emand that t wo measurement op erators act on t wo or more distant subsystems indep enden tly . Determining wh ether there is an en tangled strategy for a m ulti-pro ve r game that ac hieve s a certain winning p robabilit y is a sp ecial case of the quan tum momen t problem. Other sp ecial cases of the quant um momen t problem include the so-called classical marginal problem [41], which asks w h ether giv en certain marginal d istributions, we can ﬁn d a j oin t distrib u- tion th at has the desired marginals. Our problem is also closely related to th e quantum marginal problem in whic h the aim is to ﬁnd a d en sit y matrix for a m ultipartite quan tum system that is consisten t with a sp eciﬁed set of redu ced density matrices for s p eciﬁc su bsystems. This problem is 2 QMA-complete and has attracted a lot of interest recen tly [32]. A s p ecial case is N -represen tabilit y , an imp ortant problem with a long history in quan tum c hemistry [28]. One k ey diﬀerence b et w een our quan tum moment pr oblem an d the quan tum marginal problem is that in the latter case the dimension of the q u an tum state, and its v arious sub systems, is sp eciﬁed. In the quan tum momen t problem the aim is to ﬁnd a s tate satisfying the giv en constrain ts in a quan tum system of an y , p ossibly in ﬁ nite, dimens ion. Finally , it ma y also b e p ossible to treat games with quantum v eriﬁer within the framew ork of the quan tum moment problem. Refuting unsat isﬁable inst ances. W e describ e a general wa y of proving that an instance of a quan tum moment problem is unsatisﬁable. The pro of follo ws from a recen t r esult of Helton and McCullough [23], a P ositivstellensatz for p olynomials in noncommuting v ariables. Th e c hoice of p olynomials will deﬁne a particular instance of the qu an tum momen t problem, where th e v ariables corresp ond to measuremen t op erators. In Helton and McCullough’s result the noncomm u ting v ariables are required to satisfy certain p olynomial equalit y and inequ ality constrain ts but can b e ev aluated in any quan tum system, ev en one of inﬁn ite dimensions. Informally , the Positi vstellensatz states that any su c h p olynomial th at is p ositiv e can b e wr itten a sum of squ ares, a form that make s it ob vious that the p olynomial is p ositiv e. By p ositiv e we mean that, w h enev er the constrain ts are satisﬁed, the p olynomial is p ositiv e semideﬁnite, i.e. it has a p ositive exp ectation v alue for all quan tum states in wh atev er q u an tum system. Su ch a represen tation as a sum of squ ares acts as a certiﬁcate f or the uns atisﬁability of an instance of a quantum momen t p roblem. Certiﬁcates of this kind hav e often b een used in the theoretical ph ysics literature to p lace v ery general b ounds on quan tum corr elations (see for example [19]). Helton and McCullough’s r esu lt sho ws that s u c h certiﬁcates are all that is ev er required to d emonstrate that an instance of the quantum moment problem is unsatisﬁable. T ensor pro ducts and comm utat ion. In order to apply the Positi vstellensatz to obtain b ou n ds on the en tangled v alues of t wo- p r o v er games, w e need to incorp orate a constraint in the corre- sp ond in g qu an tum moment p roblem that ensur es Alice and Bob’s measurements act on diﬀerent subsystems H A and H B . When Alice and Bob share a quantum sys tem of some ﬁn ite d im en - sion, this means that one demands that the Hilb ert space H describing this system d ecomp oses as H = H A ⊗ H B . Alice and Bob’s measuremen ts should b e of the form A s ⊗ B t with A s ∈ H A and B t ∈ H B for all questions s and t . Unfortunately , w e can only apply the P ositivstellensatz when the constrain ts are p olynomials in A s and B t . Thus w e need an additional trick to imp ose an explicit tensor pr o duct stru ctur e. T o get arou n d this pr oblem, we demand that for all s and t w e hav e [ A s , B t ] = 0, i.e., that all measurement operators of Alice comm ute with all those of Bob. If the Hilb ert space is ﬁ nite-dimensional, th en imp osing th e comm utativit y constraints is actually e quivalent to demand ing a tensor pro d uct structure. This result is well- kn o wn in the mathemat- ical ph ysics literature [45]. Here, we p r o vide a simple v ersion of this argumen t accessible from a computer science p ersp ectiv e, wh ic h directly applies to our analysis of m ulti-pro ve r games. F rom a ph ysics p oin t of view, ho wev er, the u sual requirement on observ ables that can b e measured in space-lik e separated regions is that th ey should c ommute , n ot th at they should hav e a tensor p ro du ct factorization. Indeed, this commutativit y requirement on local observ ables is regarded by many as an axiom that shou ld b e satisﬁed b y any reasonable quantum mec han ical theory of nature [21]. Unfortunately , wh en the algebra of observ ables cannot b e represen ted on a ﬁnite-dimensional Hilb ert sp ace it is an op en question wh ether this comm utativit y prop erty 3 implies the existence of a tensor pro d uct factorizatio n. Our results w ill p ro vide b ounds on the v alues of m ulti-pro ver games that are v alid for all quan tum systems wheneve r the observ ables of diﬀeren t pla y ers commute. W e will refer to the maxim um probabilit y of winn ing a game G with (p ossibly) inﬁnite-dimensional op erators s atisfying commuta tivit y constrain ts as the ﬁeld-the or etic value ω f ( G ) of the game. It is an op en question whether this is the same as the us ual en tangled v alue of the game. Since MIP ∗ w as deﬁned with a tensor pro d uct structure in mind , we here deﬁne the class MIP f , w h ere the tensor pro ducts are replaced by the comm utativ e requir ement. Th e class MIP f seems m ore app r opriate to our main motiv ation of studying th e p o w er of m ulti-prov er ga mes where the prov ers are only limited by wh at they can achiev e physically . Restricting Alice and Bob to sharing ﬁnite-dimensional systems do es not seem natural from a ph ysical p ersp ectiv e. A hierarc hy of semideﬁnite programs. W e kno w that the Po sitivstellensatz leads to certiﬁ- cates that tell us when a particular quan tum moment p roblem is unsatisﬁable. But ho w can w e ﬁnd suc h certiﬁcates? If w e place a b ound on the size of the certiﬁcate, then the pr oblem of deter- mining w h ether there exists a certiﬁcate of that size can b e form ulated as a semideﬁnite program (SDP) [47, 6]. In particular, searching for certiﬁcates of increasing size yields a hierarc hy of SDPs. The resulting hierarch y is ve ry similar to the one presente d in a groundbr eaking pap er of Na v ascu´ es, Pironio and Ac ´ ın [36], whic h partly motiv ated this w ork. In many applicatio ns , in cluding m u lti-pr ov er games, we are not only in terested in wh ether a sp eciﬁc in s tance of the m omen t problem is satisﬁable but in ﬁndin g the b est p ossible b ound on some linear com bination of moments. Once again ﬁ xing the size of the certiﬁcate s of infeasibilit y straigh tforw ard ly leads to a hierarc hy of S DPs that p ro vide progressive ly tigh ter b ounds. F or m ulti-prov er games G , it was previously n ot kno wn whether the solutions to this hierarc hy of S DPs con v erged to the en tangled v alue of the game, whic h w e denote by ω ∗ ( G ). Here w e almost s ho w this. What we actually show is that the hierarc hy con v erges to the ﬁ eld-theoretic v alue (see ab ov e) of a non-lo cal ga me G . In the language of the quan tum moment pr oblem, we wish to know if there exists an en tangled strategy for G suc h that the prov ers win with some ﬁxed pr ob ab ility p . Ho we ver, the Po sitivstellensatz only yields a certiﬁcate that there is no ent angled strategy that wins w ith probabilit y p if there is also no su c h strategy ev en with inﬁnite- dimensional me asur ement op er ators . If the m easur emen t op erators are inﬁ nite-dimensional, then the commutati vity constraints do not necessarily imply the existence of a tensor p ro duct structure. In other words, w e sh o w that our hierarc hy co nv erges to the ﬁeld-theoretic v alue of the game. MIP ∗ is recursiv e S ince our hierarc hy conv erges, w e can compu te the v alue of an en tangled game and hen ce obtai n an algorithm f or d eciding memb ership of MIP ∗ (under the assumption th at the optimal v alue is ac hiev ed with ﬁnite-dimensional op erators) and of MIP f . Th is implies that these classes are recursiv e. Examples: The I 3322 inequalit y and Y ao’s inequality . Finally , w e demonstrate the p ow er of our tec hniqu e by pro viding an extremely s imple, algorithmically constructed, certiﬁcate b ounding the v alue of a tw o-party Bell inequalit y [4] kn o wn as the I 3322 inequalit y [16], and a multi- p la y er non-lo cal game suggested b y Y ao and collab orators [50]. 4 1.3 Op en Questions With resp ect to the ab ov e d iscussion, it would b e interesting to kno w whether there are games G su c h that ω ∗ ( G ) is strictly less than ω f ( G ). Can it really help the pro vers to hav e inﬁnite- dimensional systems when the num b er of questions and answe rs in the ga me a r e ﬁnite? On e w a y to establish that there is no adv antag e to ha ving inﬁnite-dimensional systems would b e to ‘round’ the SDP hierarc h y d irectly to a quan tum strategy with ﬁnite en tanglemen t, bypassing th e (nonconstructiv e) Posit ivstellensatz altogether. F or XOR-game s, the ﬁrst lev el of our hierarch y is tigh t and it is w ell-kno wn ho w a solution of the SDP can b e tr ansformed in to a quantum s trategy via so-calle d Tsirelson’s vect or construction [11, 12, 13]. Ho wev er, there exist many n on-lo cal games, for whic h the ﬁrst lev el of the hierarch y do es not p ro vide us with the optimal v alue of the game, but merely gives us an upp er b ound. This f act alone shows that f or general games, we cannot ﬁnd su c h a nice em b edd ing of v ectors int o observ ables as can b e done for X OR-games. Ho wev er, something similar ma y still b e p ossible f or restricted classes of games, exhibiting a lik ewise sp ecial structure. W e also do n ot esta blish an ything ab out the r ate of conv ergence of the SDP hierarc hy . In some n um er ical exp er im ents with s mall games, the lo w lev els of the S DP h ierarc hy do yield optimal solutions. Establishing this in general would provide an up p er b oun d on MIP ∗ . W e hav e made partial progress on this question b y provi n g con verge nce for a particular hierarch y of SDPs. 1.4 Related work In Ref. [36], a pap er whic h partly insp ir ed this w ork, Na v ascu´ es, Pironio, and Ac ´ ın (NP A), deﬁned a closely related semideﬁn ite programming hierarc hy . Su bsequent ly , and ind ep endently of us, NP A ha v e pro ved that their semideﬁnite p rogramming h ierarc hy con v erges to the ﬁeld-theoretic v alue of the game [34]. O ur pap er and theirs are complemen tary: While our w ork emph asizes the connection with P ositivstellensatz of Helton and McCullough, NP A prov e con v ergence directly . Their p ro of has a num b er of adv anta ges: most n otably , wh en their hierarc hy conv erges to the ﬁeld-theoretic v alue of the game at a ﬁ nite lev el, NP A ob tain a b ound on the d imension of th e s tate required to repro du ce the correlations. NP A hav e also sh o wn that their new tec h n ique for provi n g conv ergence can b e extended to general p olynomial optimizati on problems in noncomm u tative v ariables [35 ]. Finally , our tec hn iques ha v e recent ly extended by Ito, Hirotada, and Matsumoto to the ca se of games with quan tum messages b et wee n ve riﬁ er and pro v ers [24]. 1.5 Outline In Section 2, w e p ro vide an in tro d uction to non-lo cal games in cluding all necessary deﬁnitions. Section 3 then d eﬁnes the q u an tum moment problem, and Section 4 in tro d uces our main tools. In particular, Section 4.1 provi d es an explanation of why we obtain a tensor pr o duct structure from comm utation relations, and in S ection 4.2, we sho w that if a quantum momen t problem is unsatisﬁable, we can ﬁnd certiﬁcates of this fact u sing the Positivstel lensatz. W e then use th ese to ols in our SDP hierarc hy in Section 5 and conclude in Sectio n 5.2 with some explicit examples. 5 2 Preliminaries 2.1 Notation W e assume general familiarit y w ith the quan tum mo del [37]. In the follo wing, we use A † to denote the conjugate transp ose of a matrix A . A matrix is Hermitian if and only if A † = A . W e write A ≥ 0 to indicate that a matrix A is p ositive semideﬁnite , i.e., it is Hermitian and has no n egativ e eigen v alues. W e also u s e A = 0 to express that A is the all-zero matrix and A 6 = 0 to in dicate that A has at least one non-zero entry . Th e ( i, j )–en try of A will b e d enoted b y [ A ] i,j . F or t w o matrices A and B w e wr ite their comm u tator as [ A, B ] = AB − B A . W e use H to den ote a Hilb ert space and H k the Hilb ert space b elonging to su bsystem k . I k is the identi ty on system k , and B ( H ) denotes the set of all b ounded op erators on the Hilb ert space H . Unless stated otherwise, w e tak e all systems to b e ﬁnite-dimensional. W e will also emplo y the shorthand B ( H ) × n := B ( H ) × . . . × B ( H ) | {z } n for the n -fold Cartesian pro duct of B ( H ), and let [ n ] := { 1 , . . . , n } . F u rthermore, we will use |S | and |L| to denote the n um b er of elemen ts of a set S and list L resp ectiv ely . F or the purp ose of su bsequent discussion, we now note that a Hermitian p olynomial p ( X ) in noncomm utativ e v ariables X = ( X 1 , . . . , X k ) is a sum of squar es (SOS ) if there exist p olynomials (matrices) r j of appropriate dimension suc h that p ( X ) = X j r † j r j . It is imp ortant to note that if p ( X ) is an SOS p olynomial, it is also a p ositiv e semideﬁnite matrix, i.e., p ( X ) ≥ 0. 2.2 Games As an examp le application of the quan tum momen t p roblem, we will consid er co op erative games among n p arties. F or simplicit y , we ﬁrst describ e the setting for only t wo parties, henceforth called Alice and Bob. A generaliza tion is straigh tforward. Let S , T , A and B b e ﬁnite sets, and π a probabilit y distribution on S × T . Let V b e a predicate on S × T × A × B . Th en G = G ( V , π ) is the follo wing tw o-p erson coop erative game: A pair of qu estions ( s, t ) ∈ S × T is chosen at rand om according to the probabilit y d istribution π . T hen s is sen t to Alice, and t to Bob. Up on receiving s , Alice has to reply with an an s w er a ∈ A . Lik ewise, Bob h as to reply to question t w ith an answer b ∈ B . They win if V ( s, t, a, b ) = 1 and lose otherwise. Alice and Bob ma y agree on any kin d of str ategy b eforehand , but they are no longer allo wed to comm un icate once th ey h a v e receiv ed questions s and t . T he value ω ( G ) of a game G is the maximum probabilit y that Alice and Bob win the game. In w hat follo ws , w e w ill write V ( a, b | s , t ) instead of V ( s, t, a, b ) to emphasize the fact that a and b are answ ers giv en questions s and t . Here, w e are particularly int erested in non-lo cal games and w here Alice an d Bob are allo wed to share an arbitrary en tangled state | ψ i to help them win the game. Let H A and H B denote the Hilb ert spaces of Alice and Bob resp ective ly . The state | ψ i ∈ H A ⊗ H B is part of the quantum strategy that Alice and Bob can agree on b eforehand. Th is means that for eac h game, Alice and Bob can c ho ose a sp eciﬁc | ψ i to maximize their c hance of success. In addition, Alice and Bob can 6 agree on qu an tum measurements where w e ma y without loss of generalit y assume that these are pro jectiv e measuremen ts [15]. 1 F or eac h s ∈ S , Alice has a p ro jectiv e measuremen t d escrib ed by { A a s : a ∈ A } on H A . F or eac h t ∈ T , Bo b has a p ro jectiv e measurement describ ed b y { B b t : b ∈ B } on H B . F or questions ( s, t ) ∈ S × T , Alice p erf orm s the measuremen t corresp onding to s on her part of | ψ i whic h giv es her outcome a . Likewise, Bob p erforms the measuremen t corresp ond ing to t on his part of | ψ i with outcome b . Both send their outcome, a and b , back to the veriﬁer. The probabilit y that Alice and Bob answ er ( a, b ) ∈ A × B is then given b y h ψ | A a s ⊗ B b t | ψ i . W e can no w deﬁn e: Deﬁnition 2.1. The entangle d v alue of a t wo -pr o v er game with classica l veriﬁer G = G ( π , V ) is giv en by: ω ∗ ( G ) = lim d →∞ max | ψ i∈ C d ⊗ C d k | ψ i k =1 max A a s ,B b t X a,b,s,t π ( s, t ) V ( a, b | s, t ) h ψ | A a s ⊗ B b t | ψ i , (1) where A a s ∈ B ( H A ) and B b t ∈ B ( H B ) for some Hilb ert space H = H A ⊗ H B , satisfying A a s , B b t ≥ 0, P a A a s = I A , P b B b t = I B for all s ∈ S and t ∈ T . W e also deﬁne a more general v ersion of this statemen t, wh ich w ill pro vide an upp er b oun d to the quan tum v alue of the game: Deﬁnition 2.2. The ﬁeld-the or etic value of a t wo-pro ver game with classica l veriﬁer G = G ( π , V ) is giv en by: ω f ( G ) = s up A a s ,B b t    X a,b,s,t π ( s, t ) V ( a, b | s, t ) A a s B b t    , (2) where k O k is the op erator n orm of O , A a s ∈ B ( H ) and B b t ∈ B ( H ) for some Hilb ert sp ace H , satisfying A a s , B b t ≥ 0, P a A a s = P b B b t = I for all s, t , and [ A a s , B b t ] = 0 f or all s ∈ S , t ∈ T , a ∈ A , and b ∈ B . Lemma 2.3. L e t G = G ( π , V ) b e a two-pr over game with classic al ve riﬁer. Then ω ∗ ( G ) ≤ ω f ( G ) . Pr o of. Let ε > 0. Cho ose d suﬃcien tly large so that there is a normalized state | ψ i and op erators A a s , B b t deﬁning a str ategy with winn ing probabilit y at least ω ∗ ( G ) − ε . Let ˆ A a s = A a s ⊗ I B and ˆ B b t = I A ⊗ B b t . Then ˆ A a s and ˆ B b t are p ositiv e semideﬁnite op erators on C d 2 satisfying all the conditions in Deﬁnition 2.2. Finally , ω f ( G ) = s up ˜ A a s , ˜ B b t    X a,b,s,t π ( s, t ) V ( a, b | s , t ) ˜ A a s ˜ B b t    ≥    X a,b,s,t π ( s, t ) V ( a, b | s, t ) ˆ A a s ˆ B b t    ≥ h ψ |  X a,b,s,t π ( s, t ) V ( a, b | s, t ) ˆ A a s ˆ B b t  | ψ i ≥ ω ∗ ( G ) − ε. 1 By Neumark’s theorem, any generalized me asurements describ ed b y p ositive-operator-v alued meas ure can b e implemented as pro jectiv e measurements in some higher d imensional Hilb ert space. See, for example, p p. 285 of Ref. [40]. 7 Since ε w as arbitrary , the result follo w s. In our examples, w e will sometimes use the term Bel l ine quality [4 ] to refer to a particular n on- lo cal game. This is an equiv alen t formulatio n, where w e only co n s ider terms of the form h ψ | A a s B b t | ψ i . The v alue of the game can then b e obtained b y a v eraging. In inequalities wh ere Alice and Bob ha v e, r esp ectiv ely , tw o measur emen t outcomes for eac h p ossible c hoice of measurement setting (i.e., A = B = { 0 , 1 } ), their measuremen ts can b e d escrib ed b y observ ables of the form A s = A 0 s − A 1 s and B t = B 0 t − B 1 t resp ectiv ely . In this case, we state inequalitie s in the form of th e observ ables A s and B t where w e will use the shorthand h A s B s i = h ψ | A s B s | ψ i . Note that it is straigh tforwa rd to extend the ab o v e deﬁnitions to the s etting inv olving m u ltiple pla y ers, bu t the resulting terms will b e m uch hard er to read. When considering games among N pla yers P 1 , . . . , P N , let S 1 , . . . , S N and A 1 , . . . , A N b e ﬁnite sets corresp onding to th e p ossible questions and answers resp ectiv ely . Let π b e a pr ob ab ility d istr ibution on S 1 × . . . × S N , and let V b e a pr edicate on A 1 × . . . × A N × S 1 × . . . × S N . Th en G = G ( V , π ) is the follo w ing N -pla y er co op erativ e ga me: A set of questions ( s 1 , . . . , s N ) ∈ S 1 × . . . × S N is chosen at rand om according to the probability distribution π . Pla y er P j receiv es question s j , and then resp onds with an answer a j ∈ A j . The p lay ers win if and only if V ( a 1 , . . . , a N | s 1 , . . . , s N ) = 1. Let | ψ i denote the play ers’ c hoice of state, and let X j := { X a j s j | a j ∈ A j } denote the p ositive -op er ator-v alued measure(men t) (PO VM) of pla yer P j for question s j ∈ S j , i.e., P a j X a j s j = I j and X a j s j ≥ 0 for all a j . The v alue of the game can no w b e written as ω ∗ ( G ) = lim d →∞ max | ψ i∈ ( C d ) ⊗ N k | ψ i k =1 max X 1 ,...,X N X s 1 ,...,s N π ( s 1 , . . . , s N ) X a 1 ,...,a N V ( a 1 , . . . , a N | s 1 , . . . , s N ) h ψ | X a 1 s 1 ⊗ . . . ⊗ X a N s N | ψ i , where th e maximization is tak en o ver all legitimate POVMs X j for all j ∈ [ N ]. Similarly , we can no w write the ﬁeld-theoretic v alue of the game as ω f ( G ) = sup X 1 ,...,X N    X s 1 ,...,s N π ( s 1 , . . . , s N ) X a 1 ,...,a N V ( a 1 , . . . , a N | s 1 , . . . , s N ) X a 1 s 1 . . . X a N s N   , where w e now hav e P a j X a j s j = I for all a j , s j , j and [ X a j s j , X a j ′ s j ′ ] = 0 for all j 6 = j ′ . 2.3 Pro of systems In teractiv e pro of systems can b e phrased as s uc h games. F or completeness, we h er e p ro vide a deﬁnition of MIP . W e refer to the introdu ction and the previous section for an explanation of the notions of a tensor p ro du ct form vs. commutatio n relations. Deﬁnition 2.4. F or 0 ≤ s < c ≤ 1, let ⊕ MIP ∗ c,s [ k ] denote th e class of all languages L r ecognized b y a classical k -pr ov er interacti ve pro of system with en tanglemen t such that: • The interactio n b et w een the v eriﬁer and the pr o v ers is limited to one round and classical comm unication. The v eriﬁer c h o oses k questions from a ﬁnite s et of p ossible questions, according to a ﬁxed probability d istribution kn o wn to the p ro v ers, and send s one qu estion to eac h p r o v er. Afterw ards , the prov ers ma y p erform an y measuremen t that has tensor pr o duct form on a shared state | ψ i that they hav e chose n ahead of time. Eac h prov er returns an answ er to the v eriﬁer, w hose decision function is kno wn to the pro v ers. 8 • If x ∈ L then there exists a s tr ategy for the pr o v ers for which the probability that the ve riﬁ er accepts is at least c (the c ompleteness p arameter). • If x / ∈ L then, wh atev er strategy th e k pro v ers follo w, the probability that the v eriﬁer accepts is at most s (the soundness parameter). Deﬁnition 2.5. F or 0 ≤ s < c ≤ 1, let ⊕ MIP f c,s [ k ] denote the class corresp onding to a mo diﬁed v ersion of the previous d eﬁ nition: here w e merely ask that the measurements op er ators b et w een the diﬀeren t pla ye rs commute . 3 The quan tum momen t p roblem 3.1 General form Let u s no w state th e qu an tum m omen t problem in its most general form, b efore explainin g its connection to non-local games. In tuitive ly , the quan tum momen t problem states that giv en a certain probabilit y distribu tion, is it p ossible to ﬁn d q u an tum measur ements and a state that pro vide us with suc h a distrib u tion? Deﬁnition 3.1 ( Quan tum momen t problem ) . Giv en a list of num b ers M = ( m i | m i ∈ [0 , 1]), a set of p olynomial equations R = { r = 0 | r : B ( H ) ×|M| → B ( H ) } , and p olynomial inequalities S = { s ≥ 0 | s : B ( H ) ×|M| → B ( H ) } , do es there exist said Hilb ert space H , op erators M i ∈ B ( H ) and a state ρ ∈ B ( H ) such that 1. F or all m i ∈ M , T r( M i ρ ) = m i . 2. F or all r ∈ R , r ( M 1 , . . . , M |M| ) = 0. 3. F or all s ∈ S , s ( M 1 , . . . , M |M| ) ≥ 0. 3.2 Non-lo cal games In this pap er, we are particularly in terested in a sp ecial case of the quantum momen t pr oblem, where w e consider measurements on many sp ace-lik e separated systems as in the setting of non- lo cal games. F or simplicit y , w e will explain the connection to non-lo cal games for only t wo s u c h systems, Alice H A and Bob H B , wh ere it is straigh tforward to extend our arguments to more th an t w o. On eac h system H A and H B , we w ant to p erform a ﬁ n ite set of p ossible measurements S and T eac h of which has the same ﬁnite set of outcomes A and B resp ectiv ely . Let m A ( a | s ) and m B ( b | t ) denote the pr obabilit y that on s ystems H A and H B w e obtain outcomes a ∈ A and b ∈ B giv en measuremen t settings s ∈ S and t ∈ T resp ectiv ely . F ur thermore, let m AB ( a | s, b | t ) d enote the joint probabilit y of obtaining outcomes a and b giv en settings s and t when p erforming measurements on systems H A and H B . Informally , our qu estion is n o w: Giv en pr obabilities m AB ( a | s, b | t ), d o es there exist a shared state ρ such that w e can ﬁnd measur emen ts on the ind ividual systems H A and H B that lead to suc h probabilities? Let’s ﬁrst consider what p olynomial equations and inequalities w e need to express our problem in the ab o ve f orm. First of all, how can we expr ess the fact that w e wan t our measuremen t op erators to act on the individ ual systems H A and H B alone? I.e, ho w can w e ensur e that the measuremen t op erators h a v e tensor p ro du ct form ? W e will sho w in Lemma 4.1 that we 9 are guaran teed to observ e suc h a tensor pro duct form if and only if for all s ∈ S , a ∈ A , t ∈ T and b ∈ B we ha v e [ A a s , B b t ] = 0, wh ere w e used A a s and B b t to denote the measuremen t op erators of Alice and Bob corresp onding to measurement settings s and t and outcomes a and b resp ectiv ely . Hence, w e need to imp ose the p olynomial equalit y constraints of the form [ A a s , B b t ] = 0. F urthermore, w e w ant that for b oth s y s tems A and B , we obtain a v alid measuremen t for ea ch measuremen t setting s ∈ S and t ∈ T . I.e., we imp ose further p olynomial equalit y constrain ts for all s ∈ S and t ∈ T of the form X a ∈ A A a s − I = 0 and X b ∈ B B b t − I = 0 , and ﬁnally the follo win g p olynomial inequalit y constrain ts for all a ∈ A and b ∈ B A a s ≥ 0 and B b t ≥ 0 . Recall, that we ma y r estrict ourselv es to considering pr o jectiv e measuremen ts. W e m ay thus add the equalit y constraints ( A a s ) 2 = A a s and ( B b t ) 2 = B b t , whic h automatically imply th at A a s , B b t ≥ 0. F or simplicit y , we will later us e this constrain t instead of the previous one. In this p ap er, we are mainly concerned with the (we ighte d ) a ve r age of the probabilities of generating certain outcomes in a non-lo cal game. In other w ords, w e w an ted to kno w if there exist op erators of the ab o ve form su ch th at ν =    X a,b,s,t π ( s, t ) V ( a, b | s, t ) A a s B b t    for some success probability ν . S emideﬁnite programming will allo w us to turn the question of existence in to an optimization problem. 4 T o ols F or our analysis we ﬁrst need to introduce tw o k ey to ols. The ﬁrst one allo ws us to deal with the fact that we w ant measurement s to hav e tensor pro d uct form. Our second to ol is an extension of the non-comm utativ e P ositivstellensatz of Helton and McCullough to the ﬁeld of complex n umb ers, from whic h we will d er ive a conv erging hierarc hy of semid eﬁnite programs. 4.1 T ensor pro duct structure fr om comm utation relations W e no w ﬁrst sho w that imp osing comm utativit y constrain ts do es indeed give us the tensor pr o duct structure required for our analysis of n on-lo cal games. It is well-kno wn th at the follo wing statemen t holds within the framew ork of quan tum mec hanics [45 ]. 2 In Ap p end ix A, w e pr ovide a simp le version of this argumen t accessible from a computer science p ersp ectiv e, wh ic h directly applies to the task at hand. 2 an algebra of typ e-I 10 Lemma 4.1. L et H b e a ﬁnite-dimensional Hilb ert sp ac e, and let { X a j s j ∈ B ( H ) | for al l j ∈ [ N ] and for al l s j ∈ S j , a j ∈ A j } . Then the fol lowing two statements ar e e quivalent: 1. F or al l j, j ′ ∈ [ N ] , j 6 = j ′ , and al l s j ∈ S j , s j ′ ∈ S j ′ , a j ∈ A j and a j ′ ∈ A j ′ it holds that [ X a j s j , X a j ′ s j ′ ] = 0 . 2. Ther e exist Hilb ert sp ac es H 1 , . . . , H N such that H = H 1 ⊗ . . . ⊗ H N and for al l j ∈ [ N ] , al l s j ∈ S j , a j ∈ A j we have X a j s j ∈ B ( H j ) . 4.2 P ositivstellensatz Our second tool, the Po sitivstellensatz (in com bination with semideﬁn ite programming) will allo w us to ﬁnd certiﬁcates for the fact a quan tum momen t problem is in feasible. F or simplicit y , we here describ e the Posit ivstellensatz from the p ersp ectiv e of non -lo cal games. An extension to the general quan tum moment problem is p ossible and w ill b e provided in a longer version of this pap er. Our results follo w almost directly fr om Helton and McCullough’s work and our pro of closely follo ws that in Ref. [23 ]. W e ha ve chosen to provide a complete pro of of the P ositivstellensatz for thr ee reasons: (i) the pr o of is more straigh tforward in our concrete setting, (ii) Helton an d McCullough’s theorem is form ulated for symmetric op erators ov er the ﬁeld R , and we need to work with Hermitian op erators o ve r the ﬁeld C , and (iii) so w e can h ighligh t the nonconstru ctiv e steps in the pro of. W e ﬁrst deﬁne: Deﬁnition 4.2 (Conv ex Cone C P ) . Let P b e a collectio n of Hermitian p olynomials in (noncom- m utativ e) v ariables { X a j s j } . The c onvex c one C P generated by P consists of p olynomials of th e form q = M X i =1 r † i r i + N X i =1 L X j =1 s † ij p i s ij , (3) where p i ∈ P , M , N and L are ﬁnite, and r i , s ij are arbitrary p olynomials. In the follo wing, w e will call Eq. (3) a weighte d sum of squar es (WSOS) represent ation of q . The purp ose of th e set P is to k eep trac k of the constrain ts on the measuremen t op erators. Note that when considering the measurement op er ators for non-lo cal games, it is suﬃcien t for us to restrict our s elv es to considering (measurement) op erators that are p ositiv e semideﬁnite. In particular, this means that all op erators are Hermitian. T he Posit ivstellensatz as suc h d o es not require u s to deal only with Hermitian v ariables in the p olynomials, but allo w s us to use an y noncomm uting matrix v ariables. In the follo win g, we will alw a ys tak e our measurement op erators to b e of the form X a s = ( ˆ X a s ) † ˆ X a s . Clearly , X a s is itself a Hermitian p olynomial in the v ariable ˆ X a s . F or clarit y of notation, w e w ill omit this exp licit expansion in the future. Note th at w e w ill th u s not im p ose the constraint that our op erators are Hermitian, and this imp licit expan s ion do es n ot increase the size of our S DP . W e can w rite our constraints in terms of the follo wing sets of Herm itian p olynomials. In the follo w ing, we w ill u se the short han d n otation O − j := X a 1 s 1 . . . X a j − 1 s j − 1 X a j +1 s j +1 . . . X a N s N where we lea ve indices s j and a j implicit, to r efer to a pro d uct of measur emen t op erators where w e exclude p la y er j . First, we wan t measurements on diﬀeren t subsystems to comm ute. In the multi-part y case, this giv es us the set of p olynomials Q 1 = { i [ X a j s j , O − j ] | for all s j ∈ S j , a j ∈ A j and all O − j } . 11 Second, w e w ant our operators to form v alid measurements. Q 2 = [ j,s j { I − X a j X a j s j } . Finally , b y Neum ark’s th eorem [40], we ma y tak e our measurement op erators to b e pro jectors, this giv es Q 3 = [ j,s j ,a j { ( X a j s j ) 2 − X a j s j } . It’s not hard to see that these constraints actually giv e u s orthogonalit y of the pro jectors. F or clarit y , h o we ver, w e ma y also include the follo wing sets of p olynomials Q 4 = { i [ X a j s j , X a ′ j s j ] | for all s j ∈ S j and all a j 6 = a ′ j } , Q 5 = { X a j s j X a ′ j s j + X a ′ j s j X a j s j | for all s j ∈ S j and all a j 6 = a ′ j } , whic h explicitly demand that pro jectors corresp ond ing to the same s j are orthogonal. Let Q = Q 1 ∪ Q 2 ∪ Q 3 ∪ Q 4 ∪ Q 5 and let P = Q ∪ ( −Q ). Note that all p olynomials in P are Hermitian. It is clear that if the measurement op erator satisfy the constrain ts, then the term X i,j s † ij p j s ij v anishes for arb itrary p j ∈ P and arbitrary p olynomial s ij . W e are no w r eady to s tate the P osi- tivstellensatz: Theorem 4.3 (P ositivstellensatz ) . L et G = G ( π , V ) b e an N -pr over game and let C P b e the c one gener ate d b y the set P deﬁne d ab ove. Set q ν = ν I − X s 1 ,...,s N π ( s 1 , . . . , s N ) X a 1 ,...,a N V ( a 1 , . . . , a N | s 1 , . . . , s N ) X a 1 s 1 . . . X a N s N . (4) If q ν > 0 , then q ν ∈ C P , i.e., ν I − X s 1 ,...,s N π ( s 1 , . . . , s N ) X a 1 ,...,a N V ( a 1 , . . . , a N | s 1 , . . . , s N ) X a 1 s 1 . . . X a N s N = X i r † i r i + X i,j s † ij p i s ij , (5) for some p i ∈ P , and some p olynomials r i , s ij . 5 Finding upp er b ound s W e no w show h ow we can app ro ximate the optimal ﬁeld-theoretic v alue of a non-local game using semideﬁnite pr ogramming. W e th er eby construct a conv erging hierarc hy of SDPs, w here eac h lev el in this hierarch y giv es us a b etter upp er b oun d on the actual v alue of the game. T o this end w e will use the Posit ivstellensatz of Theorem 4.3 in com bin ation with the b eautiful approac h of P arrilo [38, 39]. F or simplicit y , we ﬁrst describ e ev erythin g for the t wo part y setting; a generali zation is straigh tforward. 12 Recall from Deﬁnition 2.2 that if for some r eal n umb er ν we hav e q ν = ν I − X a,b,s,t π ( s, t ) V ( a, b | s, t ) A a s B b t ≥ 0 , (6) and the op erators { A a s } and { B b t } form a v alid measuremen t, then ν ≥ ω f ( G ) gives us an upp er b ound for the optim u m v alue of the game. When trying to ﬁnd the optimal v alue of the game, our task is th us to ﬁ nd the smallest ν for which q ν ≥ 0 for any c hoice of measurement op erators. Clearly , if w e could express q ν as an S OS f or any c h oice of measur emen t op erators { A a s } and { B b t } then q ν ≥ 0 an d we wo uld also ha ve ν ≥ ω f ( G ). Luc kily , the Positivste llensatz of Theorem 4.3 giv es us almost the conv erse: if q ν > 0, then q ν can b e written as a weighte d sum of squ ares (WSOS ). Recall from the previous section, that the purp ose of the additional term in the weig hted sum s of squares r ep resen tation is to deal with the constraint that we w ould like the op erators { A a s } and { B b t } to form a v alid quan tum measurement. Note that q ν reduces to an SOS if we could expr ess q ν as a WSOS, i.e., q ν = ν I − X a,b,s,t π ( s, t ) V ( a, b | s, t ) = M X i =1 r † i r i + N X i =1 L X j =1 s † ij p i s ij , (7) for some p olynomials r i and s ij in the v ariables { A a s } and { B b t } in suc h a w a y that wheneve r th e v ariables satisfy the constrain ts th e seco n d term in the ab o ve expans ion v anishes. It is not diﬃcult to see that if q ν > 0, then there exists no strategy that ac hieves a winn ing probabilit y of ν or higher. Applied to our pr oblem, the Posit ivstellensatz thus tells us that if there exists no strategy that ac hiev es winning probabilit y ν , then q ν c an b e written as a weigh ted sum of squares. In tuitiv ely , the WSOS representa tion of q ν th us b ears witness to the fact that the set of measurement op erators and s tates giving a success p robabilit y higher than ν is emp t y . The adv antag e of this pro cedure is that semid eﬁnite programming can b e used to test whether p olynomials (suc h as q ν ) adm it a repr esen tation as WSOS. In Section 5.2, w e will look at some sp eciﬁc examples of this approac h (see also [39, 43] for the analogous treatmen t for commutat ive v ariables). When trying to ﬁ nd the optimal v alue of the game, our task is thus to ﬁnd the smallest ν for whic h q ν admits a WSOS represen tation. Hence, we wan t to minimize ν sub ject to q ν ∈ C P . Recall that if q ν ∈ C P , then q ν is of the form q ν = M X i =1 r † i r i + N X i =1 L X j =1 s † ij p i s ij , (8) for some p olynomials r i and s ij in the v ariables { A a s } and { B b t } . A p oint that is wo rth noting no w is that in the ab o ve optimization, Eq. (8) is an iden tit y true for all { A a s } , { B b t } , rather than an equation that is only true wh en { A a s } , { B b t } corresp ond to pro jectiv e measur emen ts. In this, the additional term is rather similar to the Lagrange m ultipliers in more conv entio nal constrained optimizations. 13 5.1 SDP hierarch y The main diﬃculty now is that we d o n ot know how large the WSOS representa tion of q ν has to b e. That is, we do not know ahead of time how large we need to c ho ose the degree of the p olynomials in the represen tation. The tec hniques discuss ed ab o ve are therefore not constructive and d o not lead to a direct compu tation of ω f ( G ). Ho wev er it is straightforw ard to ﬁ nd semideﬁnite relaxations that p ro vide upp er b ound s on ω f ( G ). In this w e simply apply the m etho ds of Parrilo [38, 39] for the case of p olynomials of comm utativ e v ariables. The main requirement is to ﬁx an integer n and lo ok for a sum of squares decomp osition for q ν that has a total d egree of at most 2 n . Letting ν = ω f ( G ) + ε , this means that ε may n ot b e m ade arb itrarily small b u t will alw ays result in an upp er b ound f or ω f ( G ). This up p er b ound can b e computed as an SDP using metho ds analogous to [39]. Consider the p r oblem given ab o v e for q ν as in E q. (8 ). Notice that all of the constrain t p olynomials p i deﬁned in S ection 4.2 h a v e total degree less than or equal to 2 so w e r equ ire that eac h r i is of total degree n and eac h s i is of total degree at m ost n − 1. The low est lev el of the hierarc hy has n = 1 and corresp onds to applying th e metho d of Lagrange multipliers to ﬁnd ing the quan tum v alue of the game. In the follo w in g, we use the term level n to refer to a leve l of the hierarc hy wh ere the total d egree of q ν is 2 n when concerned with a b ip artite game. F or a ga me G , denote the solution to the SDP at level n as ω sdp n ( G ). It should b e clear that if q ν has a WSO S decomp osition of degree 2 n , it m ust also hav e a WSOS decomp osition with h igher degree. As suc h, the optim um derived fr om the hierarc h y of SDPs m u s t ob ey the follo wing inequalities: ω sdp 1 ( G ) ≥ ω sdp 2 ( G ) ≥ · · · ≥ ω sdp n ( G ) . (9) Theorem 5.1. The solutio ns to the SDP hier ar c hy c onver ge to ω f ( G ) , i .e., lim n →∞ ω sdp n = ω f ( G ) . Pr o of. That ω sdp n ( G ) ≥ ω f ( G ) f ollo ws from our discussion ab o ve. T o p ro ve conv ergence, w e use the P ositivstellensatz give n by Theorem 4.3. Fix ε > 0 and let ν = ω f ( G ) + ε with q ν deﬁned as in Eq. (4). By Th eorem 4.3, q ν has a represen tation as a WSOS, q ν = M X i =1 r † i r i + N X i =1 L X j =1 s † ij p i s ij , Let 2 D b e the maxim um degree of any of the p olynomials r † i r i and s † ij p i s ij that o ccurs in the ab o ve expression. Then, if we co n s ider a lev el D SDP relaxation, w e must necessa rily arrive at an optim um suc h that ω sdp D ( G ) ≤ ω f ( G ) + ε . Lik ewise, b y choosing ε arbitrarily close to zero, there is a co rr esp ondin g SDP with total degree 2 n whose optimum ω sdp n ( G ) is arbitrarily close to ω f ( G ). In p articular, from Eq. (9 ), we can see th at as n → ∞ , the optimum of the SDP hierarch y m us t con v erge to ω f ( G ). In Section 5.2.2, we pro vide a simple example of h o w th e degree of the p olynomials can b e increased wh en goi ng from lev el 1 to lev el 2. Th ere are many conn ections of this semideﬁnite programming hierarc hy to other metho d s that can b e used to b ound the quan tum v alues of games. In particular, it can b e sho wn that the dual semideﬁnite programs to this hierarch y are equiv alen t to the moment m atrix metho ds of NP A [36 ], thus sh o wing th at the h ierarc hy of semideﬁnite pr ograms discussed in that work conv erges to the enta ngled v alue of the game. Our example of the CHS H inequalit y b elo w demonstrates this connection explicit ly . In r elation to th is, it is wo rth n oting that the d ualit y b etw een the t w o appr oac h es (sum of squares and momen t matrix) arises also in the 14 case of comm utativ e v ariables where the momen t matrix metho d s of Laserre [30] are dual to the semideﬁnite programs discussed b y P arrilo [39 ]. 5.2 Examples 5.2.1 CHSH inequalit y W e will no w lo ok at the simplest non-lo cal game th at is d eriv ed f rom the C HSH inequalit y [14]. In particular, w e will illustrate ho w the to ols that hav e we dev elop ed allo w us to prov e that [11] S CHSH = h A 1 B 1 i + h A 1 B 2 i + h A 2 B 1 i − h A 2 B 2 i ≤ 2 √ 2 , where A 1 , A 2 and B 1 , B 2 are observ ables with eigen v alues ± 1 corresp onding to Ali ce and Bob’s mea- surement settings resp ectiv ely . First of all, note that since we are only interested in th e exp ectatio n v alues of the form h A 1 B 2 i we ma y simplify our problem: instead of d ealing with th e probab ilities of in d ividual measurement outcomes, we are only in terested in whether said exp ectati on v alues can b e obtained. Here, our constrain ts b ecome muc h simpler and we only d emand that A 2 j = I , B 2 j = I and [ A j , B k ] = 0 for all j, k ∈ { 1 , 2 } . The Bell op erator for the CHS H inequalit y is giv en by [8 ] B CHSH = A 1 B 1 + A 1 B 2 + A 2 B 1 − A 2 B 2 . Hence, to ﬁnd the optim u m v alue our goal is to minimize ν sub ject to q ν = ν I − B CHSH ∈ C P . The constrain t in the ab ov e optimization thereby amoun ts to determining whether q ν as wr itten ab o ve can b e cast in the form of a WSOS , w hic h redu ces to an S OS for measurement op erators satisfying the constrain ts. The numerical pac k age SOST OOLS [42] giv es a fronte nd to other SDP solv ers and explains ho w to ap p ly these techniques in the case of comm utativ e v ariables. Similar ideas can b e applied here. Ho w ever, f or our simple example, it is not h ard to see ho w this p roblem can b e recast in a language that may b e more familiar. Since B CHSH is a noncomm utativ e p olynomial of degree 2, the lo west lev el relaxation consists of lo oking for a WSOS d ecomp osition f or q ν that is of degree 2. T o this end, we shall consider a v ector of monomial of degree 1, namely , z = ( A 1 , A 2 , B 1 , B 2 ) † . Our goal is to ﬁnd a 4 × 4 matrix Γ suc h that q ν = z † Γ z when ev er the constraints are satisﬁed. I.e. we h av e [ A j , B k ] = 0 for all j, k ∈ { 1 , 2 } , and p olynomials p ( A ) j := I − ( A j ) 2 , p ( B ) j := I − ( B j ) 2 , j = 1 , 2 , (10) and their negations v anish. Evid ently , since w e wan t q ν to b e a Hermitian p olynomial, and we w ant our co mmutation constrain ts to h old, w e ma y w ithout loss of ge nerality take Γ to b e real and symmetric. Note that this already tak es care of the commuta tion constrain ts. Moreo ve r, since all remaining constrain ts are quadr atic, when lo oking for a WSOS decomp osition for q ν , it suﬃces to consider s ij in Eq. (8) as multiples of I . Let γ ij = [Γ] i,j , then a sm all calc u lation sho ws that th is amoun ts to requirin g ν = γ 11 + γ 22 + γ 33 + γ 44 0 = γ 12 = γ 21 = γ 34 = γ 43 − 1 = 2 γ 13 = 2 γ 14 = 2 γ 23 1 = 2 γ 24 , (11) 15 so that q ν = ν I − B CHSH = z † Γ z + 2 X j =1 γ j j p ( A ) j + 4 X j =3 γ j j p ( B ) j − 2 . (12) Using the constrain ts giv en in Eq. (11), we see that Γ should b e of the form Γ = 1 2     2 γ 11 0 − 1 − 1 0 2 γ 22 − 1 1 − 1 − 1 2 γ 33 0 − 1 1 0 2 γ 44     . (13) Eﬀectiv ely , Γ is the matrix obtained by expressin g ν I − B CHSH − P 2 j =1 γ j j p ( A ) j − P 4 j =3 γ j j p ( B ) j − 2 in the form of z † Γ z . No w, if w e can ﬁnd a Γ ≥ 0 that is of this form, then whenev er the p olynomials giv en in Eq. (10 ) v anish, q ν = z † Γ z is an SOS. T o see this, note that in this case, we may wr ite Γ = U † D U , wh ere U is unitary and D = diag( d i ) only consists of nonnegativ e diagonal en tries. Then we can write q ν as P i d i ( U z ) † i ( U z ) i whic h is clearly an SOS. Conv ersely , n ote that if q ν is an SOS, w e can ﬁnd suc h a matrix Γ. Hence, w e can rephr ase our optimizat ion p roblem as the SDP minimize T r(Γ) sub ject to Γ ≥ 0. This is, in fact, exactly the d ual of the SDP corresp onding to the ﬁrst lev el of the SDP h ierarc hy giv en b y NP A [36], and the d ual of the SDP for the sp ecia l case of X OR games [49 ]. Solving this SDP , one obtains Γ = 1 2     √ 2 0 − 1 − 1 0 √ 2 − 1 1 − 1 − 1 √ 2 0 − 1 1 0 √ 2     , (14) whic h give s 2 √ 2 as an optim um. F rom here and Eq. (12), it is p ossib le to wr ite do wn a WSO S decomp osition for ν = 2 √ 2 as q 2 √ 2 = 2 √ 2 I − B CHSH = 1 2 √ 2 ( h † 1 h 1 + h † 2 h 2 ) + 1 √ 2 2 X j =1 p ( A ) j + 1 √ 2 4 X j =3 p ( B ) j − 2 , (15) with h 1 = A 1 + A 2 − √ 2 B 1 and h 2 = A 1 − A 2 − √ 2 B 2 . This immed iately implies that whenever the constraints are satisﬁed, q 2 √ 2 ≥ 0 and h ence B CHSH ≤ 2 √ 2 I . It is w ell kno wn that for the CHSH inequalit y , this b ound can b e ac hiev ed [11]. 5.2.2 The I 3322 inequalit y W e no w consider another example of a t w o-pla yer game, where the ﬁrst leve l of the hierarc hy do es not giv e a tight b ound. Th e I 3322 inequalit y [16] is a Bell inequalit y p hrased in terms of probabilities (not exp ectation v alues) whereby Alice and Bob can eac h p erform one of three possib le t wo-o utcome measuremen ts. Without loss of generalit y , the Bell op erator in this case can b e written as: B 3322 = A a 1 ( B b 1 + B b 2 + B b 3 ) + A a 2 ( B b 1 + B b 2 − B b 3 ) + A a 3 ( B b 1 − B b 2 ) − A a 1 − 2 B b 1 − B b 2 , 16 where A a i and B b j ( i, j = 1 , 2 , 3) are p ro jectors corresp ondin g to, resp ectiv ely , outcome a of Alice’s i -th m easuremen t and outcome b of Bob’s j -th measur emen t for s ome ﬁ x ed a and b . T o the b est our kno wledge, the maxim um enta ngled v alue for B 3322 , i.e., ω ∗ ( I 3322 ), is not known. The b est kn o wn lo w er b oun d on ω ∗ ( I 3322 ) is 0.25 [1 6 ]; some upp er b ounds (0.375 [31], 0.3660 [1]) are also kno wn in the literat u re. Here, w e will mak e use of the to ols that w e ha v e dev elop ed to obtain a h ierarc h y of upp er b ounds on this m axim um. In an alogous with th e CHSH scenario, this corresp onds to solving the follo wing SDP for some ﬁxed degree of q ν : minimize ν, sub ject to q ν = ν I − B 3322 ∈ C P . (16) In p articular, sin ce B 3322 is a noncommutat ive p olynomial of degree 2, th e lo west level SDP relaxation w ould corresp on d to c ho osing a v ector of monomials with degree at most one, i.e ., z † = ( I , A a 1 , A a 2 , A a 3 , B b 1 , B b 2 , B b 3 ) . (17) W e can no w pro ceed analogously to the CHSH case, wh ere we will lo ok for a particular matrix Γ restricted by our constraint s, namely , ( A a j ) 2 = A a j and ( B b j ) 2 = B b j for all j ∈ { 1 , 2 , 3 } , wh ere again for the pu rp ose of imp lemen tation, we will implicitly enforce the comm utativit y cond itions [ A i , B j ] = 0 for all i, j ∈ { 1 , 2 , 3 } . Solving the corresp onding SDP (App endix D.1), one obtains ω sdp 1 ( I 3322 ) = 3 / 8, and the matrix Γ = 1 2           3 4 0 − 1 − 1 2 1 0 − 1 2 0 2 0 0 − 1 − 1 − 1 − 1 0 2 0 − 1 − 1 1 − 1 2 0 0 1 − 1 1 0 1 − 1 − 1 − 1 2 0 0 0 − 1 − 1 1 0 2 0 − 1 2 − 1 1 0 0 0 1           , whic h provides a WSOS d ecomp osition for ν = 3 / 8, i.e., q 3 / 8 = 3 8 I − B 3322 = z † Γ z + X i s † i p i s i , (18) where p i = ( A a i − ( A a i ) 2 : i = 1 , 2 , 3 , B b i − 3 −  B b i − 3  2 : i = 4 , 5 , 6 . , s i = ( 1 : i = 1 , 2 , 4 , 5 , 1 √ 2 : i = 3 , 6 . . (19) Giv en that ω sdp 1 ( I 3322 ) is far f rom the b est kn o wn lo wer b ound on ω ∗ ( I 3322 ), it seems natural to also look at h igher lev el relaxatio n s for I 3322 . F or the next lev el, we will look for q ν that is of degree at most 4. Note that we h a v e m any options to extend the hierarc h y . The easiest wa y to pro ceed is to extend the v ector z by some monomials of degree tw o in the measurement op erators. F or this we do n ot eve n hav e to consider usin g all p ossible degree 2 monomials, but co uld consider only a subset of them suc h as that give n b y the follo wing vecto r z † = ( I , A a 1 , A a 2 , A a 3 , B b 1 , B b 2 , B b 3 , A a 1 B b 1 , A a 1 B b 2 , A a 1 B b 3 , A a 2 B b 1 , A a 2 B b 2 , A a 2 B b 3 , A a 3 B b 1 , A a 3 B b 2 , A a 3 B b 3 ) . (20) 17 In particular, s olving the corresp ond ing SDP (App endix D.2) with z given by Eq. (20) gives an optim um that is appr o ximately 0 . 251 470 90 wh ic h is signiﬁcan tly less than 3 / 8 = 0 . 375. Clearly , w e could increase the size of z further by in cluding all relev ant monomials of degree 2 or less. z † = ( I , A a 1 , A a 2 , . . . , B b 2 , B b 3 , A a 1 A a 2 , A a 1 A a 3 , . . . , A a 3 A a 2 , B b 1 B b 2 , B b 1 B b 3 , . . . , B b 3 B b 2 , A a 1 B b 1 , A a 1 B b 2 , . . . , A a 3 B b 3 ) . Pro ceeding as b efore, we end up with th e optim um of the second order relaxation ω sdp 2 ( I 3322 ) ≈ 0 . 250 939 72. In the n ext lev el, we wo uld then include all monomials of degree 3 and less, and this giv es ω sdp 3 ( I 3322 ) ≈ 0 . 250 875 56. 5.2.3 Y ao’s inequality Finally , we examine a w ell-kno wn tripartite Bell inequalit y [50] among three pr ov ers: Alice, Bob and Charlie. E ac h pr o v er p erforms thr ee p ossible measuremen ts, eac h of whic h has t w o p ossible outcomes. Similarly to the CHSH inequalit y ab o v e, we ma y thus express eac h measurement as an observ able with eigen v alues ± 1. F or simplicit y , let A 1 , A 2 , A 3 , B 1 , B 2 , B 3 and C 1 , C 2 , C 3 corresp ond to the observ ables of Alice, Bob and Charlie resp ectiv ely . Y ao’s inequalit y states that for any shared state ρ w e ha ve S Y ao = h A 1 B 2 C 3 i + h A 2 B 3 C 1 i + h A 3 B 1 C 2 i − h A 1 B 3 C 2 i − h A 2 B 1 C 3 i − h A 3 B 2 C 1 i ≤ 3 √ 3 . (21) W e now pro vide a ve ry simple p r o of of this inequalit y based on our framewo rk . First of all, note that since w e are only intereste d in the exp ectati on v alues of the form h A 1 B 2 C 3 i we ma y again r estrict ourselv es to dealing only with exp ectat ion v alues in analogy with the CHSH example present ed abov e. Our constr aints are also analogous to the CHSH case. Among them, we h av e the follo w ing constraint p olynomials p ( A ) j := I − ( A j ) 2 , p ( B ) j := I − ( B j ) 2 , p ( C ) j := I − ( C j ) 2 , (22) for j = 1 , 2 , 3. Next, n ote that the Bell op erator for Y ao’s inequalit y can b e written as [c.f. Eq. (21)] B Y ao = A 1 B 2 C 3 + A 2 B 3 C 1 + A 3 B 1 C 2 − A 1 B 3 C 2 − A 2 B 1 C 3 − A 3 B 2 C 1 , (23) whic h is a noncomm utativ e p olynomial of degree 3. F or our task at hand, w e will consider the follo wing SDP relaxation minimize ν, sub ject to q ν = ν I − B Y ao ∈ C P , with q ν b eing a p olynomial of degree at most 6. As usual, we will imp licitly enforce the comm u- tativit y constrain ts, i.e., [ A i , B j ] = 0, [ A i , C k ] = 0, and [ B j , C k ] = 0 for all i, j, k ∈ { 1 , 2 , 3 } . With this assumption, it turns out that it suﬃces to consider the follo wing 25-elemen t vec tor z =           I A 1 B 2 C 3 A 2 B 3 C 1 A 3 B 1 C 2 A 1 B 3 C 2 A 2 B 1 C 3 A 3 B 2 C 1           ⊕         A 1 B 1 C 2 A 1 B 2 C 1 A 2 B 1 C 1 A 1 B 2 C 2 A 2 B 1 C 2 A 2 B 2 C 1         ⊕         A 1 B 1 C 3 A 1 B 3 C 1 A 3 B 1 C 1 A 1 B 3 C 3 A 3 B 1 C 3 A 3 B 3 C 1         ⊕         A 2 B 2 C 3 A 2 B 3 C 2 A 3 B 2 C 2 A 2 B 3 C 3 A 3 B 2 C 3 A 3 B 3 C 2         . (24) 18 In this case, since the constraint p olynomials give n in Eq. (22) are quadratic, wh en lo oking for a WSOS decomp osition for q ν , we will need to consider s ij in Eq. (8 ) as an arbitrary p olynomial of A i , B j and C k with degree at most 2. Pro ceeding in a wa y analogous to the 2nd lev el relaxation for I 3322 inequalit y , one obtains the 25 × 25 p ositiv e semideﬁnite matrix Γ = Γ 7 × 7 L 6 i =1 Γ 3 × 3 , where Γ 7 × 7 := 1 2             3 √ 3 − 1 − 1 − 1 1 1 1 − 1 1 √ 3 0 0 − 1 3 √ 3 − 1 3 √ 3 − 1 3 √ 3 − 1 0 1 √ 3 0 − 1 3 √ 3 − 1 3 √ 3 − 1 3 √ 3 − 1 0 0 1 √ 3 − 1 3 √ 3 − 1 3 √ 3 − 1 3 √ 3 1 − 1 3 √ 3 − 1 3 √ 3 − 1 3 √ 3 1 √ 3 0 0 1 − 1 3 √ 3 − 1 3 √ 3 − 1 3 √ 3 0 1 √ 3 0 1 − 1 3 √ 3 − 1 3 √ 3 − 1 3 √ 3 0 0 1 √ 3             , Γ 3 × 3 := 1 12 √ 3   1 1 1 1 1 1 1 1 1   . (25) F rom s ome simp le calculations, it then follo ws that whenev er th e constraints A 2 i = B 2 j = C 2 k = I are satisﬁed, w e ha ve 3 √ 3 I − B Y ao = z † Γ z = 1 6 √ 3   h † 0 h 0 + 2 X j =1 h † + ,j h + ,j + 2 X j =1 h † − ,j h − ,j + 1 2 X j,k =1 , 2 , 3 h † j,k h j,k   , (26 ) where h 0 = 3 √ 3 I − B Y ao , h + ,j = A 1 B 2 C 3 + e i (2 πj / 3) A 2 B 3 C 1 + e i (4 πj / 3) A 3 B 1 C 2 , h − ,j = A 1 B 3 C 2 + e i (2 πj / 3) A 2 B 1 C 3 + e i (4 πj / 3) A 3 B 2 C 1 , h j,k = A j B j C k + A j B k C j + A k B j C j . This mak es it explicit th at whenev er the constraints are satisﬁed, 3 √ 3 I − B 3322 ≥ 0 and therefore S 3322 ≤ 3 √ 3. 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In Pr o c e e dings of ST ACS 2006 , v olume 3884 of LNCS , pages 162– 171, 2006. [49] S. W ehner. Ts ir elson b ounds for generalized Clauser-Horne-Shimony- Holt inequalities. P hysic al R eview A , 73:0221 10, 2006. [50] And rew C. Y ao. T alk at QIP 2007 , Brisbane, Australia. 2007. A T o ol 1: T ensor pro duct structure from comm utation relations W e no w pro vide a simple p ro of of Lemma 4.1 from a computer science p ersp ectiv e that is suitable to the task at hand. F or simp licit y , w e address the case of t wo-pro ver systems in detail, and merely sk etc h the extension to the multiple p ro v ers at the end. F or ease of reference, w e shall now repro du ce the Lemma for the t wo -pr o v er setting: Lemma A.1. L et H b e a ﬁnite-dimensiona l Hilb ert sp ac e, and let { A a s ∈ B ( H ) | s ∈ S } and { B b t ∈ B ( H ) | s ∈ T } . Then the fol lowing two statements ar e e quivalent: 1. F or al l s ∈ S , t ∈ T , a ∈ A and b ∈ B it holds that [ A a s , B b t ] = 0 . 22 2. Ther e exist Hilb ert sp ac es H A , H B such that H = H A ⊗ H B and for al l s ∈ S , a ∈ A we have A a s ∈ B ( H A ) and for al l t ∈ T , b ∈ B we have B b t ∈ B ( H B ) . F or our argument we will not consider individual op erators, but instead look at the C ∗ -algebra of op erators wh ich is well u n dersto o d in ﬁnite dimensions [46, 7]. The C ∗ -algebra of op erators A = { A 1 , . . . , A n } consists of all complex p olynomials in suc h op erators and their conju gate trans p ose: if A is an elemen t of the algebra, then so is A † . F or examp le, the set of all b ounded op erators B ( H ) on a Hilb ert space H is a C ∗ -algebra. F or con v enience, w e w ill also wr ite A = hAi for suc h an algebra A generated by op erators from the s et A . W e will n eed th e follo wing n otions: The c enter Z of an algebra A is the set of all elemen ts in A that comm ute with all elemen ts of A , i.e., Z = { Z | Z ∈ A , ∀ A ∈ A : [ Z, A ] = 0 } . If A ⊆ B ( H ) for some Hilb ert space H , then the c ommutant of A in B ( H ) is giv en by Comm( A ) = { X | X ∈ B ( H ) , ∀ A ∈ A : [ X, A ] = 0 } . F urthermore, an algebra A is called simple , if its only ideals 3 are { 0 } and A itself. It is easy to see that if A only has a tr ivial cen ter, i.e., Z = { c I | c ∈ C } , then A is simple [46]. Finally , A is called semisimple if it can b e decomp osed into a d irect sum of sim p le algebras. A.1 Optimizing non-lo cal games Before we sho w ho w to p r o v e Lemma 4.1, w e ﬁrst demonstrate that when considering non-lo cal games we can greatly simp lify our pr oblem and restrict ou r selv es to C ∗ -algebras th at are simp le. It is w ell kno wn that we can decomp ose an y ﬁnite dimensional algebra in to the sum of simple algebras. Lemma A.2 ([46]) . L e t A b e a ﬁnite-dimensional C ∗ -algebr a. Then ther e exists a de c omp osition A = M j A j , such that A j is simple. W e furth ermore n ote that for an y s im p le alge br a, the follo w ing holds: Lemma A.3 ([46]) . L et H b e a Hilb ert sp ac e, and let A ⊆ B ( H ) b e simple, then ther e exists a bip artite p artitioning of the Hilb ert sp ac e H such that H = H 1 ⊗ H 2 and A ∼ = B ( H 1 ) ⊗ I 2 . W e no w sho w that without loss of generalit y , we may assume that the algebras generated by Alice and Bob’s measuremen t op erators are in fact simple. Lemma A.4. L et H = H A ⊗ H B and let A = { A a s ∈ B ( H A ) } and B = { B b t ∈ B ( H B ) } b e the set of Alic e and Bob’s me asur ement op er ators r esp e ctively. L et ρ ∈ B ( H ) b e the state shar e d by Alic e and Bob. Supp ose that for such op er ators we have q = X s ∈ S,t ∈ T π ( s, t ) X a ∈ A,b ∈ B V ( a, b | s, t ) T r  ( A a s ⊗ B b t ) ρ  . Then ther e exist me asur ement op er ators ˜ A = { ˜ A a s } and ˜ B = { ˜ B b t } and a state ˜ ρ su c h q ≤ X s ∈ S,t ∈ T π ( s, t ) X a ∈ A,b ∈ B V ( a, b | s, t ) T r  ( ˜ A a s ⊗ ˜ B b t ) ˜ ρ  . and the C ∗ -algebr a gener ate d by ˜ A and ˜ B is simple. 3 An ideal I of A is a subalgebra I ⊆ A such that for all I ∈ I and A ∈ A , we hav e I A ∈ I and AI ∈ I . 23 Pr o of. Let A = hAi and B = hBi . I f A and B are simp le, w e are done. If not, w e kno w from Lemma A.2 and Lemma A.3 that there exists a decomp osition H A ⊗ H B = L j k H j A ⊗ H k B . Consider T r(( M A ⊗ M B ) ρ ), where M A ⊗ M B ∈ A ⊗ B . It follo ws f rom the ab o v e that M A ⊗ M B = L j k (Π j A ⊗ Π k B ) M A ⊗ M B (Π j A ⊗ Π k B ), where Π j A and Π k B are pro jectors on to H j A and H k B resp ectiv ely . Let ˆ ρ = L j k (Π j A ⊗ Π k B ) ρ (Π j A ⊗ Π k B ). C learly , T r(( M A ⊗ M B ) ˆ ρ ) = T r   M j k (Π j A ⊗ Π k B ) M A ⊗ M B (Π j A ⊗ Π k B ) ρ   = T r(( M A ⊗ M B ) ρ ) . The statemen t no w follo ws im m ediately by con vexit y: Alice and Bob can no w measure ρ { Π j A ⊗ Π k B } and r ecordin g the classical outcomes j, k . T he new measuremen ts will then b e ˜ A a s,j = Π j A A a s Π j A and ˜ B b t,k = Π k B B b t Π k B on state ˜ ρ j k = (Π j A ⊗ Π k B ) ρ (Π j A ⊗ Π k B ) / T r ((Π j A ⊗ Π k B ) ρ ). By constru ction, ˜ A j = { ˜ A a s,j } and ˜ B k = { ˜ B b t,k } are simple. Let q j k denote the probabilit y that w e obtain outcomes j, k , and let r j k = X s ∈ S,t ∈ T π ( s, t ) X a ∈ A,b ∈ B V ( a, b | s, t ) T r( ˜ A a s,j ⊗ ˜ B b t,j ˜ ρ j k ) . Then q = P j k q j k r j k ≤ max j k r j k . Let u, v b e suc h that r u,v = max j k r j k . Hence, we can skip the initial measurement and instead use measurement s ˜ A a s = ˜ A a s,u , ˜ B b t = ˜ B b t,v and state ˜ ρ = ˜ ρ u,v . This easy argumen t also immediately tells us that w hen A and B are ab elian, w e can ﬁnd a classical strategy that achiev es q : J ust p erform the m easuremen t as ab ov e. If A and B are ab elian, the remaining state will b e one-dimensional and hence classica l. A.2 T ensor pro duct structure W e are now ready to prov e Lemma 4.1. First, w e examine the case w here w e are giv en a simple algebra A ∈ B ( H ), for some Hilb ert sp ace H . W e will need the follo wing version of S c hur’s lemma. Lemma A.5. L et Z b e the c enter of B ( H ) . Then Z = { c I | c ∈ C } . Pr o of. Let C ∈ Z and let d = dim ( H ). Let E = { E ij | i, j ∈ [ d ] } b e a basis for B ( H ), wh ere E ij := | i ih j | is the matrix of all 0’s and a 1 at p osition ( i, j ). Sin ce C ∈ Z and E ij ∈ B ( H ) w e hav e for all i ∈ [ d ] C E ii = E ii C. Note that C E ii (or E ii C ) is the matrix of all 0’s bu t the i th column (or ro w) is determined b y the elemen ts of C . Hence all oﬀ diagonal elemen ts of C must b e 0. No w consider C ( E ij + E j i ) = ( E ij + E j i ) C. Note that C ( E ij + E j i ) (or ( E ij + E j i ) C ) is the matrix in whic h the i th and j th columns (rows) of C hav e b een sw app ed and the r emainin g element s are 0. Hence all d iagonal elemen ts of C m ust b e equal. T hus there exists some c ∈ C suc h that C = c I . Using this Lemma, w e can no w show that 24 Lemma A.6. L et C ∈ B ( H A ⊗ H B ) b e such that for al l B ∈ B ( H B ) we have [ C, ( I A ⊗ B )] = 0 Then ther e exists an A ∈ B ( H A ) such that C = A ⊗ I B . Pr o of. Let d A = dim( H A ) and d B = dim( H B ). Note that we can wr ite an y C ∈ B ( H A ⊗ H B ) as C =    C 11 . . . C 1 d A . . . . . . C d A 1 . . . C d A d A    , for d B × d B matrices C ij . W e hav e C ( I A ⊗ B ) = ( I A ⊗ B ) C if and only if for all i, j ∈ [ d A ] C ij B = B C ij , i.e., [ C ij , B ] = 0. S ince this must hold for all B ∈ B ( H B ), we h a v e by Lemma A.5 that there exists some a ij ∈ C suc h that C ij = a ij I B . Hence C = A ⊗ I B with A = [ a ij ]. F or the case that the algebra generated b y Alice and Bob’s measurement op erators is simple, Lemma 4.1 no w follo ws immediately: Pr o of of L emma 4.1 if A is simple. Let A = h{ A a s }i ⊆ B ( H ) b e th e algebra generated b y Alice’s measuremen t op erators. If A is simple, it follo ws from Lemma A.3 that A ∼ = B ( H A ) ⊗ I B for H = H A ⊗ H B . It th en follo ws from Lemma A.6 that f or all t ∈ T and b ∈ B w e m u s t hav e B b t ∈ B ( H B ). Th u s, w e obtain a tensor pro d uct structure! Recall that Lemma A.4 states th at for our appli- cation this is all w e need. In general, w hat happ ens if A is not simple? Whereas our argumen t shows that ther e alw a ys exist measuremen t op er ations such that A is simple, the solution found via optimization ma y not ha v e this prop ert y . W e no w ske tc h the argumen t in the case where the A is semisimple, which by Lemma A.2 w e may alw a ys assume in the ﬁnite-dimensional case . F ortun ately , w e can still assume that our commutatio n relations lea ve us with a bipartite structure. W e can essen tially infer this from v on Neumann’s famous Double Comm utant T heorem [46, 7], partially stated here. Theorem A.7. L et A b e a ﬁnite-dimensional C ∗ -algebr a. Then ther e exists H = H A ⊗ H B such that A ∼ = M j B ( H j A ) ⊗ I j B and Comm( A ) ∼ = M j I j A ⊗ B ( H j B ) . (27) Pr o of. (Sk etc h) W e already kno w f rom Lemma A.2 that A can b e decomp osed in to a sum of simple alge b r as. Clearly , the RHS of Eq. (27) is an elemen t of Comm( A ). T o see that the LHS is contai ned in th e RHS, consider the pro jection Π j A on to H j A . Note that Π j A ∈ A , and thus for an y X ∈ Comm( A ) we hav e [ X , Π j A ] = 0. Hence, we can w rite X = P j (Π j A ⊗ I B ) X (Π j A ⊗ I B ), and thus w e can r estrict ours elv es to considerin g eac h f actor individ ually . Th e result then follo w s immediately from Lemma A.6. 25 If w e hav e more than tw o pro vers, the argument is essen tially analogous, and w e merely sketc h it in th e r elev an t case when the algebra generated by the pr o v er’s measurements is simple, s in ce Lemma A.4 directly extends to more than tw o pro vers as w ell. S upp ose w e ha v e N prov ers P 1 , . . . , P N and let H d enote their joint Hilb ert space. Let A b e the algebra generated b y all measuremen t op erators of pro v ers P 1 , . . . . P N − 1 resp ectiv ely . Then it follo w s from Lemma A.6 and Lemma A.3 that there exists a bipartite partitioning of H suc h that H = H 1 ,...,N − 1 ⊗ H N , A ∼ = B ( H 1 ,...,N − 1 ) and for all measurement op erators M of pro ver P N w e ha ve that M ∈ B ( H N ). By app lying Lemma A.6 recurs iv ely w e obtain that there exists a w a y to p artition the Hilb ert sp ace in to su bsystems H = H 1 ⊗ . . . ⊗ H N suc h that the measurement operators of prov er P j act on H j alone. In qu an tum mec hanics, we will alw ays obtain such a tensor pr o duct structure f rom commuta- tion relations, ev en if the Hilb ert sp ace is inﬁnite-dimensional. Here, w e start out with a t yp e-I algebra, the corresp onding Hi lb ert sp ace and op erators can th en b e obtained by the famous Ge lfand - Naimark-Segal (GNS) construction [46], an approac h whic h is rather b eautiful in its abstraction. In qu an tum s tatistical mec hanics and quant u m ﬁeld theory , we will also en coun ter factors of t yp e-I I and t yp e-I I I. As it tur n s out, the ab o ve argum en t do es not generally hold in this case, how eve r, there are a n u mb er of conditions that can lead to a similar stru cture. S adly , we cannot consid er this case here and merely refer to the su r v ey article b y Summers [45]. Note that in quan tum mec h an ics itself, w e th us hav e ω ∗ ( G ) = ω f ( G ). B T o ol 2: P ositivstellensatz Here, we will pro vide the details for th e pro of of Th eorem 4.3 . F or ease of reference, w e ﬁrst repro du ce the theorem as follo ws: Theorem B.1. L et G = G ( π , V ) b e an N - pr over game and let C P b e the c one gener ate d by the set P deﬁne d in Se ction 4.2. Set q ν = ν I − X s 1 ,...,s N π ( s 1 , . . . , s N ) X a 1 ,...,a N V ( a 1 , . . . , a N | s 1 , . . . , s N ) X a 1 s 1 . . . X a N s N . (4) If q ν > 0 , then q ν ∈ C P , i.e., ν I − X s 1 ,...,s N π ( s 1 , . . . , s N ) X a 1 ,...,a N V ( a 1 , . . . , a N | s 1 , . . . , s N ) X a 1 s 1 . . . X a N s N = X i r † i r i + X i,j s † ij p i s ij , (5) for some p i ∈ P , and some p olynomials r i , s ij . W e n ow pr ov e the c ontr ap ositive statemen t of Theorem 4.3. I n particular, we show that if q ν has no represen tation as a WSOS, then q ν 6 > 0 and there exist op erators and a state on some Hilb ert space that ac hieve winn ing probabilit y ν . The pr o of pro ceeds in tw o stages. W e ﬁrst us e the Hahn-Banac h theorem to show (nonconstructive ly) th e existence of a linear function that separates q ν from the con ve x cone C P and then use a GNS construction, as describ ed b elo w. Unfortunately w e will not in general end up with op erators on a ﬁ nite-dimensional Hilb ert space. W e start by establishing some simple facts ab out C P . Lemma B.2. L et W b e the pr o duct of some numb er of variables fr om the set { X a j s j } . Then I − W † W ∈ C P , and I − W W † ∈ C P . 26 Pr o of. The pro of is b y induction on n , the n umb er of v ariables in the pr o duct W . F or n = 1, w e ha v e to sho w that for all j, a j , s j , I − ( X a j s j ) 2 ∈ C P . W e do this for I − ( X a j s j ) 2 . W riting I − ( X a j s j ) 2 = X a ′ j 6 = a j ( X a ′ j s j ) 2 − X a ′ j  ( X a ′ j s j ) 2 − X a ′ j s j  + h I − X a ′ j X a ′ j s j i , (28) mak es it clear th at I − ( X a j s j ) 2 ∈ C P . F or n ≥ 1, write W = V X a j s j , wh ere we h a v e assumed without loss of generalit y that the elemen t X a j s j is righ tmost in W , and wher e V is the pro duct of n − 1 v ariables. Then I − W † W = I − X a j s j V † V X a j s j = I − ( X a j s j ) 2 + X a j s j ( I − V † V ) X a j s j . (29) No w I − ( X a j s j ) 2 ∈ C P b y the r esult for n = 1 and I − V † V ∈ C P b y the in d uctiv e hyp othesis. Moreo ver, for an y p olynomial r ∈ C P , and an y arbitrary p olynomial s , it is easy to see that s † r s ∈ C P . Hence, this implies that I − W † W ∈ C P . The argumen t for I − W W † is analogous. Lemma B.3. L e t p b e a H ermitian p olynomial. Then ther e e xi sts a r e al numb er t ≥ 0 and an s ∈ C P such that p = s − t I . Pr o of. The p olynomial p is a ﬁ nite sum of terms of the form p ′ = w ∗ v W † V + w v ∗ V † W , wh er e V , W are p ro ducts of the v ariables, w, v ∈ C and w ∗ is the complex conju gate of w (lik ewise for v ∗ ). If w e can sh o w the result for p ′ , th en the result for general p olynomials p follo w s immediately . T o this end, note that w e can write p ′ = ( v ∗ V † + w ∗ W † )( v V + w W ) − | v | 2 V † V − | w | 2 W † W (30) so that ( | v | 2 + | w | 2 ) I + p ′ = ( v ∗ V † + w ∗ W † )( v V + w W ) + | v | 2 ( I − V † V ) + | w | 2 ( I − W † W ) (31) whic h is in C P b y Lemma B.2. T aking t = | v | 2 + | w | 2 , the result f ollo ws for p ′ , w h ic h in turn implies the result for general p olynomials p . W e n o w wan t to sho w that if q ν is a Herm itian p olynomial that d o es not lie in C P , then there exists a linear functional that separates it from C P . The follo w in g Lemma closely follo w s [23, Prop osition 3.3] where the only diﬀerence is that we consider p olynomials ov er c omplex in stead of r eal Herm itian matrices. F ortu nately , the essen tial ingredient of the p ro of, th e Hahn-Banac h theorem also holds in this case. W e state en tire pr o of f or con venience: Lemma B.4. L et M b e the sp ac e of Hermitian p olynomials over c omplex matric e s. L et q b e a Hermitian p olynomial such that q 6∈ C P . Then ther e exi sts a line ar functional λ : M → R such that λ ( C P ) ≥ 0 , λ ( I ) > 0 , λ ( q ) ≤ 0 . Pr o of. Let M b e the space of Hermitian p olynomials o ver complex matrices. Let µ : M → R b e a linear functional deﬁned as µ ( p ) := inf { t > 0 : p = s − t I for some s ∈ C P } . Note that b y Lemma B.3, we can express any p ∈ M in this form. C learly , µ is a seminorm on M . Note that for q 6∈ C P w e ha v e µ ( q − I ) ≥ 1 by deﬁ n ition. W e consider n o w a ﬁxed q 6∈ C P and let L b e the span of q − I in M , i.e., all Herm itian p olynomials t ( q − I ) with t ∈ R . Deﬁne a linear functional f : L → R so that f ( t ( q − I )) := t . It is not h ard to see that f ≤ µ on L . No w we mak e the 27 ﬁrst nonconstru ctiv e step. By the Hahn-Banac h theorem [44, Theorem 3.3], f extend s to a linear functional F : M → R such th at F ( p ) ≤ µ ( p ) for all p ∈ M . W e n o w claim that λ = − F sati sﬁ es the requirement s of the lemma: First of all, note that we ha v e for all s ∈ C P that F ( s ) − F ( q ) + 1 = F ( s − q + ( q − I )) = F ( s − I ) ≤ µ ( s − I ) ≤ 1, where the ﬁrst equalit y follo ws from the linearit y of F , and the ﬁrst in equ alit y follo ws from F b eing an extension of f . Hence , F ( s ) ≤ F ( q ). Clearly , w e also ha ve that f or all s ∈ C P and t > 0, ts ∈ C P and hence tF ( s ) = F ( ts ) ≤ F ( q ). Th us, if s ∈ C P then F ( s ) ≤ 0 and F ( q ) ≥ 0. Hence, λ = − F satisﬁes λ ( C P ) ≥ 0 and λ ( q ) ≤ 0 as required. It remains to sho w that λ ( I ) > 0. First of all, n ote that I ∈ C P . Supp ose that on the contrary w e ha v e λ ( I ) = 0. Let p ∈ C P and n ote that by Lemma B.3 we ma y write − p = s − t I for s ome t > 0 and s ∈ C P . F rom t I = s + p , w e hav e 0 = tλ ( I ) = λ ( t I ) = λ ( s ) + λ ( p ) ≥ 0 and hen ce λ ( s ) = λ ( p ) = 0 for all p ∈ C P . No w note that since I − q ∈ M w e ma y write I − q = r − s f or some r , s ∈ C P . But then 0 = λ ( r ) − λ ( s ) = λ ( I − q ) = 1 whic h is a con tradiction. The remainder of the pro of of Theorem 5.2 is no w exactly identica l to [23], which in itself is analogous to the famous GNS constru ction [46, 7] that allo ws us to ﬁn d a represent ation in terms of b ounded op er ators on a Hilb ert space. W e here provide a slight ly annotated ve rsion of this approac h in the hop e that it will b e more accessible to the presen t audience. Theorem B.5 (Helton and McCullough) . L et M b e the sp ac e of Hermitian p olynomials. L et λ : M → R b e a line ar functional such that λ ( C P ) ≥ 0 and λ ( I ) > 0 . Then ther e exists a Hi lb ert sp ac e H , b ounde d op er ators { ˆ X a j s j } on H , and a state γ ∈ H such that for al l p ∈ P we have p ( { ˆ X a j s j } ) ≥ 0 and for any Hermitian q ∈ M , h γ | q ( { ˆ X a j s j } ) | γ i = λ ( q ) . Pr o of. First, w e construct a Hilb ert space H from M : F or s, t ∈ M , deﬁne h s | t i = 1 2 λ ( s † t + t † s ) . It is easy to verify that h s | t i is sym metric, bilinear and is also p ositive s emideﬁnite whenev er s = t , since s † s ∈ C P and hence h s | s i = λ ( s † s ) ≥ 0. Note that h·| ·i is degenerate, but in ord er to obtain a Hilb ert s pace, we n eed to tu rn h·| · i into an inner p ro duct. Th is can b e d on e in the stand ard wa y b y ’mo ding out’ the degeneracy: Consider J = { s ∈ M | h s | s i = 0 } . It is not diﬃcult to v erify that J forms a linear s u bspace of M and th at J is a left ideal of M W e no w consid er the quotien t space M / J , whic h is the v ector space created b y the equiv alence classes [ x ] = { x + j | j ∈ J } . Addition and scalar multiplica tion are deﬁned via the follo wing op erations inh erited from M : for x, y ∈ M and α ∈ C , w e ha ve [ x + y ] = [ x ] + [ y ] and [ αx ] = α [ x ]. W e can now d eﬁne the inner pro du ct h [ x ] | [ y ] i = h x | y i . 28 It is imp ortan t to note that this inner p r o duct d o es not d ep end on our choic e of represen tativ e from eac h equ iv alence class, and w e ha ve eliminated the degeneracy present earlier. The Hilb ert space is no w obtained b y formin g the completion of M / J with resp ect to this inner pro duct. Second, w e no w need to sho w that th er e exists a representa tion π : M → B ( H ). W e ﬁrst deﬁne the action of π ( x ) with x ∈ M on the ve ctors [ y ] as X [ y ] = [ xy ] , where w e use the shorthand X = π ( x ). I t is straigh tforw ard to v erify that this deﬁnition is again indep endent of our c hoice of r epresen tativ e from eac h equiv alence class, and that π is a homomorphism. F or simplicit y , w e only sho w b ound edness for op erators { X a j s j } ∈ M . T o see that X = π ( x ) for x ∈ { X a j s j } is b ounded, note that b y Lemma B.2 w e ha ve I − x † x ∈ C P and that h X [ s ] | X [ s ] i = h xs | xs i = h s | s i − λ ( s † ( I − x † x ) s ) , where λ ( s † ( I − x † x ) s ) ≥ 0, since s † ( I − x † x ) s ∈ C P . F r om Lemma B.2 we also ha v e that I − xx † ∈ C P , and h ence th e argument for π ( x † ) is analogous and w e ma y wr ite X † = π ( x † ) w ithout am biguit y . Hence w e can ﬁnd claimed op erators { ˆ X a j s j } ∈ B ( H ). Third, we need to deﬁn e the v ector γ . Since I ∈ M we choose γ = [ I ]. Hence, h γ | γ i = λ ( I ) > 0, and thus γ is non-zero. Let q ∈ M and write q ( X ) for the p olynomial where v ariables x j ha v e b een substituted by their representa tions X j . Note that h q ( X ) γ | γ i = λ ( q ) , where we h a v e used the fact that q is a Hermitian p olynomial. By a similar argument, it follo ws that for p ∈ C P and r ∈ M w e hav e h p ( X )[ r ] | [ r ] i = λ ( r † pr ) ≥ 0 , since r † pr ∈ C P and hence p ( X ) ≥ 0 as promised. W e can no w complete the pro of of Theorem 4.3. Pr o of of The or e m 4.3. Reca ll that our goal w as to prov e the contrapositiv e: If q ν / ∈ C P then q ν 6 > 0. F rom Lemma B.4 w e ha ve that if q ν / ∈ C P , then there exists a lin ear fun ctional λ that separate s q ν from C P . Lemma B.5 giv es us that there exists a Hilb ert space H , m easur emen t op erators { ˜ X a j s j } , and a v ector γ suc h that λ ( q ν ) = h γ | q ν ( { ˜ X a j s j } ) | γ i ≤ 0 . and th e op erators satisfy all the constrain ts, i.e., for all p ∈ P we hav e p ( { ˜ X a j s j } ) ≥ 0. (Note that w e only hav e equalit y constrain ts, whic h w e implemen ted by including b oth p and − p in P .) Since γ is not zero, w e ha ve q ν 6 > 0 whic h completes the pro of. Unfortunately , Theorem B.5 do es not tell us w h ether the u nderlying Hilb ert space H is ﬁn ite- dimensional, or wh ether the algebra generated by the op erators { X a j s j } is t yp e-I at all. Hence, we cannot ensu re without fu rther pro of that the fact th at our measurement op erators d o satisfy the comm utation constrain ts necessarily leads to th em havi n g tensor p r o duct form. Thus, we do n ot kno w whether there exist games G for whic h ω ∗ ( G ) < ω f ( G ): f or suc h games we ma y would ha ve to get a t yp e-I I or t yp e-I I I algebra. 29 C T o ol 3: Semideﬁnite Programming A semideﬁn ite pr ogram (SDP) is an optimizat ion o ver Hermitian matrices [47]. The ob j ectiv e function d ep ends linearly on the matrix v ariable and th e optimizati on is carried out sub jected to the m atrix v ariable b eing p ositiv e semideﬁnite and satisﬁes v arious aﬃne constrain ts. Any semideﬁnite program ma y b e written in the follo wing standar d form [6]: maximize − T r [ G 0 Z ] , (32a) sub ject to T r [ G k Z ] = b k ∀ k , (32b) Z ≥ 0 , (32c) where G 0 and all the G k ’s are Herm itian matrices, and the b k are real num b ers that together sp ecify the optimization; Z is the Hermitian matrix v ariable to b e optimized. An SDP also arises in the ine quality form , whic h seeks to minimize a linear function of the optimization v ariables ( x 1 , x 2 , . . . , x n ) ∈ R n , sub jected to a linear m atrix inequalit y: minimize b ′ k x k (33a) sub ject to F 0 + X k x k F k ≥ 0 . (33b) As in the stand ard form, F 0 and all th e F k ’s are Hermitian matrices, while ( b ′ 1 , b ′ 2 , . . . , b ′ n ) is a real v ector of length n . D Some Other Miscellaneous Det ails D.1 Implemen t ing Lo w est Lev el SDP Relaxations for I 3322 Here, we will provide the explicit form for the matrices F k and constan ts b ′ k that deﬁne the SDP used in the lo w est level relaxation for ﬁnding an upp er b ound on ω ∗ ( I 3322 ). Note that as with the CHSH case, in the lo w est lev el relaxation, we shall c ho ose s ij in Eq. (8 ) as multiples of I . T o this end, w e will write the SDP in the inequalit y form as minimize b ′ ν ν + 6 X k =1 b ′ k x k (34a) sub ject to Γ = F 0 + ν F ν + 6 X k =1 x k F k ≥ 0 . (34b) In particular, w e will set F 0 = 1 2           0 1 0 0 2 1 0 1 0 0 0 − 1 − 1 − 1 0 0 0 0 − 1 − 1 1 0 0 0 0 − 1 1 0 2 − 1 − 1 − 1 0 0 0 1 − 1 − 1 1 0 0 0 0 − 1 1 0 0 0 0           , F ν =           1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           , 30 and for k = 1 , 2 , . . . , 6, we shall c h o ose F k suc h that its only nonzero entries are [ F k ] 1 ,k +1 = [ F k ] k +1 , 1 = 1 / 2 and [ F k ] k +1 ,k +1 = − 1. Corresp ondingly , b ′ k is c h osen suc h that b ′ γ = 1 and b ′ k = 0 for k = 1 , 2 , . . . , 6. With this c hoice of F k , b ′ k and with z give n by Eq. (17), it is easy to see that the matrix inequalit y constrain t, Eq. (34b), ensures that z † Γ z = z † F 0 + ν F ν + 6 X k =1 x k F k ! z = −B 3322 + ν I + 6 X k =1 x k p k = SOS where p k ’s are d eﬁned in Eq. (19 ) and th e last equ ality follo w s from the p ositive semideﬁn iteness of Γ. D.2 Implemen t ing Higher Lev el SDP Relaxations for I 3322 In wh at follo ws, w e will giv e a ske tc h of ho w the lev el 2 relaxa tion for I 3322 inequalit y with z giv en b y Eq. (20) can b e imp lemen ted as an SDP in the inequalit y form, Eq. (33). Sp eciﬁcally , we wan t to write Eq. (16) as: minimize ν, sub ject to F 0 + ν F ν + X k x k F k ≥ 0 , (35) where, as with the lo w est lev el relaxation, F 0 and F ν are real and symmetric matrices c hosen such that z † F 0 z = − B 3322 , z † F ν z = I . (36) Hereafter, we will fo cus on writing th e second sum in Eq. (8) as P k x k z † F k z f or some appropriate c hoice of Hermitian matrix F k where x k is some v ariable to b e optimized. As opp osed to the lo w est lev el relaxation, the most general second lev el relaxation w ould r equire that eac h s ij in Eq. (8) is a p olynomial of degree at most 1. Let s ij = P k λ ij k M k where M k is the k -en try of the vecto r µ = ( I , A a 1 , A a 2 , A a 3 , B b 1 , B b 2 , B b 3 ) † whic h consists of all degree 1 or lo wer monomials th at ca n b e found in z . F or a ﬁxed i and j , w e th u s hav e s † ij p i s ij = X k ,l M † k  λ ∗ ij k λ ij l  p i M l = µ † Λ ij p i µ, (37) where here λ ∗ ij k is the complex conjugate of λ ij k , p i µ is a v ector formed by multiplying eac h en try of µ b y p i and Λ ij is a 7 × 7 matrix w ith its ( k, l )-en try giv en by λ ∗ ij k λ ij l . Clearly , as it is, Λ ij is a rank 1 but otherwise arbitrary p ositiv e semideﬁnite matrix. Analogously , w e see that if w e f urther p erform a sum o v er j in Eq. (37), then we ma y write P j s † ij p i s ij = µ † Λ i p i µ wh er e Λ i = P j Λ ij is n o w an arbitrary p ositiv e semideﬁnite matrix. Moreov er, the requirement of Λ i b eing p ositiv e semideﬁnite can also b e remo v ed if w e recall the fact th at in the case of I 3322 inequalit y , if p i is in P , so is − p i . Then, wh at r emains to b e d one is to express P j s † ij p i s ij and h ence µ † Λ i p i µ in the form z † Ω i z for some Hermitian Ω i . Evidently , the ent ries in Λ i will b e related to the entries in Ω i linearly . F or example, since [Λ 1 ] 6 , 5 M † 6 p 1 M 5 =[Λ 1 ] 6 , 5 ( B b 2 ) †  ( A a 1 ) 2 − ( A a 1 )  ( B b 1 ) =[Λ 1 ] 6 , 5 h ( A a 1 B b 2 ) † ( A a 1 B b 1 ) − ( B b 2 ) † ( A a 1 B b 1 ) i , 31 w e ma y mak e the follo win g iden tiﬁcation [Ω 1 ] 9 , 8 = − [Ω 1 ] 6 , 8 = [Λ 1 ] 6 , 5 and so on. Eac h indep enden t en try of Λ i therefore corresp onds to an indep enden t optimiza tion v ariable x k and some Hermitia n matrix F k via Ω i . In the example ab o v e, the F k corresp ondin g to (the r eal part of ) [Λ 1 ] 6 , 5 w ould b e zero eve ryw here except for its (6 , 8), (9 , 8), (8 , 6) and (8 , 9) en try , which r eads as − 1 , 1 , − 1 , 1 resp ectiv ely . More intuitiv ely , eac h of these F k ’s giv es r ise to some p olynomial iden tities suc h that z † F k z = 0 wh en ev er the constrain ts are satisﬁed. Putting ev erything together, we see that the searc h for a p ositiv e semideﬁnite Γ suc h that ν I − B 3322 − P ij s † ij p i s ij = z † Γ z can also b e w ritten as the searc h for a p ositiv e semideﬁnite Γ such that ν I − B 3322 − X k x k z † F k z = z † Γ z , (38) for some appropriate c hoice of F k . C omparing this w ith Eq. (35 ) and E q . (36), w e see that eviden tly an y higher order relaxation in our hierarc hy can also b e imp lemen ted as an SDP . D.3 Y ao’s inequalit y Here, we note that for { A i , B j , C k } i,j,k =1 , 2 , 3 satisfying th e comm u tation relations [ A i , B j ] = 0, [ A i , C k ] = 0, [ B j , C k ] = 0 and for Γ giv en by Eq. (25), w e hav e 3 √ 3 I − B Y ao = z † Γ z + X i,j,k α ij k  I − t † ij k t ij k  + 1 12 √ 3 X i,j,k ′  f ( A ) ij k + f ( B ) ij k + f ( C ) ij k + f ( A ) † ij k + f ( B ) † ij k + f ( C ) † ij k  , (39) where t ij k := A i B j C k , α ij k =      1 2 √ 3 : i 6 = j 6 = k , 0 : i = j = k , 1 12 √ 3 : otherwise. f ( l ) ij k =      2 t † ij k t ik j − t † j j k t j kj − t † k j k t k k j : l = A, 2 t † ij k t k j i − t † iik t k ii − t † ik k t k k i : l = B , 2 t † ij k t j ik − t † ij j t j ij − t † ij i t j ii : l = C , and th e second sum P ′ is o ver all p ossible i, j, k su c h that i 6 = j 6 = k . In contrast with Eq. (26) , the ab o v e equalit y holds ev en if n one of the constrain ts A 2 i = B 2 j = C 2 k = I are satisﬁed. Moreov er, the ab o ve equ ality can also b e cast in the form of Eq. (8) with the help of iden tities suc h as I − t † ij k t ij k = p ( C ) k + C † k p ( B ) j C k + g † j k p ( A ) i g j k , and f ( A ) ij k + f ( A ) † ij k = 2 g † j k p ( A ) i g j k + 2 g † k j p ( A ) i g k j − g † j k  p ( A ) j + p ( A ) k  g j k − g † k j  p ( A ) j + p ( A ) k  g k j + ( g j k + g k j ) † p ( A ) j ( g j k + g k j ) + ( g j k + g k j ) † p ( A ) k ( g j k + g k j ) − 2 ( g j k + g k j ) † p ( A ) i ( g j k + g k j ) , where g j k := B j C k . 32

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