Quantum no-signalling correlations and non-local games
aa r X i v : . [ m a t h . OA ] S e p QUANTUM NO-SIGNALLING CORRELATIONS ANDNON-LOCAL GAMES
IVAN G. TODOROV AND LYUDMILA TUROWSKA
Abstract.
We introduce and examine three subclasses of the familyof quantum no-signalling (QNS) correlations introduced by Duan andWinter: quantum commuting, quantum and local. We formalise the no-tion of a universal TRO of a block operator isometry, define an operatorsystem, universal for stochastic operator matrices, and realise it as a quo-tient of a matrix algebra. We describe the classes of QNS correlations interms of states on the tensor products of two copies of the universal op-erator system, and specialise the correlation classes and their represen-tations to classical-to-quantum correlations. We study various quantumversions of synchronous no-signalling correlations and show that theypossess invariance properties for suitable sets of states. We introducequantum non-local games as a generalisation of non-local games. Wedefine the operation of quantum game composition and show that theperfect strategies belonging to a certain class are closed under channelcomposition. We specialise to the case of graph colourings, where we ex-hibit quantum versions of the orthogonal rank of a graph as the optimaloutput dimension for which perfect classical-to-quantum strategies ofthe graph colouring game exist, as well as to non-commutative graph ho-momorphisms, where we identify quantum versions of non-commutativegraph homomorphisms introduced by Stahlke.
Contents
1. Introduction 22. Preliminaries 63. Stochastic operator matrices 94. Three subclasses of QNS correlations 124.1. Quantum commuting QNS correlations 124.2. Quantum QNS correlations 144.3. Local QNS correlations 165. The operator system of a stochastic operator matrix 176. Descriptions via tensor products 227. Classical-to-quantum no-signalling correlations 287.1. Definition and subclasses 287.2. Description in terms of states 318. Classical reduction and separation 339. Quantum versions of synchronicity 36
Date : 4 September 2020.
Introduction
Non-local games [18] have in the past decade acquired significant promi-nence, demonstrating both the power and limitations of quantum entangle-ment. These are cooperative games, played by two players, Alice and Bob,against a verifier, in each round of which the verifier feeds in as an inputa pair ( x, y ), selected from the cartesian product X × Y of two finite sets,and the players produce as an output a pair ( a, b ) from a cartesian product A × B . The combinations ( x, y, a, b ) that yield a win are determined bya predicate function λ : X × Y × A × B → { , } . A probabilistic strat-egy is a family p = { ( p ( a, b | x, y )) ( a,b ) ∈ A × B : ( x, y ) ∈ X × Y } of probabilitydistributions, one for each input pair ( x, y ), where the value p ( a, b | x, y ) de-notes the probability that the players spit out the output ( a, b ) given theyhave received the input ( x, y ). Such families p are in addition required tosatisfy a no-signalling condition, which ensures no communication betweenthe players takes place during the course of the game, and are hence called no-signalling (NS) correlations .In pseudo-telepathy games [10], no deterministic perfect (that is, win-ning) strategies exist, while shared entanglement can produce perfect quan-tum strategies. Such strategies consist of two parts: a unit vector ξ in thetensor product H A ⊗ H B of two finite dimensional Hilbert spaces (repre-senting the joint physical system of the players), and local measurementoperators ( E x,a ) x,a (for Alice) and ( F y,b ) y,b (for Bob), leading to the proba-bilities p ( a, b | x, y ) = h ( E x,a ⊗ F y,b ) ξ, ξ i . Employing the commuting model ofQuantum Mechanics leads, on the other hand, to the broader set of quantumcommuting strategies, whose underlying no-signalling correlations arise frommutually commuting measurement operators (that is, E x,a F y,b = F y,b E x,a )acting on a single Hilbert space. This viewpoint leads to the following chainof classes of no-signalling correlations:(1.1) C loc ⊆ C q ⊆ C qa ⊆ C qc ⊆ C ns . The class C qa of approximately quantum correlations is the closure of thequantum class C q – known, due to the work of Slofstra [70] (see also [22]) to UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 3 be strictly larger than C q – and C ns is the class of all no-signalling correla-tions, playing a fundamental role in generalised probabilistic theories [3, 4].The long-standing question of whether C qa coincides with the class C qc ofall quantum commuting correlations, known as Tsirelson’s problem, was re-cently settled in the negative in [37]. Due to the works [38] and [56], thisalso resolved the fundamental Connes Embedding Problem [65].In this paper, we propose a quantisation of the chain of inclusions (1.1).Our motivation is two-fold. Firstly, the resolution of the Connes EmbeddingProblem in [37] follows complexity theory routes, and it remains of greatinterest if an operator algebraic approach is within reach. The classes ofcorrelations we introduce are wider and hence may offer more flexibility inlooking for counterexamples.Our second source of motivation is the development of non-local gameswith quantum inputs and quantum outputs. A number of versions of quan-tum games have already been examined. In [19], the authors studied thecomputability and the parallel repetition behaviour of the entangled valueof a rank one quantum game, where the players receive quantum inputsfrom the verifier, but a measurement is taken against a rank one projectionto determine the likelihood of winning. In [30], the focus is on multipleround quantum strategies that are available to players with quantum mem-ory, while the quantum-classical and extended non-local games consideredin [67] both have classical outputs (see also [14]). Here, we propose a frame-work for quantum-to-quantum non-local games, which generalises directly(classical) non-local games. This allows us to define a quantum versionof the graph homomorphism game (see [22, 51, 52, 62]), and leads to no-tions of quantum homomorphisms between (the widely studied at present[8, 7, 20, 21, 46, 71]) non-commutative graphs.Our starting point is the definition of quantum no-signalling correlationsgiven by Duan and Winter in [21]. Note that no-signalling (NS) correlationscorrespond precisely to (bipartite) classical information channels from X × Y to A × B with well-defined marginals. In [21], quantum no-signalling(QNS) correlations are thus defined as quantum channels M X × Y → M A × B (here M Z denotes the space of all Z × Z complex matrices) whose marginalchannels are well-defined. In Section 4, we define the quantum versions ofthe classes in (1.1), arriving at an analogous chain(1.2) Q loc ⊆ Q q ⊆ Q qa ⊆ Q qc ⊆ Q ns . The base for our definitions is a quantisation of positive operator valuedmeasures, which we develop in Section 3. The stochastic operator matrices defined therein replace the families ( E x,a ) x ∈ X,a ∈ A of measurement operatorsthat play a crucial role in the definitions of the classical classes (1.1). InSection 5, we define a universal operator system T X,A , whose concrete repre-sentations on Hilbert spaces are precisely determined by stochastic operatormatrices. Our route passes through the definition of a universal ternaryring of operators V X,A of a given A × X -block operator isometry, which is a I. G. TODOROV AND L. TUROWSKA generalisation of the Brown algebra of a unitary matrix [11] (see also [29]).We describe T X,A as a quotient of a full matrix algebra (Corollary 5.6); thisis a quantum version of a previous known result in the classical case [26].We show that any such quotient possesses the local lifting property [43].This unifies a number of results in the literature, implying in particular [35,Theorem 4.9].In Section 6, we provide operator theoretic descriptions of the classes Q loc , Q qa , Q qc and Q ns , establishing a perfect correspondence between theelements of these classes and states on operator system tensor products. Wesee that, similarly to the case of classical NS correlations [49], each QNScorrelation of the class Q qc arises from a state on the commuting tensorproduct T X,A ⊗ c T Y,B , and that similar descriptions hold for the rest of theaforementioned classes. Along with the hierarchy (1.2), we introduce anintermediate chain(1.3) CQ loc ⊆ CQ q ⊆ CQ qa ⊆ CQ qc ⊆ CQ ns , lying between (1.1) and (1.2), whose terms are classes of classical-to-quantumno-signalling (CQNS) correlations . We define their universal operator sys-tem, and provide analogous characterisations in terms of states on its tensorproducts; this is achieved in Section 7. In Section 8, we point out the canon-ical surjections Q x → CQ x → C x (where x denotes any specific correlationclass from the set { loc , q , qa , qc , ns } ). Combined with the separation resultsat each term, known for (1.1), this implies that the inclusions in (1.2) and(1.3) are proper.The class Q loc at the ground level of the chain (1.2) is in fact well-known: its elements are precisely the local operations and shared random-ness (LOSR) channels (see e.g. [73, p. 358]). Thus, the channels from Q q can be thought of as entanglement assisted LOSR transformations, and asimilar interpretation can be adopted for the higher terms of (1.2).The notion of a synchronous NS correlation [61] is of crucial importancewhen correlations are employed as strategies of non-local games. Here, weassume that X = Y and A = B . These correlations were characterised in[61] as arising from traces on a universal C*-algebra A X,A – the free prod-uct of | X | copies of the | A | -dimensional abelian C*-algebra. In Section 9,we propose two quantum versions of synchronicity. Fair correlations aredefined in operational terms, but display a lower level of relevance than tra-cial correlations, which are defined operator algebraically, via traces on theuniversal C*-algebra of a stochastic operator matrix. Tracial QNS correla-tions are closely related to factorisable channels [1] which have been used toproduce counterexamples to the asymptotic Birkhoff conjecture [31]. Moreprecisely, if one restricts attention to QNS correlations that arise from theBrown algebra as opposed to the ternary ring of operators V X,A , then thetracial QNS correlations are precisely the couplings of a pair of factorisablechannels with equal terms.
UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 5
Restricted to CQNS and NS correlations, traciality produces classes ofcorrelations that strictly contain synchronous NS correlations. The differ-ence between synchronous and tracial NS correlations can be heuristicallycompared to that between projection and positive operator valued measures.The operational significance of tracial QNS, tracial CQNS and tracial NScorrelations arises from the preservation of appropriate classes of states,which quantise the symmetry possessed by the classical pure states sup-ported on the diagonal of a matrix algebra. The ground class, of locallyreciprocal states , turns out to be a twisted version of de Finetti states [17].Thus, the higher classes of quantum reciprocal and
C*-reciprocal states canbe thought of as an entanglement assisted and a commuting model version,respectively, of de Finetti states.In Section 10, we point out how QNS and CQNS correlations can beused as strategies for quantum-to-quantum and classical-to-quantum non-local games. This is not an exhaustive treatment, and is rather intended tosummarise several directions and provide a general context that we hope toinvestigate subsequently. In Subsection 10.1, we show that, when comparedto NS correlations, CQNS correlations provide a significant advantage inthe graph colouring game [15]. Employing the CQNS classes, we define newversions of quantum chromatic numbers of a classical graph G . The class CQ loc yields the well-known orthogonal rank ξ ( G ) of G [68]; thus, the chro-matic numbers ξ q ( G ) and ξ qc ( G ), arising from CQ q and CQ qc , respectively,can be thought of as entanglement assisted and commuting model versionsof this classical graph parameter. We show that ξ qc ( G ) does not degenerate,in that it is always lower bounded by p d/θ ( G ), where d is the number ofvertices of G and θ ( G ) is its Lov´asz number.In Subsection 10.2, we define a non-commutative version of the graph ho-momorphism game [51]. We show that its perfect strategies from the class Q loc correspond precisely to non-commutative graph homomorphisms in thesense of Stahlke [71]. Thus, the perfect strategies from the larger classesin (1.2) can be thought of as quantum non-commutative graph homomor-phisms. We note that special cases have been previously considered in [8] and[55]. The treatment in the latter papers was restricted to non-commutativegraph isomorphisms, and the suggested approach was operator-algebraic.We remedy this by suggesting, up to our knowledge, the first operationalapproach to non-commutative graph homomorphisms, thus aligning the non-commutative case with the case of quantum homomorphisms between clas-sical graphs [51].Finally, in Subsection 10.3, we introduce a quantum version of non-localgames that contains as a special case the games considered in the previoussubsections. To this end, we view the rule predicate as a map between theprojection lattices of algebras of diagonal matrices. We define game composi-tion, show that the perfect strategies from a fixed class x ∈ { loc , q , qa , qc , ns } are closed under channel composition, and prove that channel composition I. G. TODOROV AND L. TUROWSKA preserves traciality. Some of these results extend results previously provedin [57] in the case of classical no-signalling strategies.2.
Preliminaries
All inner products appearing in the paper will be assumed linear in thefirst variable. Let H be a Hilbert space. We denote by B ( H ) the spaceof all bounded linear operators on H and often write L ( H ) if H is finitedimensional. If ξ, η ∈ H , we write ξη ∗ for the rank one operator given by( ξη ∗ )( ζ ) = h ζ, η i ξ . In addition to inner products, h· , ·i will denote bilineardualities between a vector space and its dual. We write B ( H ) + for thecone of positive operators in B ( H ), denote by T ( H ) its ideal of trace classoperators, and by Tr – the trace functional on T ( H ).An operator system is a self-adjoint subspace S of B ( H ) for some Hilbertspace H , containing the identity operator I H . The linear space M n ( S ) ofall n by n matrices with entries in S can be canonically identified with asubspace of B ( H n ), where H n is the direct sum of n -copies of H ; we set M n ( S ) + = M n ( S ) ∩ B ( H n ) + and write S h for the real vector space of allhermitian elements of S . If K is a Hilbert space, T ⊆ B ( K ) is an operatorsystem and φ : S → T is a linear map, we let φ ( n ) : M n ( S ) → M n ( T ) bethe (linear) map given by φ ( n ) (( x i,j ) i,j ) = ( φ ( x i,j )) i,j . The map φ is called positive (resp. unital ) if φ ( S + ) ⊆ T + (resp. φ ( I H ) = I K ), and completelypositive if φ ( n ) is positive for every n ∈ N . We call φ a complete orderembedding if it is injective and φ − | φ ( S ) : φ ( S ) → S is completely positive;we write S ⊆ c . o . i . T . We note that C is an operator system in a canonicalway; a state of S is a unital positive (linear) map φ : S → C . We denoteby S ( S ) the (convex) set of all states of S . We note that every operatorsystem is an operator space in a canonical fashion, and denote by S d thedual Banach space of S , equipped with its canonical matrix order structure.Operator systems can be described abstractly via a set of axioms [58]; werefer the reader to [23], [58] and [64] for details and for further backgroundon operator space theory.We denote by | X | the cardinality of a finite set X , let H X = ⊕ x ∈ X H anddenote by M X the space of all complex matrices of size | X |×| X | ; we identify M X with L ( C X ) and write I X = I C X . For n ∈ N , we set [ n ] = { , . . . , n } and M n = M [ n ] . We write ( e x ) x ∈ X for the canonical orthonormal basis of C X , denote by D X the subalgebra of M X of all diagonal, with respect tothe basis ( e x ) x ∈ X , matrices, and let ∆ X : M X → D X be the correspondingconditional expectation.When ω is a linear functional on M X , we often write ω = ω X . Thecanonical complete order isomorphism from M X onto M d X maps an element ω ∈ M X to the linear functional f ω : M X → C given by f ω ( T ) = Tr( T ω t )(here, and in the sequel, ω t denotes the transpose of ω in the canonicalbasis); see e.g. [63, Theorem 6.2]. We will thus consider M X as self-dual UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 7 space with pairing(2.1) ( ρ, ω ) → h ρ, ω i := Tr( ρω t ) . On the other hand, note that the Banach space predual B ( H ) ∗ can be canon-ically identified with T ( H ); every normal functional φ : B ( H ) → C thuscorresponds to a (unique) operator S φ ∈ T ( H ) such that φ ( T ) = Tr( T S φ ), T ∈ B ( H ). In the case where X is a fixed finite set (which will sometimescome in the form of a direct product), we will use a mixture of the twodualities just discussed: if ω, ρ ∈ M X , S ∈ T ( H ) and T ∈ B ( H ), it will beconvenient to continue writing h ρ ⊗ T, ω ⊗ S i = Tr( ρω t ) Tr( T S ) . If X and Y are finite sets, we identify M X ⊗ M Y with M X × Y and write M XY in its place. Similarly, we set D XY = D X ⊗ D Y . Here, and in thesequel, we use the symbol ⊗ to denote the algebraic tensor product of vectorspaces. For an element ω X ∈ M X and a Hilbert space H , we let L ω X : M X ⊗ B ( H ) → B ( H ) be the linear map given by L ω X ( S ⊗ T ) = h S, ω X i T .If H = C Y and ω Y ∈ M Y , we thus have linear maps L ω X : M XY → M Y and L ω Y : M XY → M X ; note that h L ω X ( R ) , ρ Y i = h R, ω X ⊗ ρ Y i , R ∈ M XY , ρ Y ∈ M Y , and a similar formula holds for L ω Y . We let Tr X : M XY → M Y (resp. Tr Y : M XY → M X ) be the partial trace, that is, Tr X = L I X (resp. Tr Y = L I Y ).Let X and A be finite sets. A classical information channel from X to A is a positive trace preserving linear map N : D X → D A . It is clear that if N : D X → D A is a classical channel then p ( ·| x ) := N ( e x e ∗ x ) is a probabilitydistribution over A , and that N is completely determined by the family { ( p ( a | x )) a ∈ A : x ∈ X } .A quantum channel from M X into M A is a completely positive tracepreserving map Φ : M X → M A ; such a Φ will be called ( X, A ) -classical ifΦ = ∆ A ◦ Φ ◦ ∆ X . A classical channel N : D X → D A gives rise to a ( X, A )-classical (quantum) channel Φ N : M X → M A by letting Φ N = N ◦ ∆ X .Conversely, a quantum channel Φ : M X → M A induces a classical channel N Φ : D X → D A by letting N Φ = ∆ A ◦ Φ | D X . Note that N Φ N = N for everyclassical channel N .Let X, Y, A and B be finite sets. A quantum correlation over ( X, Y, A, B )(or simply a quantum correlation if the sets are understood from the context)is a quantum channel Γ : M XY → M AB . Such a Γ is called a quantum no-signalling (QNS) correlation [21] if(2.2) Tr A Γ( ρ X ⊗ ρ Y ) = 0 whenever Tr( ρ X ) = 0and(2.3) Tr B Γ( ρ X ⊗ ρ Y ) = 0 whenever Tr( ρ Y ) = 0 . I. G. TODOROV AND L. TUROWSKA
We denote by Q ns the set of all QNS correlations; it is clear that Q ns is aclosed convex subset of the cone CP( M XY , M AB ) of all completely positivemaps from M XY into M AB . Remark 2.1.
A quantum channel Γ : M XY → M AB is a QNS correlationif and only if Tr A Γ( ρ ′ ) = 0 and Tr B Γ( ρ ′′ ) = 0provided ρ ′ , ρ ′′ ∈ M XY are such that Tr X ρ ′ = 0 and Tr Y ρ ′′ = 0. Indeed,suppose that Γ is a QNS correlation and ρ ′ ∈ M XY , Tr X ρ ′ = 0. Writing ρ ′ = P x,x ′ ,y,y ′ ρ ′ x,x ′ ,y,y ′ e x e ∗ x ′ ⊗ e y e ∗ y ′ , we have that P x ∈ X ρ ′ x,x,y,y ′ = 0 for all y , y ′ ∈ Y . Thus Tr (cid:16)P x ∈ X ρ ′ x,x,y,y ′ e x e ∗ x (cid:17) = 0, and henceTr A Γ X x ∈ X ρ ′ x,x,y,y ′ e x e ∗ x ! ⊗ e y e ∗ y ′ ! = 0 , y, y ′ ∈ Y. Since Tr e x e ∗ x ′ = δ x,x ′ , we also have Tr A Γ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) = 0 if x = x ′ , forall y, y ′ ∈ Y . It follows that Tr A Γ( ρ ′ ) = 0. The second property is verifiedsimilarly, while the converse direction of the statement is trivial.A classical correlation over ( X, Y, A, B ) is a family p = (cid:8) ( p ( a, b | x, y )) ( a,b ) ∈ A × B : ( x, y ) ∈ X × Y (cid:9) , where ( p ( a, b | x, y )) ( a,b ) ∈ A × B is a probability distribution for each ( x, y ) ∈ X × Y ; classical correlations p thus correspond precisely to classical channels N p : D XY → D AB . A classical no-signalling correlation (or simply a no-signalling (NS) correlation) is a correlation p = (( p ( a, b | x, y )) a,b ) x,y thatsatisfies the conditions(2.4) X a ′ ∈ A p ( a ′ , b | x, y ) = X a ′ ∈ A p ( a ′ , b | x ′ , y ) , x, x ′ ∈ X, y ∈ Y, b ∈ B, and(2.5) X b ′ ∈ B p ( a, b ′ | x, y ) = X b ′ ∈ B p ( a, b ′ | x, y ′ ) , x ∈ X, y, y ′ ∈ Y, a ∈ A. We denote by C ns the set of all NS correlations and identify its elementswith classical channels from D XY to D AB . Given a classical correlation p ,we write Γ p = Φ N p ; thus, Γ p : M XY → M AB is the ( X × Y, A × B )-classicalchannel given by(2.6) Γ p ( ρ ) = X x ∈ X,y ∈ Y X a ∈ A,b ∈ B p ( a, b | x, y ) h ρ ( e x ⊗ e y ) , e x ⊗ e y i e a e ∗ a ⊗ e b e ∗ b . Remark 2.2. If p is a classical correlation over ( X, Y, A, B ) then p is anNS correlation precisely when Γ p is a QNS correlation. Indeed, if Tr ρ X = 0 UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 9 and p satisfies (2.4) and (2.5) thenTr A Γ p ( ρ X ⊗ ρ Y )= X x ∈ X,y ∈ Y X a ∈ A,b ∈ B p ( a, b | x, y ) h ρ X e x , e x i h ρ Y e y , e y i e b e ∗ b = X y ∈ Y X b ∈ B X x ∈ X X a ∈ A p ( a, b | x, y ) h ρ X e x , e x i ! h ρ Y e y , e y i e b e ∗ b = 0;(2.3) is checked similarly. Conversely, assuming that Γ p satisfies (2.2) and(2.3), the relations (2.4) and (2.5) are obtained by substituting in (2.6) ρ = e x e ∗ x ⊗ e y e ∗ y − e x ′ e ∗ x ′ ⊗ e y e ∗ y and ρ = e x e ∗ x ⊗ e y e ∗ y − e x e ∗ x ⊗ e y ′ e ∗ y ′ . It followsthat if Γ is a ( X × Y, A × B )-classical QNS correlation then Γ = Γ p for someNS correlation p .Let H , . . . , H k be Hilbert spaces, at most one of which is infinite dimen-sional, T ∈ B ( H ⊗ · · · ⊗ H k ) and f be a bounded functional on B ( H i ⊗ · · · ⊗ H i k ), where k ≤ n and i , . . . , i k are distinct elements of [ n ] (not necessarilyin increasing order). We will use the expression L f ( T ), or h T, f i (in the case k = n ), without mentioning explicitly that a suitable permutation of thetensor factors has been applied before the action of f . We note that, if g is a bounded functional on B ( H j ⊗ · · · ⊗ H j l ), where l ≤ n and the subset { j , . . . , j l } does not intersect { i , . . . , i k } , then(2.7) L f L g = L g L f . Considering an element ω ∈ M X as a functional on M X via (2.1), we havethat, if E = ( E x,x ′ ) x,x ′ ∈ M X ⊗ B ( H ) then(2.8) L e x e ∗ x ′ ( E ) = E x,x ′ , x, x ′ ∈ X. Stochastic operator matrices
Let
X, Y, A and B be finite sets. A stochastic operator matrix over ( X, A )is a positive operator E ∈ M X ⊗ M A ⊗ B ( H ) for some Hilbert space H suchthat(3.1) Tr A E = I X ⊗ I H . We say that E acts on H . This terminology becomes natural after notingthat the operator stochastic matrices E ∈ D X ⊗ D A ⊗ B ( C ) coincide, afterthe natural identification of D X ⊗D A with the space of all | X |×| A | matrices,with the row-stochastic scalar-valued matrices.Let E ∈ M X ⊗ M A ⊗ B ( H ) be a stochastic operator matrix and E x,x ′ ,a,a ′ ∈B ( H ), x, x ′ ∈ X , a, a ′ ∈ A , be the operators such that E = X x,x ′ ∈ X X a,a ′ ∈ A e x e ∗ x ′ ⊗ e a e ∗ a ′ ⊗ E x,x ′ ,a,a ′ ;we write E = ( E x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ . Note that E x,x ′ ,a,a ′ = L e x e ∗ x ′ ⊗ e a e ∗ a ′ ( E ) , x, x ′ ∈ X, a, a ′ ∈ A. Set E a,a ′ = ( E x,x ′ ,a,a ′ ) x,x ′ ∈ X ∈ M X ⊗ B ( H );thus, E a,a ′ = L e a e ∗ a ′ ( E ), a, a ′ ∈ A , and hence E a,a ∈ ( M X ⊗ B ( H )) + , a ∈ A .By Choi’s Theorem, stochastic operator matrices E are precisely the Choimatrices of unital completely positive maps Φ E : M A → M X ⊗ B ( H ) definedby(3.2) Φ E ( e a e ∗ a ′ ) = E a,a ′ , a, a ′ ∈ A. Recall that a positive operator-valued measure (POVM) on a Hilbert space H , indexed by A , is a family ( E a ) a ∈ A of positive operators on H , such that P a ∈ A E a = I H . If E a is a projection for each a ∈ A , the family ( E a ) a ∈ A iscalled a projection valued measure (PVM) . Theorem 3.1.
Let H be a Hilbert space and E ∈ ( M X ⊗ M A ⊗ B ( H )) + .The following are equivalent: (i) E is a stochastic operator matrix; (ii) ( E a,a ) a ∈ A is a POVM in M X ⊗ B ( H ) ; (iii) Tr A L ω X ( E ) = I H , for all states ω X ∈ M X ; (iv) Tr A L ω X ( E ) = Tr( ω X ) I H , for all ω X ∈ M X ; (v) there exists a Hilbert space K and operators V a,x : H → K , x ∈ X , a ∈ A , such that ( V a,x ) a,x ∈ B ( H X , K A ) is an isometry and (3.3) E x,x ′ ,a,a ′ = V ∗ a,x V a ′ ,x ′ , x, x ′ ∈ X, a, a ′ ∈ A. In particular, if E is a stochastic operator matrix then ( E x,x,a,a ) a ∈ A is aPOVM for every x ∈ X .Proof. (i) ⇔ (ii) and (iv) ⇒ (iii) are trivial, while (i) ⇒ (iii) is immediate from(2.7).(iii) ⇒ (iv) By assumption, Tr A L ω ( E ) = Tr( ω ) I H for every state ω ∈ M X . Write ω = P i =1 λ i ω i , where ω i is a state in M X and λ i ∈ C , i = 1 , , , A L ω ( E ) = X i =1 λ i Tr A L ω i ( E ) = X i =1 λ i I H = Tr( ω ) I H . (iii) ⇒ (i) By (2.7), for all ω X ∈ S ( M X ) and all normal states τ on B ( H ),we have h I X ⊗ I H , ω X ⊗ τ i = 1 = h Tr A L ω X ( E ) , τ i = h L ω X Tr A ( E ) , τ i = h Tr A ( E ) , ω X ⊗ τ i . By polarisation and linearity, h Tr A ( E ) , σ i = h I X ⊗ I H , σ i for all σ ∈ ( M X ⊗ B ( H )) ∗ , and hence Tr A ( E ) = I X ⊗ I H .(i) ⇒ (v) Let Φ = Φ E be the unital completely positive map given by(3.2). By Stinespring’s Dilation Theorem, there exist a Hilbert space ˜ K , anisometry V : C X ⊗ H → ˜ K and a unital *-homomorphism π : M A → B ( ˜ K ) UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 11 such that Φ( T ) = V ∗ π ( T ) V , T ∈ M A . Up to unitary equivalence, ˜ K = C A ⊗ K for some Hilbert space K and π ( T ) = T ⊗ I K , T ∈ M A . Write V a,x : H → K , a ∈ A , x ∈ X , for the entries of V , when V is considered asa block operator matrix. For ξ, η ∈ H , x, x ′ ∈ X and a, a ′ ∈ A , we have (cid:10) E x,x ′ ,a,a ′ ξ, η (cid:11) = D L e x e ∗ x ′ ( E a,a ′ ) ξ, η E = Tr (cid:16) L e x e ∗ x ′ (Φ( e a e ∗ a ′ ))( ξη ∗ ) (cid:17) = Tr (Φ( e a e ∗ a ′ )(( e x ′ e ∗ x ) ⊗ ( ξη ∗ )))= Tr ( V ∗ (( e a e ∗ a ′ ) ⊗ I K ) V ( e x ′ ⊗ ξ )( e x ⊗ η ) ∗ )= h V ∗ (( e a e ∗ a ′ ) ⊗ I K ) V ( e x ′ ⊗ ξ ) , e x ⊗ η i = h (( e a e ∗ a ′ ) ⊗ I K ) V ( e x ′ ⊗ ξ ) , V ( e x ⊗ η ) i = h (( e a e ∗ a ′ ) ⊗ I K )(( e a ′ e ∗ a ′ ) ⊗ I K ) V ( e x ′ ⊗ ξ ) , (( e a e ∗ a ) ⊗ I K ) V ( e x ⊗ η ) i = (cid:10) V a ′ ,x ′ ξ, V a,x η (cid:11) = (cid:10) V ∗ a,x V a ′ ,x ′ ξ, η (cid:11) . (v) ⇒ (ii) Let ξ = P x ∈ X P a ∈ A e x ⊗ e a ⊗ ξ x,a , where ξ x,a ∈ H , x ∈ X , a ∈ A . Using (3.3), we have h Eξ, ξ i = X x,x ′ ∈ X X a,a ′ ∈ A (cid:10) V a ′ ,x ′ ξ x ′ ,a ′ , V a,x ξ x,a (cid:11) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈ X X a ∈ A V a,x ξ x,a (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , and thus E is positive. Since V is an isometry, we have X a ∈ A E x,x ′ ,a,a = X a ∈ A V ∗ a,x V a,x ′ = δ x,x ′ I H . (cid:3) Let ( E x,a ) a ∈ A be a POVM on a Hilbert space H for every x ∈ X . Astochastic operator matrix of the form(3.4) E = X x ∈ X X a ∈ A e x e ∗ x ⊗ e a e ∗ a ⊗ E x,a will be called classical . A general stochastic operator matrix can thus bethought of as a coordinate-free version of a finite family of POVM’s. Remarks. (i)
In view of Theorem 3.1, stochastic operator matrices areprecisely the positive completions E of partially defined diagonal block ma-trices D = ( E a,a ) a ∈ A with entries in M X ⊗ B ( H ) and Tr A ( D ) = I . (ii) The following generalisation of Naimark’s Dilation Theorem wasproved in [59]: if ( E x,a ) a ∈ A ⊆ B ( H ), x ∈ X , are POVM’s then there exista Hilbert space ˜ H , a PVM ( ˜ E a ) a ∈ A ⊆ B ( ˜ H ) and isometries V x : H → ˜ H , x ∈ X , with orthogonal ranges such that(3.5) E x,a = V ∗ x ˜ E a V x , a ∈ A, x ∈ X. This can be seen as a corollary of Theorem 3.1: given POVM’s ( E x,a ) a ∈ A ⊆B ( H ), x ∈ X , let E be the stochastic operator matrix defined by (3.4) andlet V = ( V a,x ) a,x be the isometry from Theorem 3.1. Set ˜ E a = e a e ∗ a ⊗ I H , a ∈ A , and let V x be the column isometry ( V a,x ) a ∈ A : H → K A , x ∈ X .Then ( ˜ E a ) a ∈ A is a PVM fulfilling (3.5).Let E ∈ M X ⊗ M A ⊗B ( H ) be a stochastic operator matrix and Φ = Φ E begiven by (3.2). Recall that the predual Φ ∗ : M X ⊗ T ( H ) → M A of Φ is thecompletely positive map satisfying h Φ ∗ ( ρ ) , ω i = h ρ, Φ( ω ) i , ρ ∈ M X ⊗ T ( H ), ω ∈ M A . For a state σ ∈ T ( H ), setΓ E,σ ( ρ X ) = Φ ∗ ( ρ X ⊗ σ ) , ρ X ∈ M X ;then Γ E,σ : M X → M A is a quantum channel. We have(3.6) Γ E,σ ( ρ X ) = L ρ X ⊗ σ ( E ) , ρ X ∈ M X ;indeed, if a, a ′ ∈ A then h Γ E,σ ( e x e ∗ x ′ ) , e a e ∗ a ′ i = h Φ ∗ ( e x e ∗ x ′ ⊗ σ ) , e a e ∗ a ′ i = h e x e ∗ x ′ ⊗ σ, Φ( e a e ∗ a ′ ) i = (cid:10) e x e ∗ x ′ ⊗ σ, E a,a ′ (cid:11) = (cid:10) σ, E x,x ′ ,a,a ′ (cid:11) = D L e x e ∗ x ′ ⊗ σ ( E ) , e a e ∗ a ′ E ;(3.6) now follows by linearity. By Choi’s Theorem, every quantum channelΦ : M X → M A has the form Γ E, for some stochastic operator matrix E ∈ M X ⊗ M A . Remark 3.2.
Let H be a Hilbert space and E ∈ M X ⊗ M A ⊗ B ( H ) be astochastic operator matrix. The following are equivalent:(i) E is classical;(ii) for each state σ ∈ T ( H ), the quantum channel Γ E,σ : M X → M A is( X, A )-classical.
Proof.
The channel Γ
E,σ is (
X, A )-classical if and only if Γ
E,σ ( e x e ∗ x ′ ) = 0whenever x = x ′ and h Γ E,σ ( e x e ∗ x ) , e a e ∗ a ′ i = 0 whenever a = a ′ . The latter equality holds for every σ if and only if E x,x ′ ,a,a ′ = 0 whenever x = x ′ and E x,x,a,a ′ = 0 whenever a = a ′ , that is, if and only if E isclassical. (cid:3) Three subclasses of QNS correlations
In this section, we introduce several classes of QNS correlations, whichgeneralise corresponding classes of NS correlations studied in the literature(see e.g. [49]).4.1.
Quantum commuting QNS correlations.
Let H be a Hilbert space,and E ∈ M X ⊗ M A ⊗ B ( H ) and F ∈ M Y ⊗ M B ⊗ B ( H ) be stochastic op-erator matrices. The pair ( E, F ) will be called commuting if L ω X ⊗ ω A ( E )and L ω Y ⊗ ω B ( F ) commute for all ω X ∈ M X , ω Y ∈ M Y , ω A ∈ M A and ω B ∈ M B . Writing E = ( E x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ and F = ( F y,y ′ ,b,b ′ ) y,y ′ ,b,b ′ , we havethat ( E, F ) is commuting if and only if E x,x ′ ,a,a ′ F y,y ′ ,b,b ′ = F y,y ′ ,b,b ′ E x,x ′ ,a,a ′ , x, x ′ ∈ X, y, y ′ ∈ Y, a, a ′ ∈ A, b, b ′ ∈ B. UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 13
Proposition 4.1.
Let H be a Hilbert space and E ∈ M X ⊗ M A ⊗ B ( H ) , F ∈ M Y ⊗ M B ⊗B ( H ) form a commuting pair of stochastic operator matrices.There exists a unique operartor E · F ∈ M XY ⊗ M AB ⊗ B ( H ) such that (4.1) h E · F, ρ X ⊗ ρ Y ⊗ ρ A ⊗ ρ B ⊗ σ i = h L ρ X ⊗ ρ A ( E ) L ρ Y ⊗ ρ B ( F ) , σ i , for all ρ X ∈ M X , ρ Y ∈ M Y , ρ A ∈ M A , ρ B ∈ M B and σ ∈ T ( H ) . Moreover, (i) E · F is a stochastic operator matrix; (ii) k E · F k ≤ k E kk F k ; (iii) If σ ∈ T ( H ) is a state then Γ E · F,σ is a QNS correlation.Proof.
Let E · F := (cid:0) E x,x ′ ,a,a ′ F y,y ′ ,b,b ′ (cid:1) ∈ M XY ⊗ M AB ⊗ B ( H ) . Denote by A (resp. B ) the C*-algebra, generated by E x,x ′ ,a,a ′ , x, x ′ ∈ X , a, a ′ ∈ A (resp. F y,y ′ ,b,b ′ , y, y ′ ∈ Y , b, b ′ ∈ B ); by assumption, B ⊆ A ′ . Let π A : M XA ( A ) → M XY AB ( B ( H )) (resp. π B : M Y B ( B ) → M XY AB ( B ( H )))be the *-representation given by π A ( S ) = S ⊗ I Y B (resp. π B ( T ) = T ⊗ I XA ).Then the ranges of π A and π B commute and hence the pair ( π A , π B ) givesrise to a *-representation π : M XA ( A ) ⊗ max M Y B ( B ) → M XY AB ( B ( H )) with π ( S ⊗ T ) = π A ( S ) π B ( T ), S ∈ M XA ( A ), T ∈ M Y B ( B ). Thus, E · F = π ( E ⊗ F ) ∈ M XY AB ( B ( H )) + . Inequality (ii) now follows from the contractivity of*-representations. In addition,Tr AB ( E · F ) = X a ∈ A X b ∈ B (cid:0) E x,x ′ ,a,a F y,y ′ ,b,b (cid:1) x,x ′ ,y,y ′ = (cid:0) δ x,x ′ δ y,y ′ I (cid:1) x,x ′ ,y,y ′ = I XY ⊗ I H , that is, E · F is a stochastic operator matrix. For x, x ′ ∈ X , y, y ′ ∈ Y , a, a ′ ∈ A , b, b ′ ∈ B and σ ∈ T ( H ), we have (cid:10) E · F, e x e ∗ x ′ ⊗ e y e ∗ y ′ ⊗ e a e ∗ a ′ ⊗ e b e ∗ b ′ ⊗ σ (cid:11) (4.2) = (cid:10) E x,x ′ ,a,a ′ F y,y ′ ,b,b ′ , σ (cid:11) = D L e x e ∗ x ′ ⊗ e a e ∗ a ′ ( E ) L e y e ∗ y ′ ⊗ e b e ∗ b ′ ( F ) , σ E , and (4.1) follows by linearity.To show (iii), let σ ∈ T ( H ) be a state. Suppose that ρ X ∈ M X is tracelessand ρ Y ∈ M Y . For every τ B ∈ M B , by (4.1) and Theorem 3.1, we have h Tr A Γ E · F,σ ( ρ X ⊗ ρ Y ) , τ B i = h Γ E · F,σ ( ρ X ⊗ ρ Y ) , I A ⊗ τ B i = h E · F, ρ X ⊗ ρ Y ⊗ I A ⊗ τ B ⊗ σ i = h Tr A L ρ X ( E ) L ρ Y ⊗ τ B ( F ) , σ i = 0 . Thus, (2.2) is satisfied; by symmetry, so is (2.3). (cid:3) If ξ is a unit vector in H , we set for brevity Γ E,F,ξ = Γ E · F,ξξ ∗ . Definition 4.2.
A QNS correlation of the form Γ E,F,ξ , where ( E, F ) is acommuting pair of stochastic operator matrices acting on a Hilbert space H ,and ξ ∈ H is a unit vector, will be called quantum commuting . We denote by Q qc the set of all quantum commuting QNS correlations. Proposition 4.3.
In Definition 4.2 one can assume, without gain of gen-erality, that σ is an arbitrary state.Proof. Suppose that H is a Hilbert space and E ∈ M X ⊗ M A ⊗ B ( H ), F ∈ M Y ⊗ M B ⊗B ( H ) form a commuting pair of stochastic operator matrices. Let σ be a state in T ( H ) and write σ = P ∞ i =1 λ i ξ i ξ ∗ i , where ( ξ i ) ∞ i =1 is sequence ofunit vectors and λ i ≥ i ∈ N , are such that P ∞ i =1 λ i = 1. Set ˜ H = H ⊗ ℓ and ξ = P ∞ i =1 √ λ i ξ i ⊗ e i ; then ξ is a unit vector in ˜ H and h ξξ ∗ , T ⊗ I ℓ i = h σ, T i , T ∈ B ( H ).Let ˜ E = E ⊗ I ℓ and ˜ F = F ⊗ I ℓ ; thus, ˜ E and ˜ F are stochastic operatormatrices acting on ˜ H that form a commuting pair. Moreover, if ρ X ∈ M X , ρ Y ∈ M Y , σ A ∈ M A and σ B ∈ M B then D Γ ˜ E, ˜ F ,ξ ( ρ X ⊗ ρ Y ) , σ A ⊗ σ B E = D ˜ E · ˜ F , ρ X ⊗ ρ Y ⊗ σ A ⊗ σ B ⊗ ξξ ∗ E = D L ρ X ⊗ σ A ( ˜ E ) L ρ Y ⊗ σ B ( ˜ F ) , ξξ ∗ E = h ( L ρ X ⊗ σ A ( E ) L ρ X ⊗ σ A ( F )) ⊗ I ℓ , ξξ ∗ i = h L ρ X ⊗ σ A ( E ) L ρ X ⊗ σ A ( F ) , σ i = h Γ E · F,σ ( ρ X ⊗ ρ Y ) , σ A ⊗ σ B i . (cid:3) Remark 4.4.
Recall that a classical NS correlation p over ( X, Y, A, B ) iscalled quantum commuting [61, 62] if there exist a Hilbert space H , POVM’s( E x,a ) a ∈ A , x ∈ X , and ( F y,b ) b ∈ B , y ∈ Y , on H with E x,a F y,b = F y,b E x,a forall x, y, a, b , and a unit vector ξ ∈ H , such that p ( a, b | x, y ) = h E x,a F y,b ξ, ξ i , x ∈ X, y ∈ Y, a ∈ A, b ∈ B. Suppose that the stochastic operator matrices E ∈ M X ⊗ M A ⊗ B ( H ) and F ∈ M Y ⊗ M B ⊗B ( H ) are classical, and correspond to the families ( E x,a ) a ∈ A , x ∈ X , and ( F y,b ) b ∈ B , y ∈ Y , respectively, as in (3.4). It is clear that pair( E, F ) is commuting and E · F is classical. We have that Γ p = Γ E,F,ξ .Indeed, by Remark 3.2, the QNS correlation Γ
E,F,ξ is classical; by Remark2.2, Γ
E,F,ξ = Γ p ′ for some NS correlation p ′ . It is now straightforward that p ′ = p .4.2. Quantum QNS correlations.
Let H A and H B be Hilbert spaces,and E ∈ M X ⊗ M A ⊗ B ( H A ) and F ∈ M Y ⊗ M B ⊗ B ( H B ) be stochasticoperator matrices; then E ⊗ F ∈ M X ⊗ M A ⊗ B ( H A ) ⊗ M Y ⊗ M B ⊗ B ( H B ) . Reshuffling the terms of the tensor product, we consider E ⊗ F as an elementof M XY ⊗ M AB ⊗B ( H A ⊗ H B ); to underline this distinction, the latter elementwill henceforth be denoted by E ⊙ F . Note that, if˜ E = E ⊗ I H B ∈ M X ⊗ M A ⊗ B ( H A ⊗ H B )and ˜ F = F ⊗ I H A ∈ M Y ⊗ M B ⊗ B ( H A ⊗ H B ) UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 15 (where the last containment is up to a suitable permutation of the tensorfactors), then ( ˜ E, ˜ F ) is a commuting pair of stochastic operator matrices,and E ⊙ F = ˜ E · ˜ F . By Proposition 4.1, E ⊙ F is a stochastic operatormatrix on H A ⊗ H B and, if σ ∈ T ( H A ⊗ H B ) is a state then, by Proposition4.1, Γ E ⊙ F,σ is a QNS correlation.
Remark 4.5.
It is straightforward to check that, if σ = σ A ⊗ σ B , where σ A ∈ T ( H A ) and σ B ∈ T ( H B ) are states, then Γ E,σ A ⊗ Γ F,σ B = Γ E ⊙ F,σ A ⊗ σ B . Definition 4.6. (i) A QNS correlation
Γ : M XY → M AB is called quan-tum if there exist finite dimensional Hilbert spaces H A and H B , stochasticoperator matrices E ∈ M X ⊗ M A ⊗ L ( H A ) and F ∈ M Y ⊗ M B ⊗ L ( H B ) anda pure state σ ∈ L ( H A ⊗ H B ) such that Γ = Γ E ⊙ F,σ .(ii) A QNS correlation will be called approximately quantum if it is thelimit of a sequence of quantum QNS correlations.
We denote by Q q (resp. Q qa ) the set of all quantum (resp. approximatelyquantum) QNS correlations. It is clear from the definitions that Q q ⊆ Q qc .It will be shown later that Q qc is closed, and hence contains Q qa .Similarly to Proposition 4.3, it can be shown that quantum QNS cor-relations can equivalently be defined using arbitrary, as opposed to pure,states. Remark 4.7.
Recall that a classical NS correlation p over ( X, Y, A, B ) iscalled quantum if there exist finite dimensional Hilbert spaces H A and H B ,POVM’s ( E x,a ) a ∈ A , on H A , x ∈ X , ( F y,b ) b ∈ A on H B , y ∈ Y , and a unitvector ξ ∈ H A ⊗ H B , such that(4.3) p ( a, b | x, y ) = h ( E x,a ⊗ F y,b ) ξ, ξ i , x ∈ X, y ∈ Y, a ∈ A, b ∈ B. It is easy to verify that, if the stochastic operator matrices E ∈ M X ⊗ M A ⊗B ( H A ) and F ∈ M Y ⊗ M B ⊗ B ( H B ) are classical, and determined by thefamilies ( E x,a ) a ∈ A , x ∈ X , and ( F y,b ) b ∈ B , y ∈ Y , then E ⊙ F is classical anddetermined by the family { ( E x,a ⊗ F y,b ) ( a,b ) ∈ A × B : ( x, y ) ∈ X × Y } . As inRemark 4.4, it is easy to see that Γ p = Γ E,F,ξ . Proposition 4.8.
The sets Q q and Q qa are convex.Proof. Let E i ∈ M X ⊗ M A ⊗ L ( K ,i ) (resp. F i ∈ M Y ⊗ M B ⊗ L ( K ,i )) bea stochastic operator matrix over ( X, A ) (resp. (
Y, B )) and σ i = η i η ∗ i bea pure state on K ,i ⊗ K ,i , i = 1 , . . . , n . Fix λ i ≥ i = 1 , . . . , n , with P ni =1 λ i = 1. Let K k = ⊕ ni =1 K k,i , k = 1 , E = ⊕ ni =1 E i , F = ⊕ ni =1 F i ,and η = ⊕ ni =1 √ λ i η i ∈ K ⊗ K , viewed as supported on the ( i, i )-terms of K ⊗ K ≡ ⊕ ni,j =1 K ,i ⊗ K ,j . Set σ = ηη ∗ . Using distributivity, we consider E (resp. F ) as a stochastic operator matrix in M X ⊗ M A ⊗ L ( K ) (resp. M Y ⊗ M B ⊗ L ( K )). A direct verification shows that n X i =1 λ i Γ E i ⊙ F i ,σ i = Γ E ⊙ F,σ ; thus, Q q is convex, and the convexity of Q qa follows from the fact that Q qa = Q q . (cid:3) Local QNS correlations.
It is clear that, if Φ : M X → M A andΨ : M Y → M B are quantum channels, then the quantum channel Γ := Φ ⊗ Ψis a QNS correlation.
Definition 4.9.
A QNS correlation
Γ : M XY → M AB is called local ifit is a convex combination of quantum channels of the form Φ ⊗ Ψ , where Φ : M X → M A and Ψ : M Y → M B are quantum channels. We denote by Q loc the set of all local QNS correlations. The elementsof Q loc are precisely the maps that arise via local operations and sharedrandomness (LOSR) (see e.g. [73, p. 358]). Remark 4.10.
We have that Q loc is a closed convex subset of Q q .Proof. Let Φ : M X → M A and Ψ : M Y → M B be quantum channels and E ∈ M X ⊗ M A and F ∈ M Y ⊗ M B be the Choi matrices of Φ and Ψ,respectively. By Remark 4.5,Φ ⊗ Ψ = Γ E, ⊗ Γ F, = Γ E ⊙ F, and hence Φ ⊗ Ψ ∈ Q q .Let (Γ k ) k ∈ N ⊆ Q loc be a sequence with limit Γ ∈ Q ns . Note that Γ k allare elements of a real vector space of dimension 2 | X | | Y | | A | | B | . Let L =2 | X | | Y | | A | | B | + 1. By Carath´eodory’s Theorem, Γ k = P Ll =1 λ ( k ) l Φ ( k ) l ⊗ Ψ ( k ) l as a convex combination. By compactness, we may assume, by passingto subsequences as necessary, that Φ ( k ) l → k →∞ Φ l , Ψ ( k ) l → k →∞ Ψ l and λ ( k ) l → k →∞ λ l . Thus, Γ = P Ll =1 λ l Φ l ⊗ Ψ l as a convex combination, that is,Γ ∈ Q loc , showing that Q loc is closed. (cid:3) Remark 4.11.
Recall that a classical NS correlation p over ( X, Y, A, B ) iscalled local if there exist families of probability distributions { ( p k ( a | x )) a ∈ A : x ∈ X } and { ( p k ( b | y )) b ∈ B : y ∈ Y } and positive scalars λ k , k = 1 , . . . , m ,such that P mk =1 λ k = 1 and p ( a, b | x, y ) = m X k =1 λ k p k ( a | x ) p k ( b | y ) , x ∈ X, y ∈ Y, a ∈ A, b ∈ B. It is clear that, if Φ k (resp. Ψ k ) is the ( X, A )-classical (resp. (
Y, B )-classical)channel corresponding to p k (resp. p k ) then Γ p = P mk =1 λ k Φ k ⊗ Ψ k and henceΓ p ∈ Q loc .If needed, we specify the dependence of Q x on the sets X , Y , A and B by using the notation Q x ( X, Y, A, B ), for x ∈ { loc , q , qa , qc , ns } . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 17 The operator system of a stochastic operator matrix
Recall [36, 74] that a ternary ring is a complex vector space V , equippedwith a ternary operation [ · , · , · ] : V × V × V → V , linear on the outer variablesand conjugate linear in the middle variable, such that[ s, t, [ u, v, w ]] = [ s, [ v, u, t ] , w ] = [[ s, t, u ] , v, w ] , s, t, u, v, w ∈ V . A ternary representation of V is a linear map θ : V → B ( H, K ), for someHilbert spaces H and K , such that θ ([ u, v, w ]) = θ ( u ) θ ( v ) ∗ θ ( w ) , u, v, w ∈ V . We call θ non-degenerate if span { θ ( u ) ∗ η : u ∈ V , η ∈ K } is dense in H . A concrete ternary ring of operators (TRO) [74] is a subspace U ⊆ B ( H, K )for some Hilbert spaces H and K such that S, T, R ∈ U implies ST ∗ R ∈ U .Let X and A be finite sets, and V X,A be the ternary ring, generated byelements v a,x , x ∈ X , a ∈ A , satisfying the relations(5.1) X a ∈ A [ v a ′′ ,x ′′ , v a,x , v a,x ′ ] = δ x,x ′ v a ′′ ,x ′′ , x, x ′ , x ′′ ∈ X, a ′′ ∈ A. Note that (5.1) implies(5.2) X a ∈ A [ u, v a,x , v a,x ′ ] = δ x,x ′ u, x, x ′ ∈ X, u ∈ V X,A . Indeed, suppose that (5.2) holds for u = u i , i = 1 , ,
3. Then X a ∈ A [[ u , u , u ] , v a,x , v a,x ′ ] = X a ∈ A [ u , u , [ u , v a,x , v a,x ′ ]] = δ x,x ′ [ u , u , u ];(5.2) now follows by induction.Let θ : V X,A → B ( H, K ) be a non-degenerate ternary representation.Setting V a,x = θ ( v a,x ), x ∈ X , a ∈ A , (5.2) implies(5.3) X a ∈ A V ∗ a,x V a,x ′ = δ x,x ′ I H , x, x ′ ∈ X ;conversely, a family { V a,x : x ∈ X, a ∈ A } ⊆ B ( H, K ) satisfying (5.3) clearlygives rise to a non-degenerate ternary representation θ : V X,A → B ( H, K ).We therefore call such a family a representation of the relations (5.1). Wenote that the set of representations of (5.1) is non-empty. Indeed, considerisometries V x , x ∈ X , with orthogonal ranges on some Hilbert space H ,i.e. V ∗ x V x ′ = δ x,x ′ I H , x, x ′ ∈ X . Fix a ∈ A and let V a,x = δ a,a V x . Then P a ∈ A V ∗ a,x V a,x ′ = V ∗ x V x ′ = δ x,x ′ I H .We note that (5.3) implies k V a,x k ≤ x ∈ X , a ∈ A . We identifythe family { V a,x : a ∈ A, x ∈ X } with the isometry V = ( V a,x ) a,x andwrite H V = H , K V = K and θ V = θ . Two representations V = ( V a,x ) a,x and W = ( W a,x ) a,x are called equivalent if there exist unitary operators U H : H V → H W and U K : K V → K W such that W a,x U H = U K V a,x , x ∈ X , a ∈ A . Write ˆ θ = ⊕ V θ V , where the direct sum is taken over all equivalence classesof representations of the relations (5.1). For u ∈ V X,A , let k u k := k ˆ θ ( u ) k .As k v a,x k ≤ V X,A is generated by v a,x , a ∈ A , x ∈ X , we have that k u k < ∞ for every u ∈ V X,A . It is also clear that k · k is a semi-norm on V X,A . Set N = n u ∈ V X,A : k u k = 0 o . We have that N is a ternary ideal of V X,A , that is, [ u , u , u ] ∈ N if u i ∈ N for some i ∈ { , , } . The ternaryproduct of V X,A thus induces a ternary product on V X,A /N , and ˆ θ inducesa ternary representation of V X,A /N that will be denoted in the same way.Letting k u k := k ˆ θ ( u ) k , u ∈ V X,A /N , we have that k · k is a norm on V X,A /N ,and hence V X,A /N is a ternary pre-C*-ring (see [74]). We let V X,A be thecompletion of V X,A /N ; thus, V X,A is a ternary C*-ring [74]. Note that ˆ θ extends to a ternary representation of V X,A (denoted in the same way) ontoa concrete TRO, and the equality k u k = k ˆ θ ( u ) k continues to hold for every u ∈ V X,A . We thus have that V X,A is a TRO in a canonical fashion. It isclear that each θ V induces a ternary representation of V X,A onto a TRO,which will be denoted in the same way.Let C X,A be the right C*-algebra of V X,A ; if V X,A is represented faithfullyas a concrete ternary ring of operators in B ( H, K ) for some Hilbert spaces H and K (that is, V X,A V ∗ X,A V X,A ⊆ V
X,A ), the C*-algebra C X,A may bedefined by letting C X,A = span { S ∗ T : S, T ∈ V
X,A } . Each representation V = ( V a,x ) a,x of the relations (5.1) gives rise [34] to aunital *-representation π V of C X,A on H V by letting π V ( S ∗ T ) = θ V ( S ) ∗ θ V ( T ) , S, T ∈ V X,A . Lemma 5.1.
The following hold true: (i)
Every non-degenerate ternary representation of V X,A has the form θ V , for some representation V of the relations (5.1); (ii) ˆ θ is a faithful ternary representation of V X,A ; (iii) Every unital *-representation π of C X,A has the form π V , for somerepresentation V of the relations (5.1).Proof. (i) Suppose that θ is a non-degenerate ternary representation of V X,A .Letting q : V X,A → V
X,A be the quotient map, write θ = θ ◦ q ; thus, θ is a non-degenerate ternary representation of V X,A . Letting V be therepresentation of the relations (5.1) such that θ = θ V , we have that θ = θ V .(ii) follows from the fact that k ˆ θ ( u ) k = k u k , u ∈ V X,A .(iii) Let π : C X,A → B ( H ) be a unital *-representation. Then there exists aternary representation θ : V X,A → B ( H, K ) such that π ( S ∗ T ) = θ ( S ) ∗ θ ( T ), S, T ∈ V
X,A (see e.g. [5, Theorem 3.4] and [24, p. 1636]). Since π is unital, θ is non-degenerate. By (i), there exists a representation V of the relations(5.1) such that θ = θ V , and hence π = π V . (cid:3) UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 19
Set e x,x ′ ,a,a ′ = v ∗ a,x v a ′ ,x ′ ∈ C X,A , x, x ′ ∈ X , a, a ′ ∈ A . We call the operatorsubsystem T X,A := span { e x,x ′ ,a,a ′ : x, x ′ ∈ X, a, a ′ ∈ A } of C X,A the
Brown-Cuntz operator system . Note that relations (5.1) imply(5.4) X a ∈ A e x,x ′ ,a,a = δ x,x ′ , x, x ′ ∈ X. Theorem 5.2.
Let H be a Hilbert space and φ : T X,A → B ( H ) be a linearmap. The following are equivalent: (i) φ is a unital completely positive map; (ii) (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ is a stochastic operator matrix; (iii) there exists a *-representation π : C X,A → B ( H ) such that φ = π | T X,A .Moreover, if (cid:0) E x,x ′ ,a,a ′ (cid:1) x,x ′ ,a,a ′ is a stochastic operator matrix acting on aHilbert space H then there exists a (unique) unital completely positive map φ : T X,A → B ( H ) such that φ ( e x,x ′ ,a,a ′ ) = E x,x ′ ,a,a ′ for all x, x ′ , a, a ′ .Proof. (i) ⇒ (ii) By Arveson’s Extension Theorem and Stinespring’s Theo-rem, there exist a Hilbert space K , a *-representation π : C X,A → B ( K )and an isometry W ∈ B ( H, K ), such that φ ( u ) = W ∗ π ( u ) W , u ∈ T X,A . ByLemma 5.1, π = π V for some representation V = ( V a,x ) a,x of the relations(5.1). By the proof of Theorem 3.1, E := (cid:0) π ( e x,x ′ ,a,a ′ ) (cid:1) ∈ ( M X ⊗ M A ⊗B ( K )) + , and hence (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) = ( I X ⊗ I A ⊗ W ) ∗ E ( I X ⊗ I A ⊗ W ) ∈ ( M X ⊗ M A ⊗ B ( H )) + . By (5.4) and Theorem 3.1, (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ is a stochastic operator ma-trix.(ii) ⇒ (iii) By Theorem 3.1, there exist a Hilbert space K and an isometry V = ( V a,x ) a,x ∈ B ( H X , K A ) such that φ ( e x,x ′ ,a,a ′ ) = V ∗ a,x V a ′ ,x ′ , x, x ′ ∈ X, a, a ′ ∈ A. The *-representation π V of C X,A is an extension of φ .(iii) ⇒ (i) is trivial.Suppose that E = (cid:0) E x,x ′ ,a,a ′ (cid:1) x,x ′ ,a,a ′ is a stochastic operator matrix actingon H . Letting V be the isometry, associated with E via Theorem 3.1, wehave that φ := π V | T X,A satisfies the required conditions. (cid:3)
Let S be an operator system. Recall that the pair ( C ∗ u ( S ) , ι ) is calleda universal C*-cover of S , if C ∗ u ( S ) is a unital C*-algebra, ι : S → C ∗ u ( S )is a unital complete order embedding, and whenever H is a Hilbert spaceand φ : S → B ( H ) is a unital completely positive map, there exists a *-representation π φ : C ∗ u ( S ) → B ( H ) such that π φ ◦ ι = φ . It is clear that theuniversal C*-cover is unique up to a *-isomorphism. The following corollaryis immediate from Theorem 5.2. Corollary 5.3.
The pair ( C X,A , ι ) , where ι is the inclusion map of T X,A into C X,A , is the universal C*-cover of T X,A . We will need the following slight extension of the equivalence (i) ⇔ (ii) ofTheorem 5.2. Proposition 5.4.
Let H be a Hilbert space and φ : T X,A → B ( H ) be a linearmap. The following are equivalent: (i) φ is a completely positive map; (ii) (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ ∈ ( M X ⊗ M A ⊗ B ( H )) + .Proof. (i) ⇒ (ii) It follows from Theorem 3.1 and Lemma 5.1 (iii), with π V a faithful *-representation of C X,A , that ( e x,x ′ ,a,a ′ ) ∈ ( M X ⊗ M A ⊗ C X,A ) + .Since T X,A ⊆ C
X,A as an operator subsystem, we have(5.5) ( e x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ ∈ ( M X ⊗ M A ⊗ T X,A ) + and (ii) follows.(ii) ⇒ (i) Write E = (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ and let T = φ (1). Since 1 = P a ∈ A e x,x,a,a (where x is any element of X ), we have that T ≥
0. Note alsothat if x, x ′ ∈ X and x = x ′ then(5.6) X a ∈ A E x,x ′ ,a,a = φ X a ∈ A e x,x ′ ,a,a ! = 0 . Assume first that T is invertible. Let ψ : T X,A → B ( H ) be the map givenby(5.7) ψ ( u ) = T − / φ ( u ) T − / , u ∈ T X,A . Setting F = (cid:0) ψ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ , we have that F = (cid:16) I XA ⊗ T − / (cid:17) E (cid:16) I XA ⊗ T − / (cid:17) ≥ . Let ω = ( ω x,x ′ ) ∈ M X and σ be a state in T ( H ). Using (5.6), we have h Tr A L ω ( F ) , σ i = h F, ω ⊗ I A ⊗ σ i = D(cid:16) I XA ⊗ T − / (cid:17) E (cid:16) I XA ⊗ T − / (cid:17) , ω ⊗ I A ⊗ σ E = D E, ω ⊗ I A ⊗ T − / σT − / E = X x,x ′ ∈ X X a ∈ A ω x,x ′ D E x,x ′ ,a,a , T − / σT − / E = X x ∈ X X a ∈ A ω x,x D E x,x,a,a , T − / σT − / E = X x ∈ X ω x,x D T, T − / σT − / E = X x ∈ X ω x,x = Tr( ω ) . By Theorem 3.1, F is a stochastic operator matrix; by Theorem 5.2, ψ iscompletely positive. Since φ ( · ) = T / ψ ( · ) T / , so is φ . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 21
Now relax the assumption that T be invertible. For every ǫ >
0, let φ ǫ : T X,A → B ( H ) be the map, given by φ ǫ ( u ) = φ ( u ) + ǫI . By the previousparagraph, φ ǫ is completely positive. Since φ = lim ǫ → φ ǫ in the point-normtopology, φ is completely positive. (cid:3) Let L X,A = ( ( λ x,x ′ ,a,a ′ ) ∈ M XA : ∃ c ∈ C s.t. X a ∈ A λ x,x ′ ,a,a = δ x,x ′ c, x, x ′ ∈ X ) ;we consider L X,A as an operator subsystem of M XA . For the next propo-sition, note that, by [16, Corollary 4.5], if T is a finite dimensional opera-tor system than its (matrix ordered) dual T d is an operator system, whenequipped with any faithful state of T as an Archimedean order unit. It isstraightforward to verify that, in this case, T dd ∼ = c . o . i . T . Proposition 5.5.
The linear map
Λ : T d X,A → L
X,A , given by (5.8) Λ( φ ) = (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ∈ X,a,a ′ ∈ A is a well-defined complete order isomorphism.Proof. By Proposition 5.4, if φ ∈ T X,A → C is a positive functional then (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ ∈ L + X,A . Thus, the map Λ + : (cid:16) T d X,A (cid:17) + → L + X,A , givenby Λ + ( φ ) = (cid:0) φ ( e x,x ′ ,a,a ′ ) (cid:1) x,x ′ ,a,a ′ , φ ∈ (cid:16) T d X,A (cid:17) + , is well-defined. It is clear that Λ + is additive andΛ + ( λφ ) = λ Λ + ( φ ) , λ ≥ , φ ∈ (cid:16) T d X,A (cid:17) + . Suppose that φ ∈ T d X,A is a hermitian functional. If φ = φ − φ , where φ and φ are positive functionals on T X,A , setΛ h ( φ ) = Λ + ( φ ) − Λ + ( φ ) . The map Λ h : ( T X,A ) d h → L X,A is well-defined: if φ − φ = ψ − ψ , where φ , φ , ψ and ψ are positive functionals then, by the additivity of Λ + , wehave that Λ + ( φ ) + Λ + ( ψ ) = Λ + ( ψ ) + Λ + ( φ ), that is, Λ + ( φ ) − Λ + ( φ ) =Λ + ( ψ ) − Λ + ( ψ ). It is straightforward that the map Λ h is R -linear, andthus it extends to a ( C -)linear map Λ : T d X,A → L
X,A .Suppose that ( φ i,j ) mi,j =1 ∈ M m (cid:16) T d X,A (cid:17) + ; thus, the map Φ : T X,A → M m ,given by Φ( u ) = ( φ i,j ( u )) mi,j =1 , is completely positive. By Proposition 5.4,( φ i,j ( e x,x ′ ,a,a ′ )) i,j ∈ ( M XA ⊗ M m ) + . This shows that Λ is completely posi-tive.If Λ( φ ) = 0 then φ ( e x,x ′ ,a,a ′ ) = 0 for all x, x ′ ∈ X and all a, a ′ ∈ A ,implying φ = 0; thus, Λ is injective. Since L X,A is an operator subsystem of M XA , it is spanned by the positive matrices it contains. Using Theorem 5.2, we see that every positive element of L X,A is in the range of Λ; it followsthat Λ is surjective.Finally, suppose that φ i,j ∈ T d X,A , i, j = 1 , . . . , m , are such that the matrix(Λ( φ i,j )) mi,j =1 is a positive element of M m ( L X,A ). Let Φ : T X,A → M m begiven by Φ( u ) = ( φ i,j ( u )) mi,j =1 . Then (cid:0) Φ( e x,x ′ ,a,a ′ ) (cid:1) ∈ M m ( L X,A ) + . ByProposition 5.4, Φ is completely positive, that is, ( φ i,j ) mi,j =1 ∈ M m (cid:16) T d X,A (cid:17) + .Thus, Λ − is completely positive, and the proof is complete. (cid:3) Let S be an operator system. A kernel in S [43] is a linear subspace J ⊆ S , for which there exists an operator system T and a unital completelypositive map ψ : S → T such that J = ker( ψ ). If J is a kernel in S , thequotient space S /J can be equipped with a unique operator system structurewith the property that, whenever T is an operator system and φ : S → T isa completely positive map annihilating J , the induced map ˜ φ : S /J → T iscompletely positive. If T is an operator system, a surjective map φ : S → T is called complete quotient , if the map ˜ φ is a complete order isomorphism.We refer the reader to [43] for further details.Let J X,A = { ( µ x,x ′ ,a,a ′ ) ∈ M XA : µ x,x ′ ,a,a ′ = 0 and µ x,x ′ ,a,a = µ x,x ′ ,a ′ ,a ′ , a = a ′ , and X x ∈ X µ x,x,a,a = 0 , a ∈ A } . Corollary 5.6.
The space J X,A is a kernel in M XA and the operator system T X,A is completely order isomorphic to the quotient M XA /J X,A .Proof.
By Proposition 5.5, the map Λ : T d X,A → M XA is a complete orderembedding. By [28, Proposition 1.8], the dual Λ ∗ : M d XA → T X,A of Λ isa complete quotient map. Identifying M d XA with M XA canonically, for anelement f ∈ M d XA , we haveΛ ∗ ( f ) = 0 ⇐⇒ h Λ ∗ ( f ) , φ i = 0 for all φ ∈ T X,A ⇐⇒ h f, Λ( φ ) i = 0 for all φ ∈ T X,A ⇐⇒ h f, T i = 0 for all T ∈ L X,A ⇐⇒ f ∈ J X,A . Thus, ker(Λ ∗ ) = J X,A . (cid:3) Descriptions via tensor products
In this section, we provide a description of the classes of QNS correlations,introduced in Section 4, analogous to the description of the classes of NScorrelations given in [49] (see also [26] and [62]). We will use the tensortheory of operator systems developed in [42]. If S and T are operatorsystems, S⊗ min T denotes the minimal tensor product of S and T : if A and B are unital C*-algebras, A⊗ min B is the spatial tensor product of A and B , and UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 23
S ⊆ c . o . i . A and T ⊆ c . o . i . B , then S ⊗ min
T ⊆ c . o . i . A ⊗ min B . The commutingtensor product S ⊗ c T sits completely order isomorphically in the maximaltensor product C ∗ u ( S ) ⊗ max C ∗ u ( T ) of the universal C*-covers of S and T ,while the maximal tensor product S ⊗ max T is characterised by the propertythat it linearises jointly completely positive maps θ : S × T → B ( H ). Werefer the reader to [42] for more details and further background.Let X , Y , A and B be finite sets. As in Section 5, we write e x,x ′ ,a,a ′ , x, x ′ ∈ X , a, a ′ ∈ A , for the canonical generators of T X,A . Similarly, wewrite f y,y ′ ,b,b ′ , y, y ′ ∈ Y , b, b ′ ∈ B , for the canonical generators of T Y,B .Given a linear functional s : T X,A ⊗ T
Y,B → C (or a linear functional s : C X,A ⊗ C
Y,B → C ), we let Γ s : M XY → M AB be the linear map given by(6.1) Γ s (cid:0) e x e ∗ x ′ ⊗ e y e ∗ y ′ (cid:1) = X a,a ′ ∈ A X b,b ′ ∈ B s (cid:0) e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ (cid:1) e a e ∗ a ′ ⊗ e b e ∗ b ′ . Remark 6.1.
The correspondence s → Γ s is a linear map from the dual( T X,A ⊗ T
Y,B ) d of T X,A ⊗ T
Y,B into the space L ( M XY , M AB ) of all lineartransformations from M XY into M AB . Theorem 6.2.
Let
X, Y, A, B be finite sets and
Γ : M XY → M AB be alinear map. The following are equivalent: (i) Γ is a QNS correlation; (ii) there exists a state s : T X,A ⊗ max T Y,B → C such that Γ = Γ s .Proof. (i) ⇒ (ii) Let Γ : M XY → M AB be a QNS correlation and writeΓ (cid:0) e x e ∗ x ′ ⊗ e y e ∗ y ′ (cid:1) = X a,a ′ ∈ A X b,b ′ ∈ B C x,x ′ ,y,y ′ a,a ′ ,b,b ′ e a e ∗ a ′ ⊗ e b e ∗ b ′ , for some C x ′ ,x,y ′ ,ya,a ′ ,b,b ′ ∈ C , x, x ′ ∈ X , y, y ′ ∈ Y , a, a ′ ∈ A , b, b ′ ∈ B . It followsfrom (2.2) and (2.3) that the Choi matrix C := (cid:16) C x,x ′ ,y,y ′ a,a ′ ,b,b ′ (cid:17) of Γ satisfies thefollowing conditions (see also [21]):(a) C ∈ M + XY AB ;(b) there exists c y,y ′ b,b ′ ∈ C such that P a ∈ A C x,x ′ ,y,y ′ a,a,b,b ′ = δ x,x ′ c y,y ′ b,b ′ , y, y ′ ∈ Y , b, b ′ ∈ B ;(c) there exists d x,x ′ a,a ′ ∈ C such that P b ∈ B C x,x ′ ,y,y ′ a,a ′ ,b,b = δ y,y ′ d x,x ′ a,a ′ , x, x ′ ∈ X , a, a ′ ∈ A .By condition (b), L ω Y B ( C ) ∈ L X,A for every ω Y B ∈ M Y B , while by condition(c), L ω XA ( C ) ∈ L Y,B for every ω XA ∈ M XA . Thus, C ∈ ( L X,A ⊗ L
Y,B ) ∩ M + XY AB ;by the injectivity of the minimal operator system tensor product, C ∈ ( L X,A ⊗ min L Y,B ) + .By [28, Proposition 1.9] and Proposition 5.5,(6.2) ( T X,A ⊗ max T Y,B ) d ∼ = c . o . i . L X,A ⊗ min L Y,B , via the identification Λ given by (5.8). The state s of T X,A ⊗ max T Y,B corre-sponding to C via (6.2) satisfies(6.3) C x,x ′ ,y,y ′ a,a ′ ,b,b ′ = s (cid:0) e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ (cid:1) , x, x ′ ∈ X, y, y ′ ∈ Y, a, a ′ ∈ A, b, b ′ ∈ B. Thus, Γ = Γ s .(ii) ⇒ (i) Let s be a state of T X,A ⊗ max T Y,B , and define C x,x ′ ,y,y ′ a,a ′ ,b,b ′ via (6.3);thus, C is the Choi matrix of Γ s . By (5.5) and and the definition of themaximal tensor procuct, (cid:0) e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ (cid:1) ∈ M XY AB ( T X,A ⊗ max T Y,B ) + , and hence the matrix C := (cid:16) C x,x ′ ,y,y ′ a,a ′ ,b,b ′ (cid:17) is positive; by Choi’s Theorem, Γ s is completely positive. Relations (5.4) imply that Γ s is trace preserving andthat conditions (b) and (c) hold. Suppose that ρ X = ( ρ x,x ′ ) x,x ′ ∈ M X haszero trace and ρ Y = ( ρ y,y ′ ) y,y ′ ∈ M Y . We have X x,x ′ ∈ X X y,y ′ ∈ Y X b,b ′ ∈ B X a ∈ A C x,x ′ ,y,y ′ a,a,b,b ′ ρ x,x ′ ρ y,y ′ e b e ∗ b ′ = X x ∈ X ρ x,x ! X y,y ′ ∈ Y X b,b ′ ∈ B ρ y,y ′ c y,y ′ b,b ′ e b e ∗ b ′ = 0 , that is, (2.2) holds; similarly, (c) implies (2.3). (cid:3) Theorem 6.3.
Let
X, Y, A, B be finite sets and
Γ : M XY → M AB be alinear map. The following are equivalent: (i) Γ is a quantum commuting QNS correlation; (ii) there exists a state s : T X,A ⊗ c T Y,B → C such that Γ = Γ s ; (iii) there exists a state s : C X,A ⊗ max C Y,B → C such that Γ = Γ s .Proof. (i) ⇒ (iii) Let H be a Hilbert space, E ∈ M X ⊗ M A ⊗B ( H ), F ∈ M Y ⊗ M B ⊗ B ( H ) form a commuting pair of stochastic operator matrices, and τ ∈T ( H ) + be such that Γ = Γ E · F,τ . By Theorem 5.2, there exist representations π X and π Y of C X,A and C Y,B , respectively, such that E x,x ′ ,a,a ′ = π X ( e x,x ′ ,a,a ′ )and F y,y ′ ,b,b ′ = π Y ( e y,y ′ ,b,b ′ ) for all x, x ′ ∈ X , y, y ′ ∈ Y , a, a ′ ∈ A , b, b ′ ∈ B .Since C X,A (resp. C Y,B ) is generated by the elements e x,x ′ ,a,a ′ , x, x ′ ∈ X , a, a ′ ∈ A (resp. f y,y ′ ,b,b ′ , y, y ′ ∈ Y , b, b ′ ∈ B ), π X and π Y have commutingranges. Let π X × π Y be the (unique) *-representation C X,A ⊗ max C Y,B → B ( H )such that ( π X × π Y )( u ⊗ v ) = π X ( u ) π Y ( v ), u ∈ C X,A , v ∈ C Y,B . By (4.2), (cid:10) Γ E · F,τ ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) , e a e ∗ a ′ ⊗ e b e ∗ b ′ (cid:11) = (cid:10) E x,x ′ ,a,a ′ F y,y ′ ,b,b ′ , τ (cid:11) = (cid:10) ( π X × π Y )( e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ ) , τ (cid:11) . Letting s ( w ) = h ( π X × π Y )( w ) , τ i , w ∈ C X,A ⊗ max C Y,B , we have Γ = Γ s .(iii) ⇒ (i) Let s be a state on C X,A ⊗ max C Y,B and write π s and ξ s forthe corresponding GNS representation of C X,A ⊗ max C Y,B and for its cyclicvector, respectively. Then E := ( π s ( e x,x ′ ,a,a ′ ⊗ x,x ′ ,a,a ′ and F := ( π s (1 ⊗ UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 25 f y,y ′ ,b,b ′ ) y,y ′ ,b,b ′ form a commuting pair of stochastic operator matrices; more-over, for x, x ′ ∈ X , y, y ′ ∈ Y , a, a ′ ∈ A and b, b ′ ∈ B , we have (cid:10) Γ s ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) , e a e ∗ a ′ ⊗ e b e ∗ b ′ (cid:11) = s ( e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ )= h π s ( e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ ) ξ s , ξ s i = h E x,x ′ ,a,a ′ F y,y ′ ,b,b ′ ξ s , ξ s i = (cid:10) Γ E,F,ξ s ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) , e a e ∗ a ′ ⊗ e b e ∗ b ′ (cid:11) . (ii) ⇔ (iii) By Corollary 5.3 and [42, Theorem 6.4], T X,A ⊗ c T Y,B sits com-pletely order isomorphically in C X,A ⊗ max C Y,B ; thus the states of T X,A ⊗ c T Y,B are precisely the restrictions of the states of C X,A ⊗ max C Y,B . (cid:3) Corollary 6.4.
The set Q qc is closed and convex.Proof. By Theorem 6.3 and Remark 6.1, the map s → Γ s is an affine bijec-tion from the state space of T X,A ⊗ c T Y,B onto Q qc . It is straightforward thatit is also a homeomorphism, when its domain is equipped with the weak*topology. Since the state space of T X,A ⊗ c T Y,B is weak* compact, its rangeis (convex and) closed. (cid:3)
Theorem 6.5.
Let
X, Y, A, B be finite sets and
Γ : M XY → M AB be alinear map. The following are equivalent: (i) Γ is an approximately quantum QNS correlation; (ii) there exists a state s : T X,A ⊗ min T Y,B → C such that Γ = Γ s ; (iii) there exists a state s : C X,A ⊗ min C Y,B → C such that Γ = Γ s .Proof. The proof is along the lines of the proof of [62, Theorem 2.8]; weinclude the details for the convenience of the reader.(iii) ⇒ (i) Let π X : C X,A → B ( H X ) and π Y : C Y,B → B ( H Y ) be faithful*-representations. Then π X ⊗ π Y : C X,A ⊗ min C Y,B → B ( H X ⊗ H Y ) is afaithful *-representation of C X,A ⊗ min C Y,B . Let s be a state satisfying (iii).By [39, Corollary 4.3.10], s can be approximated in the weak* topology byelements of the convex hull of vector states on ( π X ⊗ π Y )( C X,A ⊗ min C Y,B );thus, given ε >
0, there exist unit vectors ξ , . . . , ξ n ∈ H X ⊗ H Y and positivescalars λ , . . . , λ n with P ni =1 λ i = 1 such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s ( e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ ) − n X i =1 λ i (cid:10)(cid:0) π X ( e x,x ′ ,a,a ′ ) ⊗ π Y ( f y,y ′ ,b,b ′ ) (cid:1) ξ i , ξ i (cid:11)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε, for all x, x ′ ∈ X , y, y ′ ∈ Y , a, a ′ ∈ A and b, b ′ ∈ B . Let ξ = ⊕ ni =1 √ λ i ξ i ∈ C n ⊗ ( H X ⊗ H Y ); then k ξ k = 1. Set E x,x ′ ,a,a ′ = I n ⊗ π X ( e x,x ′ ,a,a ′ ) and F y,y ′ ,b,b ′ = π Y ( f y,y ′ ,b,b ′ ). Then ( E x,x ′ a,a ′ ) x,x ′ ,a,a ′ (resp. ( F y,y ′ ,b,b ′ ) y,y ′ ,b,b ′ ) is astochastic operator matrix on C n ⊗ H X (resp. H Y ), and (cid:12)(cid:12) s (cid:0) e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ (cid:1) − (cid:10) E x,x ′ a,a ′ ⊗ F y,y ′ ,b,b ′ ξ, ξ (cid:11)(cid:12)(cid:12) < ε. It follows that Γ s is in the closure of the set of correlations of the formΓ E ⊙ F,ξ , where E and F act on, possibly infinite dimensional, Hilbert spaces H and K . Given such a correlation Γ E ⊙ F,ξ , let ( P α ) α (resp. ( Q β ) β ) be a net of finite rank projections on H (resp. K ) such that P α → α I H (resp. Q β → β I K ) in the strong operator topology. Set H α = P α H (resp. K β = Q β K ), E α = ( I X ⊗ I A ⊗ P α ) E ( I X ⊗ I A ⊗ P α ) (resp. F β = ( I Y ⊗ I B ⊗ Q β ) F ( I Y ⊗ I B ⊗ Q β )), and ξ α,β = k ( P α ⊗ Q β ) ξ k ( P α ⊗ Q β ) ξ (note that ξ α,β is eventuallywell-defined). Then E α (resp. F β ) is a stochastic operator matrix acting on H α (resp. K β ), and Γ E α ⊙ F β ,ξ α,β → ( α,β ) Γ E ⊙ F,ξ along the product net. Itfollows that Γ s ∈ Q qa .(i) ⇒ (iii) Given ε >
0, let E and F be stochastic operator matrices actingon finite dimensional Hilbert spaces H X and H Y , respectively, and ξ ∈ H X ⊗ H Y be a unit vector, such that (cid:12)(cid:12)(cid:10) Γ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) , e a e a ′ ⊗ e b e ∗ b ′ (cid:11) − (cid:10)(cid:0) E x,x ′ ,a,a ′ ⊗ F y,y ′ ,b,b ′ (cid:1) ξ, ξ (cid:11)(cid:12)(cid:12) < ε, for all x, x ′ ∈ X , y, y ′ ∈ Y , a, a ′ ∈ A and b, b ′ ∈ B . By Lemma 5.1, thereexists a *-representation π X (resp. π Y ) of C X,A (resp. C Y,B ) on H X (resp. H Y ) such that E x,x ′ ,a,a ′ = π X ( e x,x ′ ,a,a ′ ) (resp. F y,y ′ ,b,b ′ = π Y ( f y,y ′ ,b,b ′ )), x, x ′ ∈ X , a, a ′ ∈ A (resp. y, y ′ ∈ Y , b, b ′ ∈ B ). Let s ε be the state on C X,A ⊗ min C Y,B given by s ε ( u ⊗ v ) = h ( π X ( u ) ⊗ π Y ( v )) ξ, ξ i , and s be a cluster point of the sequence { s /n } n in the weak* topology. Then s (cid:0) e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ (cid:1) = lim n →∞ s /n (cid:0) e x,x ′ ,a,a ′ ⊗ f y,y ′ ,b,b ′ (cid:1) = (cid:10) Γ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) , e a e a ′ ⊗ e b e ∗ b ′ (cid:11) , giving Γ = Γ s .(ii) ⇔ (iii) follows from the fact that T X,A ⊗ min T Y,B ⊆ c . o . i C X,A ⊗ min C Y,B . (cid:3) Recall [63] that, given any Archimedean ordered unit (AOU) space V ,there exists a (unique) operator system OMIN( V ) (resp. OMAX( V )) withunderlying space V , called the minimal operator system (resp. the maximaloperator system ) of V that has the property that every positive map froman operator system T into V (resp. from V into an operator system T ) isautomatically completely positive as a map from T into OMIN( V ) (resp.from OMAX( V ) into T ). If V is in addition an operator system, we denoteby OMIN( V ) (resp. OMAX( V )) the minimal (resp. maximal) operatorsystem of the AOU space, underlying V . Lemma 6.6.
Let V and W be finite dimensional AOU spaces with units e and f , respectively. An element u ∈ OMAX( V ) ⊗ max OMAX( W ) is positiveif and only if u = P ki =1 v i ⊗ w i , for some v i ∈ V + , w i ∈ W + , i = 1 , . . . , k .Proof. Let D be the set of all sums of elementary tensors v ⊗ w with v ∈ V + and w ∈ W + . We claim that if, for every ǫ >
0, there exists u ǫ ∈ D such that k u ǫ k → ǫ → u + u ǫ ∈ D for every ǫ >
0, then u ∈ D . Assume, without UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 27 loss of generality, that k u ǫ k ≤ ǫ >
0. Set L = 2 dim( V ) dim( W ) + 1and, using Carath´eodory’s Theorem, write u + u ǫ = L X j =1 v ( ǫ ) j ⊗ w ( ǫ ) j , where v ( ǫ ) j ∈ V + , w ( ǫ ) j ∈ W + and k v ( ǫ ) j k = k w ( ǫ ) j k for all j = 1 , . . . , L and all ǫ >
0. Since v ( ǫ ) j ⊗ w ( ǫ ) j ≤ u + u ǫ and k u + u ǫ k ≤ k u k + 1 for all ǫ >
0, we have k v ( ǫ ) j k ≤ p k u k + 1 and k w ( ǫ ) j k ≤ p k u k + 1, j = 1 , . . . , L . By compactness,we may assume that v ( ǫ ) j → ǫ → v j and w ( ǫ ) j → ǫ → w j for all j = 1 , . . . , L . Itfollows that u = P Lj =1 v j ⊗ w j ∈ D .Let(6.4) S = l X p =1 a p ⊗ v p and T = r X q =1 b q ⊗ w q , for some a p ∈ M n , v p ∈ V + , p = 1 , . . . , l , and b q ∈ M + m , w q ∈ W + , q =1 , . . . , r . If α ∈ M ,nm then α ( S ⊗ T ) α ∗ = l X p =1 r X q =1 ( α ( a p ⊗ b q ) α ∗ ) v p ⊗ w q ∈ D. Suppose that S ∈ M n (OMAX( V )) + and α ∈ M ,nm . By the definition ofthe maximal tensor product [42], if ǫ > S + ǫ n has the form of S in (6.4). Hence α ( S ⊗ T ) α ∗ + ǫα (1 n ⊗ T ) α ∗ = α (( S + ǫ n ) ⊗ T ) α ∗ ∈ D. Since α (1 n ⊗ T ) α ∗ ∈ D , the previous paragraph shows that α ( S ⊗ T ) α ∗ ∈ D. Now let T ∈ M m (OMAX( W )) + , and write T + ǫ m in the form of T in(6.4). Then α ( S ⊗ T ) α ∗ + ǫα ( S ⊗ m ) α ∗ = α ( S ⊗ ( T + ǫ m )) α ∗ ∈ D. By the previous paragraph, α ( S ⊗ m ) α ∗ ∈ D ; by the first paragraph, α ( S ⊗ T ) α ∗ ∈ D .Let u ∈ (OMAX( V ) ⊗ max OMAX( W )) + . By the definition of the max-imal tensor product [42], for every ǫ >
0, there exist n, m ∈ N , S ∈ M n (OMAX( V )) + , T ∈ M m (OMAX( W )) + and α ∈ M ,nm , such that u + ǫ α ( S ⊗ T ) α ∗ . By the previous and the first paragraph, u ∈ D . (cid:3) Theorem 6.7.
Let
X, Y, A, B be finite sets and
Γ : M XY → M AB be alinear map. The following are equivalent: (i) Γ is a local QNS correlation; (ii) there exists a state s : OMIN( T X,A ) ⊗ min OMIN( T Y,B ) → C such that Γ = Γ s . Proof. (ii) ⇒ (i) Let s : OMIN( T X,A ) ⊗ min OMIN( T Y,B ) → C be a state.Using [40, Theorem 9.9] and [28, Proposition 1.9], we can identify s withan element of (cid:16) OMAX( T d X,A ) ⊗ max OMAX( T d Y,B ) (cid:17) + . By Lemma 6.6, thereexist states φ i ∈ (cid:16) T d X,A (cid:17) + and ψ i ∈ (cid:16) T d Y,B (cid:17) + , and non-negative scalars λ i , i = 1 , . . . , m , such that s ≡ P mi =1 λ i φ i ⊗ ψ i . Set E i = ( φ i ( e x,x ′ ,a,a ′ )) x,x ′ ,a,a ′ (resp. F i = ( ψ i ( f y,y ′ ,b,b ′ )) y,y ′ ,b,b ′ ), and let Φ i : M X → M A (resp. Ψ i : M Y → M B ) be the quantum channel with Choi matrix E i (resp. F i ), i = 1 , . . . , m .Then Γ s = P mi =1 λ i Φ i ⊗ Ψ i .(i) ⇒ (ii) Write Γ = P mi =1 λ i Φ i ⊗ Ψ i as a convex combination of quantumchannels Φ i : M X → M A and Ψ i : M Y → M B , i = 1 , . . . , m , and let s be a functional on T X,A ⊗ T
Y,B such that Γ = Γ s . Let E i ∈ ( M X ⊗ M A ) + (resp. F i ∈ ( M Y ⊗ M B ) + ) be the Choi matrix of Φ i (resp. Ψ i ); thus, E i (resp. F i ) is a stochastic operator matrix acting on C . By Theorem 5.2,there exist positive functionals φ i : T X,A → C and ψ i : T Y,B → C suchthat ( φ i ( e x,x ′ ,a ′ ,a )) x,x ′ ,a,a ′ = E i and ( ψ i ( f y,y ′ ,b ′ ,b )) y,y ′ ,b,b ′ = F i , i = 1 , . . . , m .It is now straightforward to see that s is the functional corresponding to P mi =1 λ i φ i ⊗ ψ i and is hence, by Lemma 6.6, a state on OMIN( T X,A ) ⊗ min OMIN( T Y,B ). (cid:3) Classical-to-quantum no-signalling correlations
In this section, we consider the set of classical-to-quantum no-signallingcorrelations, and provide descriptions of its various subclasses in terms ofcanonical operator systems.7.1.
Definition and subclasses.
Let X , Y , A and B be finite sets and H be a Hilbert space. Definition 7.1.
A family
Θ = ( σ x,y ) x ∈ X,y ∈ Y of states in M AB is called a classical-to-quantum no-signalling (CQNS) correlation if (7.1) Tr A σ x ′ ,y = Tr A σ x ′′ ,y and Tr B σ x,y ′ = Tr B σ x,y ′′ , for all x, x ′ , x ′′ ∈ X and all y, y ′ , y ′′ ∈ Y . A stochastic operator matrix E ∈ M X ⊗ M A ⊗ B ( H ) will be called semi-classical if L e x e ∗ x ′ ( E ) = 0 whenever x = x ′ , that is, if E = X x ∈ X e x e ∗ x ⊗ E x , for some E x ∈ ( M A ⊗ B ( H )) + with Tr A E x = I H , x ∈ X . We write E =( E x ) x ∈ X ; note that, in its own right, E x is a stochastic operator matrix in L ( C ) ⊗ M A ⊗ B ( H ), x ∈ X .Suppose that E = ( E x ) x ∈ X and F = ( F y ) y ∈ Y form a commuting pair ofsemi-classical stochastic operator matrices, acting on a Hilbert space H and σ is a vector state on B ( H ). The family formed by the states(7.2) σ x,y = L σ ( E x · F y ) , x ∈ X, y ∈ Y, UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 29 is a CQNS correlation; indeed, by Proposition 4.1, Tr A σ x,y = L σ ( F y ) andTr B σ x,y = L σ ( E x ) for all x, y . We call the CQNS correlations of thisform quantum commuting . Similarly, if ( E x ) x ∈ X (resp. ( F y ) y ∈ Y ) is a semi-classical stochastic operator matrix on H A (resp. H B ) and σ is a vectorstate on L ( H A ⊗ H B ), the family formed by σ x,y = L σ ( E x ⊙ F y ) , x ∈ X, y ∈ Y, will be called a quantum CQNS correlation. A CQNS correlation Θ =( σ x,y ) x ∈ X,y ∈ Y will be called approximately quantum if there exist quantumCQNS correlations Θ n = ( σ ( n ) x,y ) x ∈ X,y ∈ Y , n ∈ N , such that σ ( n ) x,y → n →∞ σ x,y , x ∈ X, y ∈ Y. Finally, Θ will be called local if there exist states σ Ai,x (resp. σ Bi,y ) in M A (resp. M B ) and scalars λ i > i = 1 , . . . , m , such that σ x,y = m X i =1 λ i σ Ai,x ⊗ σ Bi,y x ∈ X, y ∈ Y. If E : D XY → M AB is a (classical-to-quantum) channel, we set Γ E = E ◦ ∆ XY ; thus, Γ E is a (quantum) channel from M XY to M AB . Given aCQNS correlation Θ = ( σ x,y ) x ∈ X,y ∈ Y , we let E Θ : D XY → M AB be thechannel given by E Θ (cid:0) e x e ∗ x ⊗ e y e ∗ y (cid:1) = σ x,y , x ∈ X, y ∈ Y, and Γ Θ = Γ E Θ . In the sequel, we will often identify Θ with the channel E Θ .For x ∈ { loc , q , qa , qc , ns } , we write CQ x for the set of all CQNS correlationsof class x; thus, the elements of CQ x will often be considered as channelsfrom D XY to M AB . Similarly to the proof of Proposition 4.3, it can beshown that quantum and quantum commuting CQNS correlations can bedefined using normal (not necessarily vector) states.In the next lemma, for (finite) sets X and A and a Hilbert space H , welet for brevity˜∆ X := ∆ X ⊗ id A ⊗ id B ( H ) : M X ⊗ M A ⊗ B ( H ) → D X ⊗ M A ⊗ B ( H )and˜∆ X,A := ∆ X ⊗ ∆ A ⊗ id B ( H ) : M X ⊗ M A ⊗ B ( H ) → D X ⊗ D A ⊗ B ( H ) . Lemma 7.2.
Let H be a Hilbert space, E ∈ M X ⊗ M A ⊗B ( H ) be a stochasticoperator matrix and σ ∈ T ( H ) be a state. Set E ′ = ˜∆ X ( E ) and E ′′ =˜∆ X,A ( E ) . Then E ′ (resp. E ′′ ) is a semi-classical (resp. classical) stochasticoperator matrix, (7.3) Γ E,σ ◦ ∆ X = Γ E ′ ,σ and ∆ A ◦ Γ E,σ ◦ ∆ X = Γ E ′′ ,σ . Moreover, if F ∈ M Y ⊗ M B ⊗ B ( H ) is a stochastic operator matrix thatforms a commuting pair with E then (7.4) ˜∆ XY ( E · F ) = ˜∆ X ( E ) · ˜∆ Y ( F ) . Proof.
Note that, if E = P x,x ′ ∈ X P a,a ′ ∈ A e x e ∗ x ′ ⊗ e a e ∗ a ′ ⊗ E x,x ′ ,a,a ′ then˜∆ X ( E ) = X x ∈ X X a,a ′ ∈ A e x e ∗ x ⊗ e a e ∗ a ′ ⊗ E x,x,a,a ′ . We now have h Γ E,σ (∆ X ( e x e ∗ x ′ )) , e a e ∗ a ′ i = δ x,x ′ h E x,x ′ ,a,a ′ , σ i = (cid:10) Γ E ′ ,σ ( e x e ∗ x ′ ) , e a e ∗ a ′ (cid:11) for all x, x ′ ∈ X and all a, a ′ ∈ A . The second identity in (7.3) is equallystraightforward. Finally, for (7.4), notice that, if E = (cid:0) E x,x ′ ,a,a ′ (cid:1) and F = (cid:0) F y,y ′ ,b,b ′ (cid:1) then both sides of the identity are equal to X x ∈ X X y ∈ Y X a,a ′ ∈ A X b,b ′ ∈ B e x e ∗ x ⊗ e y e ∗ y ⊗ e a e ∗ a ′ ⊗ e b e ∗ b ′ ⊗ E x,x,a,a ′ F y,y,b,b ′ . (cid:3) Theorem 7.3.
Fix x ∈ { loc , q , qa , qc , ns } . If Γ ∈ Q x then Γ | D XY ∈ CQ x ;conversely, if E ∈ CQ x then Γ E ∈ Q x . Moreover, for a channel E : D XY → M AB , we have that (i) E ∈ CQ qc if and only if Γ E = Γ E · F,σ , where ( E, F ) is a commut-ing pair of semi-classical stochastic operator matrices, acting on aHilbert spaces H , and σ is a normal state on B ( H ) ; (ii) E ∈ CQ q if and only if Γ E = Γ E ⊙ F,σ , where E and F are semi-classical stochastic operator matrices, acting on finite dimensionalHilbert spaces H A and H B , respectively, and σ is a state on L ( H A ⊗ H B ) .Proof. It is trivial that if Γ ∈ Q ns then Γ | D XY ∈ CQ ns . Conversely, supposethat E ∈ CQ ns , and let ρ X ∈ M X and ρ Y ∈ M Y be states, with Tr( ρ X ) = 0.By (7.1),Tr A Γ E ( ρ X ⊗ ρ Y ) = X x ∈ X X y ∈ Y h ρ X e x , e x ih ρ Y e y , e y i Tr A σ x,y = 0and, by symmetry, Γ E ∈ Q ns .Let E ∈ M X ⊗ M A ⊗ B ( H ) and F ∈ M Y ⊗ M B ⊗ B ( H ) form a commutingpair of stochastic operator matrices and σ ∈ T ( H ) be a state. It followsfrom Lemma 7.2 thatΓ E · F,σ | D XY = Γ ∆ XY ( E · F ) ,σ | D XY = Γ ∆ X ( E ) · ∆ Y ( F ) ,σ | D XY ∈ CQ qc . Conversely, suppose that E Θ ∈ CQ qc , where Θ = ( σ x,y ) x ∈ X,y ∈ Y is a CQNScorrelation. Let H , σ , E and F be such that (7.2) holds; then Γ Θ = Γ E · F,σ .A similar argument applies in the case x = q, and the case x = qa fol-lows from the fact that the map
E → E ◦ ∆ XY , from L ( D XY , M AB ) into L ( M XY , M AB ), is continuous. Finally, if σ x,y = σ x ⊗ σ y , where σ x ∈ M A (resp. σ y ∈ M B ) is a state, x ∈ X (resp. y ∈ Y ), and Φ : M X → M A (resp. Ψ : M Y → M B ) is the channel given by Φ( e x e ∗ x ′ ) = δ x,x ′ σ x (resp.Ψ( e y e ∗ y ′ ) = δ y,y ′ σ y ), then Γ E = Φ ⊗ Ψ, and the case x = loc follows. (cid:3)
UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 31
Description in terms of states.
We next introduce an operator sys-tem, universal for classical-to-quantum no-signalling correlations in a similarmanner that T X,A is universal for the (fully) quantum correlations, and de-scribe the subclasses of CQNS correlations via states on tensor products ofits copies.Let B X,A = M A ∗ · · · ∗ M A | {z } | X | times , a free product, amalgamated over the unit. For each x ∈ X , write { e x,a,a ′ : a, a ′ ∈ A } for the canonical matrix unit system of the x -th copy of M A , andlet R X,A = span { e x,a,a ′ : x ∈ X, a, a ′ ∈ A } , considered as an operator subsystem of B X,A .Given operator systems S , . . . , S n , their coproduct S = S ⊕ · · · ⊕ S n isan operator system, equipped with complete order embeddings ι i : S i → S ,characterised by the universal property that, whenever R is an operatorsystem and φ i : S i → R is a unital completely positive map, i = 1 , . . . , n ,there exists a unique unital completely positive map φ : S → R such that φ ◦ ι i = φ i , i = 1 , , . . . , n . We refer the reader to [41, Section 8] for a detailedaccount of the coproduct of operator systems. Remark 7.4.
Let A i , i = 1 , . . . , n , be unital C*-algebras and S = span { a i : a i ∈ A i , i = 1 , . . . , n } , considered as an operator subsystem of the freeproduct A ∗ · · · ∗ A n , amalgamated over the unit. It was shown in [27,Theorem 5.2] that S ∼ = c . o . i . A ⊕ · · · ⊕ A n . In particular, we have(7.5) R X,A ∼ = M A ⊕ · · · ⊕ M A | {z } | X | times . An application of [62, Lemma 2.8] now shows that(7.6) R X,A ⊗ c R Y,B ⊆ c . o . i . B X,A ⊗ max B Y,B . Theorem 7.5.
Let H be a Hilbert space and φ : R X,A → B ( H ) be a linearmap. The following are equivalent: (i) φ is a unital completely positive map; (ii) (cid:16)(cid:0) φ ( e x,a,a ′ ) (cid:1) a,a ′ ∈ A (cid:17) x ∈ X is a semi-classical stochastic operator ma-trix.Proof. (i) ⇒ (ii) The restriction φ x of φ to the x -th copy of M A is a unitalcompletely positive map. By Choi’s Theorem, (cid:0) φ x ( e x,a,a ′ ) (cid:1) a,a ′ is a stochasticoperator matrix in M A ⊗B ( H ) for every x ∈ X ; thus, (cid:16)(cid:0) φ ( e x,a,a ′ ) (cid:1) a,a ′ ∈ A (cid:17) x ∈ X is a semi-classical stochastic operator matrix.(ii) ⇒ (i) For each x ∈ X , let φ x : M A → B ( H ) be the linear map defined byletting φ x ( e a e ∗ a ′ ) = φ ( e x,a,a ′ ). By Choi’s Theorem, φ x is a (unital) completelypositive map. By the universal property of the coproduct, there exists a (unique) unital completely positive map ψ : R X,A → B ( H ) whose restrictionto the x -th copy of M A coincides with φ x . It follows that ψ = φ , and hence φ is completely positive. (cid:3) Remark 7.6.
By [27, Theorem 5.1], R X,A is an operator system quotientof M XA . Now [28, Proposition 1.8] shows that, if Q X,A = {⊕ x ∈ X T x ∈ ⊕ x ∈ X M A : ∃ c ∈ C s.t. Tr T x = c, x ∈ X } , then the linear map Λ cq : R d X,A → Q
X,A , given byΛ cq ( φ ) = ⊕ x ∈ X (cid:0) φ ( e x,a,a ′ ) (cid:1) a,a ′ , is a well-defined unital complete order isomorphism.We denote the canonical generators of R Y,B by f y,b,b ′ , y ∈ Y , b, b ′ ∈ B .Given a functional t : R X,A ⊗ R
Y,B → C , we let E t : D XY → M AB be thelinear map defined by E t (cid:0) e x e ∗ x ⊗ e y e ∗ y (cid:1) = X a,a ′ ∈ A X b,b ′ ∈ B t (cid:0) e x,a,a ′ ⊗ f y,b,b ′ (cid:1) e a e ∗ a ′ ⊗ e b e ∗ b ′ . We note that t → E t is a linear map from ( R X,A ⊗R Y,B ) ∗ into L ( D XY , M AB ).Theorem 7.5 and the universal property of the coproduct imply the exis-tence of a unital completely positive map β X,A : R X,A → T
X,A such that β X,A ( e x,a,a ′ ) = e x,x,a,a ′ , x ∈ X, a, a ′ ∈ A. Similarly, the matrix ( δ x,x ′ e x,a,a ′ ) x,x ′ ,a,a ′ is stochastic, and Theorem 5.2 im-plies the existence of a unital completely positive map β ′ X,A : T X,A → R
X,A such that β ′ X,A ( e x,x ′ ,a,a ′ ) = δ x,x ′ e x,a,a ′ , x, x ′ ∈ X, a, a ′ ∈ A. It is clear that β ′ X,A ◦ β X,A = id R X,A . Theorem 7.7.
The map t → E t is an affine isomorphism (i) from the state space of R X,A ⊗ max R Y,B onto CQ ns ; (ii) from the state space of R X,A ⊗ c R Y,B onto CQ qc ; (iii) from the state space of R X,A ⊗ min R Y,B onto CQ qa ; (iv) from the state space of OMIN ( R X,A ) ⊗ min OMIN ( R Y,B ) onto CQ loc .Proof. It is clear that the map t → E t is bijective. It is also straightforwardto see that, for a linear functional s : T X,A ⊗T Y,B → C , we have Γ s | D XY = E t ,where t = s ◦ ( β X,A ⊗ β Y,B ). The claims now follow from Theorems 6.2, 6.3,6.5, 6.7, 7.3 and the functoriality of the involved tensor products. (cid:3)
As a consequence of Theorem 7.7, we see that the sets CQ qc and CQ loc are closed (as are CQ ns and CQ qa ). Remark.
As in Theorems 6.3 and 6.5, the classes CQ qc and CQ qa canbe equivalently described via states on the C*-algebraic tensor products B X,A ⊗ max B Y,B and B X,A ⊗ min B Y,B , respectively. For the class CQ qa , this is UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 33 a direct consequence of the injectivity of the minimal tensor product in theoperator system category, while for the class CQ qc , this is a consequence ofRemark 7.4. 8. Classical reduction and separation
Let X and A be finite sets. We let A X,A = ℓ ∞ A ∗ · · · ∗ ℓ ∞ A | {z } | X | times , where the free product is amalgamated over the unit, and S X,A = ℓ ∞ A ⊕ · · · ⊕ ℓ ∞ A | {z } | X | times , the operator system coproduct of | X | copies of ℓ ∞ A . Note that, by [27,Theorem 5.2] (see Remark 7.4), S X,A is an operator subsystem of A X,A . Welet ( e x,a ) a ∈ A be the canonical basis of the x -th copy of ℓ ∞ A inside S X,A ; thus, S X,A is generated, as a vector space, by { e x,a : x ∈ X, a ∈ A } , and therelations X a ∈ A e x,a = 1 , x ∈ X, are satisfied. Note that, by the universal property of the operator systemcoproduct, S X,A is characterised by the following property: whenever H is aHilbert space and { E x,a : x ∈ X, a ∈ A } is a family of positive operators on H such that ( E x,a ) a ∈ A is a POVM for every x ∈ X , there exists a (unique)unital completely positive map φ : S X,A → B ( H ) such that φ ( e x,a ) = E x,a , x ∈ X , a ∈ A .We denote by E the map sending a quantum channel Γ : M XY → M AB to Γ | D XY (and recall that N stands for the map sending Γ to N Γ =∆ AB ◦ Γ | D XY ); Remark 8.1 below justifies calling E and N classical reduc-tion maps . The forward implications all follow similarly to the one in (ii),which was shown in Theorem 7.3, while the reverse ones can be seen afteran application of Lemma 7.2. We recall that we identify C ns with the set {N p : p an NS correlation } . Remark 8.1.
Let X , Y , A and B be finite sets, x ∈ { loc , q , qa , qc , ns } , p ∈ C x and E ∈ CQ x . The following hold: (i) p ∈ C x ⇔ E p ∈ CQ x ⇔ Γ p ∈ Q x ; (ii) E ∈ CQ x ⇔ Γ E ∈ Q x .Moreover, the maps E : Q x → CQ x and N : CQ x → C x are well-defined andsurjective. We identify an element N of C x with the corresponding classical-to-quantum channel from D XY into M AB , and an element E of CQ x with thecorresponding quantum channel from M XY into M AB . The subsequent tablesummarises the inclusions between the various classes of correlations: C loc ⊂ C q ⊂ C qa ⊂ C qc ⊂ C ns ∩ ∩ ∩ ∩ ∩CQ loc ⊂ CQ q ⊂ CQ qa ⊂ CQ qc ⊂ CQ ns ∩ ∩ ∩ ∩ ∩Q loc ⊂ Q q ⊂ Q qa ⊂ Q qc ⊂ Q ns . By Bell’s Theorem, C loc = C q for all subsets X, Y, A, B of cardinality atleast 2. By Remark 8.1, we have that CQ loc = CQ q and Q loc = Q q . By[70], C q = C qa for some finite sets X , Y , A and B (see also [22]) and hence CQ q = CQ qa and Q q = Q qa for a suitable choice of sets. The inequality C qc = C ns is well-known (it follows e.g. from [26, Theorem 7.11]), implyingthat CQ qc = CQ ns and Q qc = Q ns .It was recently shown [37] that the inequality C qa = C qc also holds truefor suitable sets X , Y , A and B , thus resolving the long-standing TsirelsonProblem and, by [38] and [56], the Connes Embedding Problem, in thenegative. It thus follows from Remark 8.1 that, for this choice of sets, CQ qa = CQ qc and Q qa = Q qc . We next strengthen these inequalities. Lemma 8.2.
Let X i and A i be finite sets, i = 1 , , with X ⊆ X and A ⊆ A . There exist unital completely positive maps ι : S X ,A → S X ,A and ι : S X ,A → S X ,A such that ι ◦ ι = id .Proof. Denote the canonical generators of S X ,A by e x,a , and of S X ,A –by f x,a . By induction, it suffices to prove the claim in two cases. Case 1. X = X and A = A ∪ { a } , where a A .Let a ∈ A . Define the maps ι and ι by setting ι ( e x,a ) = (cid:26) f x,a if a ∈ A \ { a } , f x,a + f x,a a = a ,and ι ( f x,a ) = (cid:26) e x,a if a ∈ A \ { a } , e x,a a ∈ { a , a } . Case 2. A = A and X = X ∪ { x } , where x X .Let x ∈ X . Define ι ( e x,a ) = f x,a , x ∈ X , a ∈ A , and ι ( f x,a ) = (cid:26) e x,a if x ∈ X , e x ,a x = x .By the universal property of the operator systems S X,A , ι and ι areunital completely positive maps, and the condition ι ◦ ι = id is readilyverified. (cid:3) Theorem 8.3.
For all finite sets X , Y , A and B of sufficiently large car-dinality, the following hold true: (i) Q qa ( X, Y, A, B ) = Q qc ( X, Y, A, B ) ; (ii) CQ qa ( X, Y, A, B ) = CQ qc ( X, Y, A, B ) ; (iii) T X,A ⊗ min T Y,B = T X,A ⊗ c T Y,B ; UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 35 (iv) R X,A ⊗ min R Y,B = R X,A ⊗ c R Y,B ; (v) B X,A ⊗ min B Y,B = B X,A ⊗ max B Y,B ; (vi) C X,A ⊗ min C Y,B = C X,A ⊗ max C Y,B .Proof.
By [37], there exist (finite) sets X , Y , A and B and an NS corre-lation p ∈ C qc ( X , Y , A , B ) \ C qa ( X , Y , A , B ) . Using [49, Corollary 3.2], let s be a state on S X ,A ⊗ c S Y ,B such that(8.1) p ( a, b | x, y ) = s ( e x,a ⊗ e y,b )for all x ∈ X , y ∈ Y , a ∈ A and b ∈ B . Assume that X ⊆ X , Y ⊆ Y , A ⊆ A and B ⊆ B . Write ι Ai (resp. ι Bi ), i = 1 ,
2, for themaps arising from Lemma 8.2 for the operator systems S X ,A and S X,A (resp. S Y ,B and S Y,B ). By the functoriality of the commuting tensorproduct, the map t := s ◦ ( ι A ⊗ ι B ) is a state on S X,A ⊗ c S Y,B . The NScorrelation q ∈ C qc ( X, Y, A, B ) arising from t as in (8.1) does not belong tothe class C qa . Indeed, if q ∈ C qa then, by [49, Corollary 3.3], t is a state on S X,A ⊗ min S Y,B , and hence s = t ◦ ( ι A ⊗ ι B ) (see Lemma 8.2) is a state on S X ,A ⊗ min S Y ,B which, in view of [49, Corollary 3.3], contradicts the factthat p is not approximately quantum.It follows that C qa ( X, Y, A, B ) = C qc ( X, Y, A, B ) for all sets X , Y , A and B of sufficiently large cardinality. Parts (i) and (ii) now follow from Remark8.1. Claim (iii) follows from Theorems 6.3 and 6.5, while (iv) – from (ii)and Theorem 7.7. Finally, (v) follows from (iv) and Remark 7.4, and (vi)follows from (iii), Corollary 5.3 and [42, Theorem 6.4]. (cid:3) Recall that an operator system S is said to possess the operator systemlocal lifting property (OSLLP) [43] if, whenever A is a unital C*-algebra, I ⊆ A is a two-sided ideal,
T ⊆ S is a finite dimensional operator subsystemand ϕ : T → A / I is a unital completely positive map, there exists a unitalcompletely positive map ψ : T → A such that ϕ = q ◦ ψ (here q : A → A / I denotes the quotient map). We conclude this section with showing that theoperator systems we introduced possess OSLPP. Proposition 8.4.
Let S be an operator system quotient of M k , for some k ∈ N , and H be a Hilbert space. Then S ⊗ min B ( H ) ∼ = c . o . i . S ⊗ max B ( H ) ,and hence S possesses OSLLP.Proof. Let J ⊆ M k be a kernel such that S = M k /J ; write q : M k → S for the quotient map. By [28, Proposition 1.8], the dual q ∗ : S d → M d k is acomplete order embedding.Fix u ∈ M n ( S ⊗ min B ( H )) + ; after a canonical identification, we consider u as an element of ( S ⊗ min M n ( B ( H ))) + . Let { S , . . . , S m } be a basis of S , and write u = P mi =1 S i ⊗ T i , for some T i ∈ M n ( B ( H )), i = 1 , . . . , m .By [41, Proposition 6.1], the map φ u : S d → M n ( B ( H )), given by φ u ( f ) = P mi =1 f ( S i ) T i , is completely positive. By Arveson’s Extension Theorem,there exists a completely positive map ψ : M d k → M n ( B ( H )) with ψ ◦ q ∗ = φ u . Let S ′ i ∈ M k be such that q ( S ′ i ) = S i , i = 1 , . . . , m , and let { S ′ i : i = m + 1 , . . . , k } be a basis of J . Then { S ′ , . . . , S ′ m , S ′ m +1 , . . . , S ′ k } is a basisof M k . Let v = k X i =1 S ′ i ⊗ T ′ i ∈ M k ⊗ M n ( B ( H ))be an element such that ψ ( g ) = k X i =1 g ( S ′ i ) T ′ i , g ∈ M d k ;by [41, Proposition 6.1], v ∈ ( M k ⊗ min M n ( B ( H ))) + . Since M k is nuclear, v belongs to ( M k ⊗ max M n ( B ( H ))) + . Let w = ( q ⊗ id)( v ); by the functorialityof the maximal tensor product, w ∈ ( S ⊗ max M n ( B ( H ))) + . We have w = k X i =1 q ( S ′ i ) ⊗ T ′ i = m X i =1 S i ⊗ T ′ i . For all f ∈ S d , we have m X i =1 f ( S i ) T ′ i = k X i =1 q ∗ ( f )( S ′ i ) T ′ i = ψ ( q ∗ ( f )) = φ u ( f ) = m X i =1 f ( S i ) T i . It follows that T i = T ′ i , i = 1 , . . . , m , and hence u = w . Thus, u ∈ M n ( S ⊗ max B ( H )) + , and it follows from [43, Theorem 8.6] that S possessesOSLLP. (cid:3) Proposition 8.4, combined with Corollary 5.6 and Remark 7.6, yield thefollowing corollary.
Corollary 8.5.
Let X and A be finite sets. Then T X,A and R X,A possessOSLPP.
Remark.
It is worth noting the different nature of the C*-algebras A X,A and B X,A on one hand, and C X,A on the other. This is best seen in the specialcase where | X | = 1, when A X,A ∼ = D A , B X,A ∼ = M A and C X,A ∼ = C ∗ u ( M A ).9. Quantum versions of synchronicity
Let X and A be finite sets, Y = X and B = A . We will often distinguishthe notation for X vs. Y (resp. A vs. B ) although they coincide, in orderto make clear with respect to which term in a tensor product a partial traceis taken. An NS correlation p = n ( p ( a, b | x, y )) a,b ∈ A : x, y ∈ X o is called synchronous [61] if p ( a, b | x, x ) = 0 x ∈ X, a, b ∈ A, a = b. In this section, we examine possible quantum versions of the notion of syn-chronicity. Our main motivation is the following result, which was provedin [61].
UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 37
Theorem 9.1.
Let p be an NS correlation. Then (i) p is synchronous and quantum commuting if and only if there existsa trace τ : A X,A → C such that (9.1) p ( a, b | x, y ) = τ ( e x,a e y,b ) , x, y ∈ X, a, b ∈ A ;(ii) p is synchronous and quantum if and only if there exist a finite di-mensional C*-algebra A , a trace τ A on A and a *-homomorphism π : A X,A → A such that (9.1) holds for the trace τ = τ A ◦ π ; (iii) p is synchronous and local if and only if there exist an abelian C*-algebra A , a trace τ A on A and a *-homomorphism π : A X,A → A such that (9.1) holds for the trace τ = τ A ◦ π . Fair correlations. If A is a unital C*-algebra, we write A op for theopposite C*-algebra of A ; recall that A op has the same underlying set (whoseelements will be denoted by u op , for u ∈ A ), the same involution, linearstructure and norm, and multiplication given by u op v op = ( vu ) op , u, v ∈ A .For a subset S ⊆ A , we let S op = { u op : u ∈ S} .For a Hilbert space H , we denote by H d its Banach space dual; if K is a(nother) Hilbert space and T ∈ B ( H, K ), we denote by T d its adjoint,acting from K d into H d . We note the relation(9.2) ( T ∗ ) d = ( T d ) ∗ , T ∈ B ( H, K ) . It is straightforward to see that if A is a C*-algebra and π : A → B ( H )is a (faithful) *-representation then the map π op : A op → B ( H d ), given by π op ( u op ) = π ( u ) d , is a (faithful) *-representation. Note that the transpo-sition map u → ( u t ) op is a *-isomorphism between M X and M op X . It wasshown in [44] that there exists a *-isomorphism ∂ A : A X,A → A op X,A suchthat ∂ A ( e x,a ) = e op x,a , x ∈ X , a ∈ A . The following analogous statements for C X,A and B X,A will be needed later.
Lemma 9.2.
Let X and A be finite sets. (i) There exists a *-isomorphism ∂ : C X,A → C op X,A such that ∂ ( e x,x ′ ,a,a ′ ) = e op x ′ ,x,a ′ ,a , x, x ′ ∈ X, a, a ′ ∈ A. (ii) There exists a *-isomorphism ∂ B : B X,A → B op X,A such that ∂ B ( e x,a,a ′ ) = e op x,a ′ ,a , x ∈ X, a, a ′ ∈ A. Proof. (i) Let π : C X,A → B ( H ) be a faithful *-representation. Write E x,x ′ ,a,a ′ = π ( e x,x ′ ,a,a ′ ), x, x ′ ∈ X , a, a ′ ∈ A . Using Theorem 3.1, let K be a Hilbert space and ( V a,x ) a,x : H X → K A be an isometry such that E x,x ′ ,a,a ′ = V ∗ a,x V a ′ ,x ′ , x, x ′ ∈ X , a, a ′ ∈ A . Let W a,x = (cid:0) V d a,x (cid:1) ∗ ; thus, W a,x ∈ B ( H d , K d ), x ∈ X , a ∈ A . Using (9.2), we have X a ∈ A W ∗ a,x ′ W a,x = X a ∈ A V d a,x ′ (cid:0) V ∗ a,x (cid:1) d = X a ∈ A (cid:0) V ∗ a,x V a,x ′ (cid:1) d = δ x,x ′ I d ; thus, ( W a,x ) a,x is an isometry. By Theorem 3.1, if F x,x ′ ,a,a ′ = W ∗ a,x W a ′ ,x ′ , x, x ′ ∈ X , a, a ′ ∈ A , then (cid:0) F x,x ′ ,a,a ′ (cid:1) x,x ′ ,a,a ′ is a stochastic operator matrix.Note that F x,x ′ ,a,a ′ = V d a,x (cid:16) V d a ′ ,x ′ (cid:17) ∗ = (cid:0) V ∗ a ′ ,x ′ V a,x (cid:1) d = E d x ′ ,x,a ′ ,a . By the universal property of C X,A , there exists a *-homomorphism π ′ : π ( C X,A ) → B (cid:0) H d (cid:1) such that π ′ (cid:0) E x,x ′ ,a,a ′ (cid:1) = E d x ′ ,x,a ′ ,a , x, x ′ ∈ X, a, a ′ ∈ A. By the paragraph before Lemma 9.2, π ′ ◦ π can be regarded as a *-homo-morphism from C X,A into C op X,A , which maps e x,x ′ ,a,a ′ to e op x ′ ,x,a ′ ,a . The claimfollows by symmetry.(ii) The words of the form e x ,a ,a ′ . . . e x k ,a k ,a ′ k span a dense ∗ -subalgebraof B X,A . As u ( u t ) op is a *-isomorphism from M A to M op A that mapsthe matrix unit e a e ∗ a ′ to ( e a ′ e ∗ a ) op , the universal property of the free productimplies that the map ∂ B given by ∂ B ( e x ,a ,a ′ . . . e x k ,a k ,a ′ k ) = e op x ,a ′ ,a . . . e op x k ,a ′ k ,a k extends to the desired *-isomorphism. (cid:3) If U is a subspace of a C ∗ -algebra A , we call a linear functional s : U ⊗ U op → C fair if(9.3) s ( u ⊗
1) = s (1 ⊗ u op ) for all u ∈ U . It will be convenient to write t Y for the transpose map on M Y . A state ρ ∈ M XY will be called fair if Tr X ((id ⊗ t Y )( ρ )) = Tr Y ((id ⊗ t Y )( ρ )). Wewrite Σ X = { ρ ∈ M + XY : ρ a fair state } , and observe that an element ρ =( ρ x,x ′ ,y,y ′ ) ∈ M + XY belongs to Σ X precisely when X x ∈ X X y,y ′ ∈ Y ρ x,x,y,y ′ e y ′ e ∗ y = X x,x ′ ∈ X X y ∈ Y ρ x,x ′ ,y,y e x e ∗ x ′ , that is, when(9.4) X x ∈ X ρ x,x,z,z ′ = X y ∈ X ρ z ′ ,z,y,y , z, z ′ ∈ X. We let Σ cl X = Σ X ∩D XY ; thus, a state ρ = ( ρ x,y ) x,y ∈ D + XY is in Σ cl X preciselywhen(9.5) X x ∈ X ρ x,z = X y ∈ X ρ z,y , z ∈ X. It follows from (9.4) and (9.5) that(9.6) ∆ XY (Σ X ) = Σ cl X . Definition 9.3.
A QNS correlation
Γ : M XY → M AB (resp. a CQNScorrelation E : D XY → M AB , an NS correlation N : D XY → D AB ) is called fair if Γ (Σ X ) ⊆ Σ A (resp. E (cid:0) Σ cl X (cid:1) ⊆ Σ A , N (cid:0) Σ cl X (cid:1) ⊆ Σ cl A ). UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 39
Theorem 9.4.
Let Γ be a QNS correlation. (i) Γ is fair if and only if there exists a state s : T X,A ⊗ max T X,A → C such that Γ = Γ s and the state s ◦ (id ⊗ ∂ ) − is fair; (ii) Γ is fair and belongs to Q qc if and only if there exists a state s : T X,A ⊗ c T X,A → C such that Γ = Γ s and the state s ◦ (id ⊗ ∂ ) − isfair; (iii) Γ is fair and belongs to Q qa if and only if there exists a state s : T X,A ⊗ min T X,A → C such that Γ = Γ s and the state s ◦ (id ⊗ ∂ ) − isfair; (iv) Γ is fair and belongs to Q loc if and only if there exists a state s :OMIN( T X,A ) ⊗ min OMIN( T Y,B ) → C such that Γ = Γ s and the state s ◦ (id ⊗ ∂ ) − is fair.Proof. We only show (i); the proofs of (ii)-(iv) are similar. Let Γ be a QNScorrelation. By Theorem 6.2, there exists a state s ∈ T X,A ⊗ max T X,A → C such that Γ = Γ s . The conditionTr A ((id ⊗ t B )(Γ( ρ )) = Tr B ((id ⊗ t B )(Γ( ρ ))is equivalent to(9.7) X a ∈ A h (id ⊗ t B (Γ( ρ )) e a ⊗ e b ′ , e a ⊗ e b i = X a ∈ A h (id ⊗ t B (Γ( ρ )) e b ′ ⊗ e a , e b ⊗ e a i ,b, b ′ ∈ B . Note thatΓ( ρ ) = X a,a ′ ∈ A X b,b ′ ∈ A X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ s ( e x,x ′ ,a,a ′ ⊗ e y,y ′ ,b,b ′ ) e a e ∗ a ′ ⊗ e b e ∗ b ′ and hence(id ⊗ t B )(Γ( ρ ))= X a,a ′ ∈ A X b,b ′ ∈ A X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ s ( e x,x ′ ,a,a ′ ⊗ e y,y ′ ,b,b ′ ) e a e ∗ a ′ ⊗ e b ′ e ∗ b . Thus, letting µ (1) y,y ′ = P x ∈ X ρ x,x,y,y ′ , we have that the left hand side of (9.7)coincides with X a ∈ A X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ s (cid:0) e x,x ′ ,a,a ⊗ e y,y ′ ,b ′ ,b (cid:1) = X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ s X a ∈ A e x,x ′ ,a,a ! ⊗ e y,y ′ ,b ′ ,b ! = X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ δ x,x ′ s (cid:0) ⊗ e y,y ′ ,b ′ ,b (cid:1) = s ⊗ X y,y ′ ∈ X µ (1) y,y ′ e y,y ′ ,b ′ ,b = s ◦ (id ⊗ ∂ ) − ⊗ X y,y ′ ∈ X µ (1) y,y ′ e op y ′ ,y,b,b ′ . Similarly, letting µ (2) x,x ′ = P y ∈ X ρ x,x ′ ,y,y , we have that the right hand side of(9.7) coincides with X a ∈ A X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ s (cid:0) e x,x ′ ,b,b ′ ⊗ e y,y ′ ,a,a (cid:1) = X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ s e x,x ′ ,b,b ′ ⊗ X a ∈ A e y,y ′ ,a,a !! = X x,x ′ ∈ X X y,y ′ ∈ X ρ x,x ′ ,y,y ′ δ y,y ′ s (cid:0) e x,x ′ ,b,b ′ ⊗ (cid:1) = s X x,x ′ ∈ X µ (2) x,x ′ e x,x ′ ,b,b ′ ⊗ , that is, with s ◦ (id ⊗ ∂ ) − X y,y ′ ∈ X µ (2) y ′ ,y e y ′ ,y,b,b ′ ⊗ . Let now ρ ∈ Σ X . By (9.4), µ (1) y,y ′ = µ (2) y ′ ,y . Hence, if s ◦ (id ⊗ ∂ ) − is fair thenΓ( ρ ) ∈ Σ A , that is, Γ is fair.Conversely, assuming that Γ is fair, the previous paragraph shows that(9.8) s ◦ (id ⊗ ∂ ) − ( u ⊗
1) = s ◦ (id ⊗ ∂ ) − (1 ⊗ u op )for any u of the form u = P y,y ′ ∈ X ( P x ρ x,x,y,y ′ ) e y,y ′ ,b,b ′ with ρ ∈ Σ X . Letting ρ = e x e ∗ x ⊗ e x e ∗ x ∈ Σ X we conclude that (9.8) holds for u = e x,x,b,b ′ , x ∈ X , b, b ′ ∈ A . Letting ρ = 1 ⊗ ω t + ω ⊗
1, where ω = α ( e z e ∗ z + e z ′ e ∗ z ′ ) + βe z e ∗ z ′ + ¯ βe z ′ e ∗ z , z = z ′ , with α ≥ | β | , we obtain that (9.8) holds for u = α (2 P y ∈ X e y,y,b,b ′ + | X | e z,z,b,b ′ + | X | e z ′ ,z ′ ,b,b ′ ) + β | X | e z ′ ,z,b,b ′ + ¯ β | X | e z,z ′ ,b,b ′ .From this we deduce that (9.8) holds for any u = e y,y ′ ,b,b ′ , y, y ′ ∈ X , b, b ′ ∈ A . (cid:3) Let
S ⊆ B ( K ) be an operator system. We let S op = { u d : u ∈ S} ,considered as an operator subsystem of B ( K d ). Note that S op is well-defined:if φ : S → B ( ˜ K ) is a unital complete isometry, then the map ˜ φ : S op →B ( ˜ K d ), given by ˜ φ ( u d ) = φ ( u ) d , is also unital and completely isometric. Wethus write u op = u d in the (abstract) operator system S op .For a linear map Φ : M X → M A , let Φ ♯ : M X → M A be the linear mapgiven by Φ ♯ ( ω ) = Φ( ω t ) t . Lemma 9.5.
Let S be an operator system. (i) If φ : S → B ( H ) be a unital completely positive map then the map φ op : S op → B ( H d ) , given by φ op ( u op ) = φ ( u ) d , is unital and com-pletely positive. (ii) Up to a canonical *-isomorphism, C ∗ u ( S op ) = C ∗ u ( S ) op . (iii) If Φ : M X → M A is a completely positive map then so is Φ ♯ . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 41
Proof. (i) Represent
S ⊆ B ( K ) as a concrete operator system. Then S op ⊆B ( K d ). Suppose that u i,j ∈ S , i, j = 1 , . . . , n , are such that ( u d i,j ) i,j ∈ M n ( B ( K d )) + . Then ( u j,i ) i,j = ( u d i,j ) d i,j ∈ M n ( B ( K )) + and hence ( φ ( u j,i )) i,j ∈ M n ( B ( H )) + . Thus, (cid:16) φ op ( u op i,j ) (cid:17) i,j = (cid:16) φ ( u i,j ) d (cid:17) i,j ∈ M n (cid:16) B ( H d ) (cid:17) + . (ii) Suppose that ψ : S op → B ( H ) is a unital completely positive map.By (i), ψ op : S → B ( H d ) is (unital and) completely positive. By the uni-versal property of the maximal C*-cover, there exists a *-homomorphism π : C ∗ u ( S ) → B ( H d ) extending ψ op . It follows that π op : C ∗ u ( S ) op → B ( H )is a *-homomorphism that extends ψ . Thus, C ∗ u ( S ) op satisfies the universalproperty of the C*-cover of S op .(iii) The transposition is a (unital) complete order isomorphism from M X onto M op X . The statement follows after observing that, under the latteridentification, Φ ♯ coincides with Φ op . (cid:3) Corollary 9.6.
A local QNS correlation Γ is fair if and only if Γ = P mi =1 λ i Φ i ⊗ Ψ i for some quantum channels Φ i , Ψ i : M X → M A and scalars λ i ≥ , i = 1 , . . . , m , P mi =1 λ i = 1 , such that (9.9) m X i =1 λ i Φ i = m X i =1 λ i Ψ ♯i . Proof.
Suppose that Γ is fair and, using Theorem 9.4, write Γ = Γ s , where s is a state on OMIN( T X,A ) ⊗ min OMIN( T Y,B ) such that s ◦ (id ⊗ ∂ ) − isfair. As in the proof of Theorem 6.7, identify s with a convex combination P mi =1 λ i φ i ⊗ ψ i , where φ i and ψ i are states on T X,A , i = 1 , . . . , m ; then thefairness condition is equivalent to(9.10) m X i =1 λ i φ i ( u ) = m X i =1 λ i ψ i ( ∂ − ( u op )) , u ∈ T X,A . Let Φ i and Ψ i be the quantum channels from M X to M A , corresponding to φ i and ψ i , respectively; then Γ = P mi =1 λ i Φ i ⊗ Ψ i . Let ˜ ψ i : u ψ i (( ∂ − ( u op )), u ∈ T X,A . By Lemma 9.2, ˜ ψ i is a state. Moreover, h Ψ ♯i ( e x e ∗ x ′ ) , e a e ∗ a ′ i = h Ψ i ( e x ′ e ∗ x ) t , e a e ∗ a ′ i = h Ψ i ( e x ′ e ∗ x ) , e a ′ e ∗ a i = ψ i ( e x ′ ,x,a ′ ,a ) = ψ i ( ∂ − ( e op x,x ′ ,a,a ′ )) = ˜ ψ i ( e x,x ′ ,a,a ′ ) , that is, the quantum channel Ψ ♯i corresponds to ˜ ψ i . Identity (9.9) now followsfrom (9.10). The converse implication follows by reversing the previoussteps. (cid:3) Corollary 9.7. (i)
A CQNS correlation E is fair if and only if there isa state t : R X,A ⊗ max R X,A → C such that t ◦ (id ⊗ ∂ B ) − is fair and E = E t . Similar descriptions hold for fair correlations in the classes CQ qc , CQ qa and CQ loc . (ii) An NS correlation p is fair if and only if there is a state t : S X,A ⊗ max S X,A → C such that t ( u ⊗
1) = t (1 ⊗ u ) , u ∈ S X,A , and p ( a, b | x, y ) = t ( e x,a ⊗ e y,b ) , x, y ∈ X, a, b ∈ A. Similar descriptions hold for fair correlations in the classes C qc , C qa and C loc .Proof. We only give details for (i). Let E : D XY → M AB be a fair CQNScorrelation. By (9.6), E ◦ ∆ XY : M XY → M AB is a fair QNS correlation. ByTheorem 9.4 (i), E ◦ ∆ XY = Γ s , for some state s on T X,A ⊗ max T X,A suchthat s ◦ (id ⊗ ∂ ) − is fair. It follows that E = E t , where t := s ◦ ( β X,A ⊗ β X,A )is a state on R X,A ⊗ max R X,A and t ◦ (id ⊗ ∂ B ) − is fair. Conversely, if E = E t for some state t on R X,A ⊗ max R X,A such that t ◦ (id ⊗ ∂ B ) − is fair thenΓ E = Γ s , where s := t ◦ ( β ′ X,A ⊗ β ′ X,A ) and s ◦ (id ⊗ ∂ ) − is fair. By Theorem9.4 (i), Γ E is fair, and hence so is E . The statements regarding CQ qc , CQ qa and CQ loc follow after a straightforward modification of the argument. (cid:3) Remark.
It follows from Theorem 9.1, Theorem 9.4 and Corollary 9.7 thatfair correlations can be viewed as a non-commutative, and less restrictive,version of synchronous correlations.9.2.
Tracial QNS correlations.
Let A be a unital C*-algebra, τ : A → C be a state and A op be the opposite C*-algebra of A . By the paragraphbefore Theorem 6.2.7 in [12], the linear functional s τ : A ⊗ max A op → C ,given by s τ ( u ⊗ v op ) = τ ( uv ), is a state.A positive element E ∈ M X ⊗ M A ⊗ A will be called a stochastic A -matrix if (id ⊗ id ⊗ π )( E ) is a stochastic operator matrix for some faithful*-representation of A . Such an E will be called semi-classical if it belongsto D X ⊗ M A ⊗ A .Let E = ( g x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ be a stochastic A -matrix, and set E op = ( g op x ′ ,x,a ′ ,a ) x,x ′ ,a,a ′ ∈ M X ⊗ M A ⊗ A op ;Lemma 9.2 shows that E op is a stochastic A op -matrix. Thus, after a per-mutation of the tensor factors, we can consider E ⊗ E op as an elementof ( M XA ⊗ M XA ⊗ ( A ⊗ max A op )) + . By Theorem 5.2, there exists a *-homomorphism π E : C X,A → A , such that π E ( e x,x ′ ,a,a ′ ) = g x,x ′ ,a,a ′ for all x, x ′ , a, a ′ . By Corollary 5.3 and Lemma 9.5, C ∗ u ( T op X,A ) ≡ C op X,A ; thus, T X,A ⊗ c T op X,A ⊆ c . o . i . C X,A ⊗ max C op X,A . Write(9.11) f E,τ = s τ ◦ ( π E ⊗ π op E ) ◦ (id ⊗ ∂ );we have that f E,τ is a state on T X,A ⊗ c T X,A , and f E,τ ( e x,x ′ ,a,a ′ ⊗ e y,y ′ ,b,b ′ ) = τ ( g x,x ′ ,a,a ′ g y ′ ,y,b ′ ,b ) , x, x ′ , y, y ′ ∈ X, a, a ′ , b, b ′ ∈ A. UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 43
In the sequel, we write Γ
E,τ = Γ f E,τ ; by Theorem 6.3, Γ
E,τ ∈ Q qc . ByTheorem 5.2, we may assume, without loss of generality, that A = C X,A and E = ( e x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ . In this case, we will abbreviate Γ E,τ to Γ τ . Definition 9.8.
A QNS correlation Γ is called (i) tracial if Γ = Γ τ , where τ : C X,A → C is a trace; (ii) quantum tracial if there exists a finite dimensional C*-algebra A ,a trace τ A on A and a *-homomorphism π : C X,A → A such that
Γ = Γ τ A ◦ π ; (iii) locally tracial if there exists an abelian C*-algebra A , a state τ A on A and a *-homomorphism π : C X,A → A such that
Γ = Γ τ A ◦ π . Theorem 9.9.
Let X and A be finite sets. (i) If Γ is a quantum tracial QNS correlation then Γ ∈ Q q ; (ii) A QNS correlation
Γ : M XX → M AA is locally tracial if and only ifthere exists quantum channels Φ j : M X → M A , j = 1 , . . . , k , suchthat (9.12) Γ = k X j =1 λ j Φ j ⊗ Φ ♯j as a convex combination. In particular, if Γ is a locally tracial QNScorrelation then Γ ∈ Q loc .Proof. (i) Suppose that H is a finite dimensional Hilbert space on which A acts faithfully and let π : C X,A → A be as in Definition 9.8 (ii). Let E x,x ′ ,a,a ′ = π ( e x,x ′ ,a,a ′ ) and E = (cid:0) E x,x ′ ,a,a ′ (cid:1) x,x ′ ,a,a ′ . By the proof of Lemma9.2, E op := (cid:16) E d x ′ ,x,a ′ ,a (cid:17) x,x ′ ,a,a ′ is a stochastic operator matrix. Let σ beany positive functional on L ( H ⊗ H d ) that extends the state s τ A which, bynuclearity, may be considered as a state on A⊗ min A op . Then Γ τ = Γ E ⊙ E op ,σ and, by the paragraph before Remark 4.7, Γ τ ∈ Q q .(ii) Suppose that Φ j : M X → M A , j = 1 , . . . , k , are quantum channelsand Γ is the convex combination (9.12). Letting (cid:16) λ ( j ) x,x ′ ,a,a ′ (cid:17) a,a ′ = Φ j (cid:0) e x e ∗ x ′ (cid:1) , x, x ′ ∈ X , we have that the matrix C j = (cid:16) λ ( j ) x,x ′ ,a,a ′ (cid:17) x,x ′ ,a,a ′ is a stochastic C -matrix. By Theorem 5.2, there exists a (unique) *-representation π j : C X,A → C such that π j ( e x,x ′ ,a,a ′ ) = λ ( j ) x,x ′ ,a,a ′ , x, x ′ ∈ X , a, a ′ ∈ A . Let π : C X,A → D k be the *-representation given by π ( u ) = k X j =1 π j ( u ) e j e ∗ j , u ∈ C X,A , and let τ k : D k → C be the state defined by τ k (cid:16) ( µ j ) kj =1 (cid:17) = P kj =1 λ j µ j . Wehave Γ τ k ◦ π (cid:0) e x e ∗ x ′ ⊗ e y e ∗ y ′ (cid:1) = X a,a ′ ∈ A X b,b ′ ∈ B ( τ k ◦ π )( e x,x ′ ,a,a ′ e y ′ ,y,b ′ ,b ) e a e ∗ a ′ ⊗ e b e ∗ b ′ = X a,a ′ ∈ A X b,b ′ ∈ B τ k k X j =1 π j ( e x,x ′ ,a,a ′ e y ′ ,y,b ′ ,b ) e j e ∗ j e a e ∗ a ′ ⊗ e b e ∗ b ′ = k X j =1 λ j X a,a ′ ∈ A X b,b ′ ∈ B λ ( j ) x,x ′ ,a,a ′ λ ( j ) y ′ ,y,b ′ ,b e a e ∗ a ′ ⊗ e b e ∗ b ′ = k X j =1 λ j X a,a ′ ∈ A λ ( j ) x,x ′ ,a,a ′ e a e ∗ a ′ ⊗ X b,b ′ ∈ B λ ( j ) y ′ ,y,b ′ ,b e b e ∗ b ′ = k X j =1 λ j Φ j ( e x e ∗ x ′ ) ⊗ Φ ♯j (cid:0) e y e ∗ y ′ (cid:1) . Conversely, let A be a unital abelian C*-algebra, τ A : A → C a state,and π : C X,A → A a *-homomorphism such that Γ = Γ τ A ◦ π . Withoutloss of generality, assume that A = C (Ω), where Ω is a compact Hausdorfftopological space, and µ is a Borel probability measure on Ω such that τ A ( f ) = R Ω f dµ , f ∈ A . Set h x,x ′ ,a,a ′ = π ( e x,x ′ ,a,a ′ ), x, x ′ ∈ X , a, a ′ ∈ A .For each s ∈ Ω, let Φ( s ) : M X → M A be the quantum channel given byΦ( s ) (cid:0) e x e ∗ x ′ (cid:1) = (cid:0) h x,x ′ ,a,a ′ ( s ) (cid:1) a,a ′ . We haveΓ (cid:0) e x e ∗ x ′ ⊗ e y e ∗ y ′ (cid:1) = X a,a ′ ∈ A X b,b ′ ∈ A (cid:18)Z Ω h x,x ′ ,a,a ′ ( s ) h y ′ ,y,b ′ ,b ( s ) dµ ( s ) (cid:19) e a e ∗ a ′ ⊗ e b e ∗ b ′ = Z Ω Φ( s ) ( e x e ∗ x ′ ) ⊗ Φ( s ) ♯ (cid:0) e y e ∗ y ′ (cid:1) dµ ( s ) . It follows that Γ is in the closed hull of the set of all correlations of the form(9.12). An argument using Carath´eodory’s Theorem, similar to the one inthe proof of Remark 4.10, shows that Γ has the form (9.12). (cid:3)
Remark 9.10. (i)
Every tracial QNS correlation Γ = Γ
E,τ is fair. Indeed,writing E = ( g x,x ′ ,a,a ′ ), we have f E,τ ◦ (id ⊗ ∂ ) − ( e x,x ′ ,a,a ′ ⊗
1) = τ ( g x,x ′ ,a,a ′ )= f E,τ ◦ (id ⊗ ∂ ) − (1 ⊗ e op x,x ′ ,a,a ′ ) . It can be seen from Corollary 9.6 and Theorem 9.9 (see the closing remarksof this section) that the converse does not hold true.
UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 45 (ii)
The set of all tracial (resp. quantum tracial, locally tracial) QNScorrelations over (
X, A ) is convex. Indeed, suppose that A (resp. B ) is aunital C*-algebra, τ A (resp. τ B ) a trace on A and E (resp. F ) a stochastic A -matrix (resp. a stochastic B -matrix). Let λ ∈ (0 , C = A⊕B , τ : C → C be given by τ ( u ⊕ v ) = λτ A ( u ) + (1 − λ ) τ B ( v ), and G = E ⊕ F , consideredas an element of M X ⊗ M A ⊗ C . Then G is a stochastic C -matrix and λ Γ E,τ A + (1 − λ )Γ F,τ B = Γ G,τ . (iii) It is straightforward from Theorem 9.1 that, if p ∈ C qc (resp. p ∈ C q , p ∈ C loc ) is synchronous then Γ p is a tracial (resp. quantum tracial, locallytracial) QNS correlation. By [22, Theorem 4.2], the set C sq of synchronousquantum NS correlations is not closed if | X | = 5 and | A | = 2. Let p ∈ C sq \C sq .Then p is a synchronous NS correlation and does not lie in C q . Assume thatΓ p is quantum tracial. By Theorem 9.9, Γ p ∈ Q q and hence, by Remark8.1, p ∈ C q , a contradiction. It follows that the set of quantum tracial NScorrelations is not closed. (iv) The set of all tracial QNS correlations is closed; this can be seen viaa standard argument (see e.g. [54]): Assuming that (Γ n ) n ∈ N is a sequenceof tracial QNS correlations converging to the QNS correlation Γ, let A n bea unital C*-algebra with a trace τ n , and E n = (cid:16) g ( n ) x,x ′ ,a,a ′ (cid:17) be a stochastic A n -matrix such that Γ n = Γ E n ,τ n . Let A be the tracial ultraproduct of thefamily { ( A n , τ n ) } n ∈ N with respect to a non-trivial ultrafilter u [33, Section4]. Write τ for the trace on A and E = ( g x,x ′ ,a,a ′ ) for the class of ⊕ n ∈ N E n in A . Then (cid:10) Γ( e x e ∗ x ′ ⊗ e y e ∗ y ′ ) , e a e ∗ a ′ ⊗ e b e ∗ b ′ (cid:11) = lim n →∞ τ n (cid:16) g ( n ) x,x ′ ,a,a ′ g ( n ) y ′ ,y,b ′ ,b (cid:17) = τ (cid:0) g x,x ′ ,a,a ′ g y ′ ,y,b ′ ,b (cid:1) . We next show that the class of all tracial QNS correlations, as well as eachof the the subclasses of quantum tracial and locally tracial QNS correlations,have natural classes of invariant states. Given a unital C*-algebra A , a trace τ : A → C and a stochastic A -matrix E = ( g z,z ′ ) z,z ′ ∈ L ( C ) ⊗ M Z ⊗ A , let ω E,τ = (cid:16) ω E,τz,z ′ ,u,u ′ (cid:17) ∈ M ZZ be defined by ω E,τz,z ′ ,u,u ′ = τ ( g z,z ′ g u ′ ,u ) , z, z ′ , u, u ′ ∈ Z. Equivalently, let E op be the stochastic A op -matrix (cid:16) g op u ′ ,u (cid:17) , and recall that s τ : A ⊗ max A op → C is the state given by s τ ( u ⊗ v op ) = τ ( uv ). Then ω E,τ = L s τ ( E ⊗ E op ) , where L s τ : M ZZ ⊗ ( A ⊗ max A op ) → M ZZ is the corresponding slice. It follows that ω E,τ is a state.
Definition 9.11.
Let Z be a finite set. A state ω ∈ M ZZ is called (i) C*-reciprocal if there exists a unital C*-algebra algebra A , a trace τ on A and a stochastic A -matrix E ∈ M Z ⊗ A such that ω = ω E,τ ; (ii) quantum reciprocal if it is C*-reciprocal, and the C*-algebra A from(i) can be chosen to be finite dimensional; (iii) locally reciprocal if it is C*-reciprocal, and the C*-algebra A from(i) can be chosen to be abelian. We will denote by Υ( Z ) (resp. Υ q ( Z ), Υ loc ( Z )) the set of all C*-reciprocal(resp. quantum reciprocal, locally reciprocal) states in M ZZ . Theorem 9.12.
Let Γ be a QNS correlation. (i) If Γ is tracial then Γ (Υ( X )) ⊆ Υ( A ) ; (ii) If Γ is quantum tracial then Γ (Υ q ( X )) ⊆ Υ q ( A ) ; (iii) If Γ is locally tracial then Γ (Υ loc ( X )) ⊆ Υ loc ( A ) .Proof. (i) Let τ be a trace on C X,A , A be a C*-algebra, τ A be a trace on A ,and E = ( g x,x ′ ) x,x ′ ∈ M X ⊗ A be a stochastic A -matrix. Set ω = Γ τ (cid:0) ω E,τ A (cid:1) and write ω = (cid:0) ω a,a ′ ,b,b ′ (cid:1) a,a ′ ,b,b ′ . Let B = A ⊗ max C X,A and τ B = τ A ⊗ τ bethe product trace on B [12, Proposition 3.4.7]. Set h a,a ′ = X x,x ′ ∈ X g x,x ′ ⊗ e x,x ′ ,a,a ′ , a, a ′ ∈ A ;thus, F := ( h a,a ′ ) a,a ′ ∈ M A ⊗ B . Moreover,Tr A F = X a ∈ A h a,a = X x,x ′ ∈ X g x,x ′ ⊗ X a ∈ A e x,x ′ ,a,a ! = X x,x ′ ∈ X g x,x ′ ⊗ δ x,x ′ X x ∈ X g x,x ⊗ B . To see that F is positive, we assume that A and C X,A are faithfully repre-sented and let V x and V a,x be operators such that ( V x ) x is a row isometry,( V a,x ) a,x is an isometry, g x,x ′ = V ∗ x V x ′ and e x,x ′ ,a,a ′ = V ∗ a,x V a ′ ,x ′ , x, x ′ ∈ X , a, a ′ ∈ A . Letting W = (cid:0)P x ∈ X V x ⊗ V a,x (cid:1) a ∈ A , considered as row operator,we have that F = W ∗ W . Hence F is a stochastic B -matrix. In addition, for a, a ′ , b, b ′ ∈ A we have τ B (cid:0) h a,a ′ h b ′ ,b (cid:1) = X x,x ′ ∈ X X y,y ′ ∈ X τ A (cid:0) g x,x ′ g y ′ ,y (cid:1) τ (cid:0) e x,x ′ ,a,a ′ e y ′ ,y,b ′ ,b (cid:1) = ω a,a ′ ,b,b ′ , implying that ω = ω F,τ B .(ii) and (iii) follow from the fact that if the C*-algebra A is finite di-mensional (resp. abelian) and τ factors through a finite-dimensional (resp.abelian) C*-algebra then so does τ B . (cid:3) UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 47
Remark 9.13. (i)
The state | X | I XX is locally reciprocal, and hence itfollows from Theorem 9.12 thatΥ loc ( A ) = (cid:26) | X | Γ( I XX ) : X finite, Γ loc. tracial QNS correlation (cid:27) = { Γ(1) : Γ : C → M AA loc. tracial QNS correlation } . (9.13)Similar descriptions hold for Υ q ( A ) and Υ( A ). Remark 9.10 thus impliesthat the sets Υ( A ), Υ q ( A ) and Υ loc ( A ) are convex and the sets Υ( A ) andΥ loc ( A ) are closed. (ii) Recall that a state ρ ∈ M XX is called de Finetti [17] if there existstates ω i ∈ M Z , i = 1 , . . . , k , such that ρ = P kj =1 λ j ω j ⊗ ω j as a convexcombination. By (9.13) and Theorem 9.9,Υ loc ( X ) = conv (cid:8) ω ⊗ ω t : ω a state in M X (cid:9) . Thus, the locally reciprocal states can be viewed as twisted de Finetti states.The presence of the transposition in our case is required in view of thenecessity to employ opposite C*-algebras. Thus, quantum reciprocal statescan be viewed as an entanglement assisted version of (twisted) de Finettistates, while C*-reciprocal states – as their commuting model version. (iii)
C*-reciprocal states are closely related to factorisable channels intro-duced in [1] (see also [32, 53], to which we refer the reader for the definitionused here). Indeed, factorisable channels have Choi matrices of the form τ ( g x,x ′ h y ′ ,y ) x,x ′ ,y,y ′ , where τ is a faithful normal trace on a von Neumannalgebra A , and ( g x,x ′ ) x,x ′ and ( h y,y ′ ) y,y ′ are matrix unit systems – a specialtype of stochastic A -matrices (see [53, Proposition 3.1]). Equivalently, theChoi matrices of factorisable channels Φ : M X → M X can be described[32, Definition 3.1] as the matrices of the form (cid:0) τ ( v ∗ a,x v a ′ ,x ′ ) (cid:1) x,x ′ ,a,a ′ , where V = ( v a,x ) a,x ∈ M X ( A ) is a unitary matrix. Note that, if E is the stochasticoperator matrix corresponding to V , then the QNS correlation Γ = Γ E,τ hasmarginal channels Γ A ( · ) = Γ( · ⊗ I ) and Γ B ( · ) = Γ( I ⊗ · ) that coincide withΦ. We can thus view tracial QNS correlations as generalised couplings offactorisable channels. Here, by a coupling of the pair (Φ , Ψ) of channels, wemean a channel Γ with Γ A = Φ and Γ B = Ψ – a generalisation of classicalcoupling of probability distributions in the sense of optimal transport [72].9.3. Tracial CQNS correlations.
In this subsection, we define a tracialversion of CQNS correlations. Let A be a unital C*-algebra, τ : A → C be atrace and E ∈ D X ⊗ M A ⊗ A be a semi-classical stochastic A -matrix. Write E = ( g x,a,a ′ ) x,a,a ′ ; thus, ( g x,a,a ′ ) a,a ′ ∈ ( M A ⊗ A ) + and P a ∈ A g x,a,a = 1, foreach x ∈ X . Set E op = ( g op x,a ′ ,a ) x,a,a ′ ; thus, E op ∈ D X ⊗ M A ⊗ A op andLemma 9.2 shows that E op is a semi-classical stochastic A op -matrix. Let φ E,x : M A → A be the unital completely positive map given by φ E,x ( e a e ∗ a ′ ) = g x,a,a ′ . By Boca’s Theorem [6], there exists a unital completely positivemap φ E : B X,A → A such that φ E ( e x,a,a ′ ) = g x,a,a ′ , x ∈ X , a, a ′ ∈ A . Let φ op E : B op X,A → A op be the map given by φ op E ( u op ) = φ E ( u ) op , which iscompletely positive by Lemma 9.5. Write f E,τ = s τ ◦ ( φ E ⊗ φ op E ) ◦ (id ⊗ ∂ B );thus, by (7.6) f E,τ is a state on R X,A ⊗ c R X,A . Note that f E,τ (cid:0) e x,a,a ′ ⊗ e y,b,b ′ (cid:1) = τ (cid:0) g x,a,a ′ g y,b ′ ,b (cid:1) , x, y ∈ X, a, a ′ , b, b ′ ∈ A. In the sequel, we write E E,τ = E f E,τ ; by Theorem 7.7, E E,τ ∈ CQ qc . Definition 9.14.
A CQNS correlation E is called (i) tracial if E = E E,τ , where E ∈ D X ⊗ M A ⊗ A is a semi-classicalstochastic A -matrix for some unital C*-algebra A and τ : A → C isa trace; (ii) quantum tracial if it is tracial and the C*-algebra as in (i) can bechosen to be finite dimensional; (iii) locally tracial if it it is tracial and the C*-algebra as in (i) can bechosen to be abelian. Proposition 9.15.
Let E : D XX → M AA be a CQNS correlation. (i) If E is quantum tracial then E ∈ CQ q ; (ii) E is locally tracial if and only if there exist channels E j : D X → M A , j = 1 , . . . , k , such that (9.14) E = k X j =1 λ j E j ⊗ E ♯j . In particular, if E is locally tracial then E ∈ CQ loc .Proof. (i) Suppose that E is quantum tracial and write E = E E,τ , where E = ( g x,a,a ′ ) x,a,a ′ ∈ D X ⊗ M A ⊗ A is a semi-classical stochastic A -matrix forsome finite dimensional C*-algebra A and a trace τ : A → C . The matrix˜ E = (cid:0) δ x,x ′ g x,a,a ′ (cid:1) x,x ′ ,a,a ′ is a stochastic matrix in M X ⊗ M A ⊗ A and hencegives rise, via Theorem 5.2, to a canonical *-homomorphism π ˜ E : C X,A → A .Letting ˜ τ = τ ◦ π ˜ E , we have that ˜ τ is a trace on C X,A and Γ E = Γ ˜ τ . Thus,Γ E ∈ Q q . By Remark 8.1, E ∈ CQ q .(ii) We fix A , τ and E as in (i), with A abelian. The trace ˜ τ , defined inthe proof of (i), now factors through an abelian C*-algebra, and hence Γ E islocally tracial. By Theorem 9.9, there exists quantum channels Φ j : M X → M A , j = 1 , . . . , k , such that Γ E = P kj =1 Φ j ⊗ Φ ♯j as a convex combination.Letting E j = Φ j | D X , j = 1 , . . . , k , we see that E has the form (9.14).Conversely, suppose that E has the form (9.14). By Theorem 9.9, thereexists an abelian C*-algebra A , a *-representation π : C X,A → A and atrace τ on A such that Γ E = Γ τ ◦ π . The stochastic operator matrix E = (cid:0) π ( e x,x,a,a ′ ) (cid:1) x,a,a ′ is semi-classical and E = E E,τ . (cid:3) We now specialise Definition 9.11 to states in D XX , that is, bipartiteprobability distributions. A probability distribution q = ( q ( x, y )) x,y ∈ X on UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 49 X × X will be called C*-reciprocal if there exists a C*-algebra A , a POVM( g x ) x ∈ X in A and a trace τ : A → C such that q ( x, y ) = τ ( g x g y ), x, y ∈ X . If A can be chosen to be finite dimensional (resp. abelian), we call q quantumreciprocal (resp. locally reciprocal ). We denote by Υ cl ( X ) (resp. Υ clq ( X ),Υ clloc ( X )) the (convex) set of all C*-reciprocal (resp. quantum reciprocal,locally reciprocal) probability distributions on X × X .It can be seen as in Remark 9.13 that the class of locally reciprocal proba-bility distributions coincides with the well-known class of exchangeable prob-ability distributions, that is, the convex combinations of the form q ( x, y ) = n X i =1 λ i q i ( x ) q i ( y ) , x, y ∈ X, where q i is a probability distribution on X , i = 1 , . . . , n . Thus, C*-reciprocaland quantum reciprocal probability distributions can be viewed as quantumversions of exchangeable distributions.It is straightforward to see that, writing ∆ = ∆ XX , we have∆(Υ( X )) = Υ cl ( X ) , ∆(Υ q ( X )) = Υ clq ( X ) and ∆(Υ loc ( X )) = Υ clloc ( X ) . These relations, combined with Theorem 9.12, easily yield the followingproposition, whose proof is omitted.
Proposition 9.16.
Let E : D XX → M AA be a CQNS correlation. (i) If E is tracial then E (cid:0) Υ cl ( X ) (cid:1) ⊆ Υ( A ) ; (ii) If E is quantum tracial then E (cid:0) Υ clq ( X ) (cid:1) ⊆ Υ q ( A ) ; (iii) If E is locally tracial then E (cid:0) Υ clloc ( X ) (cid:1) ⊆ Υ loc ( A ) . Tracial NS correlations.
The correlation classes introduced in Sec-tions 9.2 and 9.3 have a natural NS counterpart. For a C*-algebra A ,equipped with a trace τ , and a classical stochastic A -matrix E ∈ D X ⊗D A ⊗ A , say, E = ( g x,a ) x,a (so that g x,a ∈ A + for all x ∈ X and all a ∈ A and P a ∈ A g x,a = 1, x ∈ X ), write p E,τ ( a, b | x, y ) = τ ( g x,a g y,b ) , x, y ∈ X, a, b ∈ A. Similar arguments to the ones in Sections 9.2 and 9.3 show that p E,τ ∈ C qc . Definition 9.17.
An NS correlation p is called (i) tracial if it is of the form p E,τ , where E is a classical stochastic A -matrix for some unital C*-algebra A and τ : A → C is a trace; (ii) quantum tracial if it is tracial and the C*-algebra A in (i) can bechosen to be finite dimensional; (iii) locally tracial if it it is tracial and the C*-algebra A in (i) can bechosen to be abelian. The next two propositions are analogous to Theorem 9.9 and 9.12, re-spectively, and their proofs are omitted.
Proposition 9.18.
Let p be an NS correlation. (i) If p is quantum tracial then p ∈ C q ; (ii) p is locally tracial if and only if p = P kj =1 λ j q j ⊗ q j , where q j = { q j ( ·| x ) : x ∈ X } , is a family of probability distributions, j =1 , . . . , k . In particular, if p is locally tracial then p ∈ C loc . Proposition 9.19.
Let N : D XX → D AA be an NS correlation. (i) If N is tracial then N (cid:0) Υ cl ( X ) (cid:1) ⊆ Υ cl ( A ) . (ii) If N is quantum tracial then N (cid:0) Υ clq ( X ) (cid:1) ⊆ Υ clq ( A ) . (iii) If N is locally tracial then N (cid:0) Υ clloc ( X ) (cid:1) ⊆ Υ clloc ( A ) . Reduction for tracial correlations.
We next specialise the state-ments contained in Remark 8.1 to tracial correlations.
Theorem 9.20.
Let X and A be finite sets, p be an NS correlation and E be a CQNS correlation. The following hold: (i) p is tracial (resp. quantum tracial, locally tracial, fair) if and only if E p is tracial (resp. quantum tracial, locally tracial, fair), if and onlyif Γ p is tracial (resp. quantum tracial, locally tracial, fair); (ii) E is tracial (resp. quantum tracial, locally tracial, fair) if and onlyif Γ E is tracial (resp. quantum tracial, locally tracial, fair).Moreover, (iii) the map N is a surjection from the class of all tracial (resp. quantumtracial, locally tracial, fair) CQNS correlations onto the class of alltracial (resp. quantum tracial, locally tracial) NS correlations; (iv) the map C is a surjection from the class of all tracial (resp. quan-tum tracial, locally tracial, fair) QNS correlations onto the class ofall tracial (resp. quantum tracial, locally tracial, fair) CQNS corre-lations.Proof. We prove first the statements about tracial correlations.(i) Suppose that the NS correlation p is tracial, and write p ( a, b | x, y ) = τ ( g x,a g y,b ), x, y ∈ X , a, b ∈ A , for some trace τ on a unital C*-algebra A and matrix F = ( g x,a ) x,a ∈ ( D X ⊗ D A ⊗ A ) + with P a ∈ A g x,a = 1, x ∈ X . The matrix F ′ = ( δ a,a ′ g x,a ) x,a,a ′ ∈ D X ⊗ M A ⊗ A is a semi-classicalstochastic A -matrix and, trivially, E p = E F ′ ,τ . Similarly, the family F ′′ =( δ a,a ′ δ x,x ′ g x,a ) x,x ′ ,a,a ′ ∈ M X ⊗ M A ⊗ A is a stochastic A -matrix and Γ p =Γ F ′′ ,τ .Conversely, suppose that Γ p = Γ E,τ , where E = ( g x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ is astochastic A -matrix and τ is a trace on the unital C*-algebra A . Then E ′ := ( g x,x,a,a ′ ) x,a,a ′ (resp. E ′′ := ( g x,x,a,a ) x,a ) is a semi-classical (resp.classical) stochastic A -matrix such that E p = E E ′ ,τ (resp. p = p E ′′ ,τ ).(ii) is similar to (i).(iii) follows from the fact that, if E is a stochastic A -matrix and τ is atrace on A such that Γ = Γ E,τ then C (Γ) = E E ′ ,τ , where E ′ is given as inthe second paragraph of the proof. UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 51 (iv) is similar to (iii). All remaining statements about quantum tracialand locally tracial correlations are analogous.Turning to the case of fair correlations, (ii) follows from the equivalence E (Σ cl X ) ⊆ Σ A ⇐⇒ Γ E (Σ X ) = E (∆ X,Y (Σ X )) . For (i), observe that E p = ∆ A,B ◦ E p and hence E p (Σ cl X ) ⊆ Σ A ⇐⇒ E p (Σ cl X ) ⊆ Σ cl A ⇐⇒ N p (Σ cl X ) ⊆ Σ cl A , showing that p is fair if and only if so is E p . As Γ p = Γ E p , the equivalencewith fairness of Γ p follows from (ii). (cid:3) We conclude this section with a comparison between the different classesof correlations of synchronous type. Note first that, if p is a synchronousquantum commuting NS correlations then, by Theorem 9.1, N p is a tracialNS correlation. In fact, the synchronous quantum commuting NS corre-lations arise precisely from classical stochastic A -matrices ( g x,a ) x,a , whereeach ( g x,a ) a ∈ A is a PVM, as opposed to POVM. Theorem 9.4 implies thattracial QNS correlations are necessarily fair. We summarise these inclusionsbelow:synch. C loc ⊂ loc. tr. NS ⊂ loc. tr. CQNS ⊂ loc. tr. QNS ∩ ∩ ∩ ∩ synch. C q ⊂ q. tr. NS ⊂ q. tr. CQNS ⊂ q. tr. QNS ∩ ∩ ∩ ∩ synch. C qc ⊂ tracial NS ⊂ tracial CQNS ⊂ tracial QNS ∩ ∩ ∩ fair NS ⊂ fair CQNS ⊂ fair QNSThe inclusions in the table are all strict. Indeed, for the first column thisfollows from [22]. It can be shown, using results on the completely positivesemidefinite cone of matrices [46, 13] that Υ clloc = Υ clq [2]. The properness ofthe first inclusion in the second column now follows from Remark 9.13. Theproperness of the second inclusion in the second column was pointed out inRemark 9.10 (iii), and Theorem 9.20 implies that the first and the secondinclusions in the third and the fourth column are proper.Let p = { p ( ·| x ) : x ∈ X } and q = { q ( ·| x ) : x ∈ X } be families ofdistributions so that, for some x ∈ X , we have that supp p ( ·| x ) ∩ supp q ( ·| x ) = ∅ . Then ˜ p = 1 / p ⊗ q + q ⊗ p ) is a fair NS correlation. However, ˜ p is nottracial; indeed, assuming the contrary, we have that ˜ p = P mj =1 λ j p j ⊗ p j as a convex combination, where { p j } mj =1 consists of families of probabilitydistributions indexed by X . Since˜ p ( a, a | x, x ) = 12 ( p ( a | x ) q ( a | x ) + q ( a | x ) p ( a | x )) = 0 , a ∈ A, we have P mj =1 λ j p j ( a | x ) = 0, and hence p j ( a | x ) = 0, for all a ∈ A and all j , a contradiction. Thus, the last inclusion in the second column is strict, and by Theorem 9.20 so are the last inclusions in the third and the fourthcolumn.Using Theorem 9.9 and Proposition 9.15, one can easily see that the sec-ond and third inclusion on the first row are strict, and hence these inclusionsare strict on all other rows as well. Any NS correlation of the form q ⊗ q ,where q = { q ( ·| x ) : x ∈ X } is a family of probability distributions withat least one x having | supp q ( ·| x ) | >
1, is not synchronous, but is locallytracial; thus, the first inclusion in the first, second and third rows are strict.10.
Correlations as strategies for non-local games
In this section, we discuss how QNS correlations can be viewed as perfectstrategies for quantum non-local games, extending the analogous viewpointon NS correlations to the quantum case. Let X , Y , A and B be finitesets. A non-local game on ( X, Y, A, B ) is a cooperative game, played by twoplayers against a verifier, determined by a rule function (which will oftenbe identified with the game) λ : X × Y × A × B → { , } . The set X (resp. Y ) is interpreted as a set of questions to, while the set A (resp. B ) as aset of answers of, player Alice (resp. Bob). In a single round of the game,the verifier feeds in a pair ( x, y ) ∈ X × Y and the players produce a pair( a, b ) ∈ A × B ; they win the round if and only if λ ( x, y, a, b ) = 1. An NScorrelation p on X × Y × A × B is called a perfect strategy for the game λ if λ ( x, y, a, b ) = 0 = ⇒ p ( a, b | x, y ) = 0 . The terminology is motivated by the fact that if, given a pair ( x, y ) ofquestions, the players choose their answers according to the probability dis-tribution p ( · , ·| x, y ), they will win every round of the game.10.1. Quantum graph colourings.
Let G be a simple graph on a finiteset X . For x, y ∈ X , we write x ∼ y if { x, y } is an edge of G . By assumption, x ∼ y implies x = y ; we write x ≃ y if x ∼ y or x = y . A classical colouringof G is a map f : X → A , where A is a finite set, such that x ∼ y = ⇒ f ( x ) = f ( y ) . The chromatic number χ ( G ) of G is the minimal cardinality | A | of a set A for which a classical colouring f : X → A of G exists.The graph colouring game for G (called henceforth the G -colouring game)[15] is the non-local game with Y = X , B = A , and rules(i) x = y = ⇒ a = b ;(ii) x ∼ y = ⇒ a = b .Thus, an NS correlation p = { ( p ( a, b | x, y )) a,b ∈ A : x, y ∈ X } is a perfect strat-egy of the G -colouring game if(S) p is synchronous;(P) x ∼ y ⇒ p ( a, a | x, y ) = 0 for all a . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 53
It is easy to see that if p is a perfect strategy of the G -colouring game fromthe class C loc then G possesses a classical colouring from the set A . Thus, theperfect strategies for the G -colouring game from C x , where x ∈ { loc , q , qc } can be thought of as classical x -colourings of G . The x-chromatic numberof G is the parameter χ x ( G ) = min {| A | : G has a classical x-colouring by A } ;in particular, χ loc ( G = χ ( G ) (see [15, 50, 62] and the references therein).We call p a G -proper correlation if condition (P) is satisfied. For a finiteset A , we let Ω A be the non-normalised maximally entangled matrix in M AA ,namely, Ω A = X a,b ∈ A e a e ∗ b ⊗ e a e ∗ b . Remark 10.1.
Let G be a graph with vertex set X . An NS correlation p over ( X, X, A, A ) is G -proper if and only x ∼ y = ⇒ (cid:10) E p (cid:0) e x e ∗ x ⊗ e y e ∗ y (cid:1) , Ω A (cid:11) = 0 . Proof.
The claim is immediate from the fact that (cid:10) E p (cid:0) e x e ∗ x ⊗ e y e ∗ y (cid:1) , Ω A (cid:11) = X a,b ∈ A X a ′ ,b ′ ∈ A p ( a, b | x, y ) h e a e ∗ a ⊗ e b e ∗ b , e a ′ e ∗ b ′ ⊗ e a ′ e ∗ b ′ i = X a,b ∈ A X a ′ ,b ′ ∈ A p ( a, b | x, y ) h e a e ∗ a , e a ′ e ∗ b ′ i h e b e ∗ b , e a ′ e ∗ b ′ i = X a ∈ A p ( a, a | x, y ) . (cid:3) Remark 10.1 allows to generalise the classical x-colourings of a graph G to the quantum setting as follows. Definition 10.2.
Let G be a graph with vertex set X . A CQNS correlation E : D XX → M AA is called G -proper if x ∼ y = ⇒ (cid:10) E ( e x e ∗ x ⊗ e y e ∗ y ) , Ω A (cid:11) = 0 . A G -proper CQNS correlation E is called (i) a quantum loc-colouring of G by A if E is locally tracial; (ii) a quantum q-colouring of G by A if E is quantum tracial; (iii) a quantum qc-colouring of G by A if E is tracial. For x ∈ { loc , q , qc } , let ξ x ( G ) = min {| A | : ∃ a quantum x-colouring of G by A } be the quantum x -chromatic number of G . Recall [68] that an orthogonal representation of a graph G with vertex set X is a family ( ξ x ) x ∈ X of unit vectors in C k such that x ∼ y = ⇒ h ξ x , ξ y i = 0 . The orthogonal rank ξ ( G ) of G is given by ξ ( G ) = min n k : ∃ an orthogonal representation of G in C k o . Proposition 10.3.
Let G be a graph with vertex set X . The following areequivalent: (i) the graph G has an orthogonal representation in C k ; (ii) there exists a quantum loc -colouring of G by a set A with | A | = k .Proof. (i) ⇒ (ii) Suppose that ( ξ x ) x ∈ X ⊆ C k is an orthogonal representationof G . Let E : D X → M A be the quantum channel given by E ( e x e ∗ x ) = ξ x ξ ∗ x , x ∈ X, and set E = E ⊗ E ♯ ; by Proposition 9.15, E is locally tracial. If x ∼ y then (cid:10) E ( e x e ∗ x ⊗ e y e ∗ y ) , Ω A (cid:11) = X a,b ∈ A D ξ x ξ ∗ x ⊗ (cid:0) ξ y ξ ∗ y (cid:1) t , e a e ∗ b ⊗ e a e ∗ b E = X a,b ∈ A Tr (cid:0) ( ξ x ξ ∗ x ) ( e a e ∗ b ) t (cid:1) Tr (cid:16)(cid:0) ξ y ξ ∗ y (cid:1) t ( e a e ∗ b ) t (cid:17) = X a,b ∈ A Tr (( ξ x ξ ∗ x ) ( e b e ∗ a )) Tr (cid:0)(cid:0) ξ y ξ ∗ y (cid:1) ( e a e ∗ b ) (cid:1) = X a,b ∈ A h ξ x , e a i h e b , ξ x i h ξ y , e b i h e a , ξ y i = X a ∈ A h ξ x , e a i h e a , ξ y i ! X b ∈ A h ξ y , e b i h e b , ξ x i ! = |h ξ x , ξ y i| = 0;thus, E is a quantum loc-colouring of G .(ii) ⇒ (i) Suppose that E : D XX → M AA is a quantum loc-colouring of G , and write E = P kj =1 λ j E j ⊗ E ♯j as a convex combination with positivecoefficients, where E j : D X → M A is a quantum channel, j = 1 , . . . , k .Suppose that x ∼ y . Then k X j =1 λ j D(cid:16) E j ⊗ E ♯j (cid:17) (cid:0) e x e ∗ x ⊗ e y e ∗ y (cid:1) , Ω A E = 0and hence, by the non-negativity of each of the terms of the sum,(10.1) D E ( e x e ∗ x ) ⊗ E ♯ ( e y e ∗ y ) , Ω A E = 0 . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 55
Let ξ x be a unit eigenvector of E ( e x e ∗ x ), corresponding to a positive eigen-value, x ∈ X . Condition (10.1) implies that D ξ x ξ ∗ x ⊗ (cid:0) ξ y ξ ∗ y (cid:1) t , Ω A E = 0,which in turn means, by the arguments in the previous paragraph, that h ξ x , ξ y i = 0. (cid:3) By Proposition 10.3, ξ loc ( G ) = ξ ( G ). Thus, the parameters ξ q and ξ qc can be viewed as quantum versions of the orthogonal rank. Proposition 10.4.
Let G be a graph. Then (i) ξ qc ( G ) ≤ ξ q ( G ) ≤ ξ loc ( G ) , and (ii) ξ x ( G ) ≤ χ x ( G ) for x ∈ { loc , q , qc } .Proof. (i) The inequalities follow from the fact that CQ loc ⊆ CQ q ⊆ CQ qc .(ii) Let p be a synchronous NS correlation that is an x-colouring of G bya set A . By Theorem 9.20, E p ∈ CQ x . By Remark 10.1, E p is G -proper.Thus, ξ x ( G ) ≤ χ x ( G ). (cid:3) Remarks. (i)
There exist graphs G for which ξ ( G ) < χ ( G ) (see e.g. [68]).By Proposition 10.3, for such G we have a strict inequality in Proposition10.4 (ii) in the case x = loc. In [52], an example of a graph G on 13 verticeswas exhibited with the property that ξ ( G ) < χ q ( G ). By Proposition 10.4(i), for this graph G , we have a strict inequality in Proposition 10.4 (ii) inthe case x = q. We do not know if a strict inequality can occur in the casex = qc. (ii) It was shown in [52] that there exists a graph G such that χ q ( G ) <ξ ( G ). By Proposition 10.4 (ii), this implies ξ q ( G ) < ξ ( G ). We do notwhether ξ qc ( G ) can be strictly smaller than ξ q ( G ).We next exhibit a lower bound on ξ qc ( G ) in terms of the Lov´asz number θ ( G ) of G . We refer the reader to [48] for the definition and properties ofthe latter parameter. We denote by K d the complete graph on d vertices.We will need some notation, which will also be essential in Subsection 10.2.If κ ⊆ X × X , let S κ = span (cid:8) e x e ∗ y : ( x, y ) ∈ κ (cid:9) ;thus, S κ is a linear subspace of M X which is a bimodule over the diagonalalgebra D X . We write E ( G ) = { ( x, y ) ∈ X × X : x ≃ y } and E ( G ) = { ( x, y ) ∈ X × X : x ∼ y } , and let S G := S E ( G ) be the graph operator system of G [20], and S G := S E ( G ) be the graph operator anti-system of G [71] (here we use the terminology of[7]). Proposition 10.5.
Let G be a graph with vertex set X . Then ξ qc ( G ) ≥ q | X | θ ( G ) . Moreover, ξ q ( K d ) = ξ qc ( K d ) = d . Proof.
Let A be a C*-algebra, τ : A → C be a trace, ( E x,a,a ′ ) ∈ D X ⊗ M A ⊗A be a semi-classical stochastic A -matrix, and Θ = ( ω x,y ) x,y ∈ X be a quantumqc-colouring of G , such that ω x,y = (cid:0) τ (cid:0) E x,a,a ′ E y,b ′ ,b (cid:1)(cid:1) a,a ′ ,b,b ′ , x, y ∈ X. Assume, without loss of generality, that
A ⊆ B ( H ) as a unital C*-subalgebraand that ξ ∈ H is a unit vector with τ ( u ) = h uξ, ξ i , u ∈ A . Set ξ x,a,a ′ = E x,a,a ′ ξ , x ∈ X , a, a ′ ∈ A ; then ω x,y = (cid:0) h ξ x,a,a ′ , ξ y,b,b ′ i (cid:1) a,a ′ ,b,b ′ , x, y ∈ X. We note that(10.2) X a ∈ A ξ x,a,a = ξ, x ∈ X. In addition, if x ∼ y then(10.3) X a,b ∈ A h ξ x,a,b , ξ y,a,b i = X a,b ∈ A h ω x,y , e a e ∗ b ⊗ e a e ∗ b i = h ω x,y , Ω A i = 0 . Let Q a,a ′ ,b,b ′ = (cid:0) h ξ x,a,a ′ , ξ y,b,b ′ i (cid:1) x,y ∈ X , a, a ′ , b, b ′ ∈ A. Note that, up to an application of the canonical shuffle, (cid:0) Q a,a ′ ,b,b ′ (cid:1) a,a ′ ,b,b ′ = ( ω x,y ) x,y = (cid:0) E Θ ( e x e ∗ x ⊗ e y e ∗ y ) (cid:1) x,y , and hence, after another application of the canonical shuffle, Choi’s Theoremimplies that the linear map Ψ : M AA → M X , given byΨ ( e a e ∗ b ⊗ e a ′ e ∗ b ′ ) = Q a,a ′ ,b,b ′ , a, a ′ , b, b ′ ∈ A, is completely positive. We haveΨ( I AA ) = X a,b ∈ A Ψ ( e a e ∗ a ⊗ e b e ∗ b ) = X a,b ∈ A Q a,b,a,b . By (10.3), x ∼ y = ⇒ h Ψ( I AA ) e x , e y i = 0 . By Theorem 3.1, there exist operators V a,x such that ( V a,x ) a,x is an isom-etry and E x,a,a ′ = V ∗ a,x V a ′ ,x , x ∈ X , a, a ′ ∈ A . Thus, if x ∈ X then h Ψ( I AA ) e x , e x i = X a,b ∈ A h ξ x,a,b , ξ x,a,b i = X a,b ∈ A k E x,a,b ξ k = X a,b ∈ A (cid:13)(cid:13) V ∗ a,x V b,x ξ (cid:13)(cid:13) ≤ X a ∈ A X b ∈ A k V b,x ξ k = | A | X b ∈ A h V b,x ξ, V b,x ξ i = | A | *X b ∈ A V ∗ b,x V b,x ξ, ξ + = | A | . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 57
Write Ψ( I AA ) = D + T , where D is diagonal and T ⊥ S G . We have shownthat D ≤ | A | I X ; thus | A | I X + T ∈ M + X . It follows that k Ψ( I AA ) k ≤ k| A | I X + T k≤ max n k| A | I X + S k : S ∈ S ⊥ G , | A | I X + S ∈ M + X o = | A | θ ( G ) . (10.4)Let J X be the matrix in M X all of whose entries are equal to one. By(10.2), Ψ(Ω A ) = X a,b ∈ A Ψ ( e a e ∗ b ⊗ e a e ∗ b ) = X a,b ∈ A ( h ξ x,a,a , ξ y,b,b i ) x,y = *X a ∈ A ξ x,a,a , X b ∈ A ξ y,b,b +! x,y = J X . (10.5)By (10.4) and (10.5), | X | = k J X k ≤ k Ω A k k Ψ k = | A | k Ψ( I AA ) k ≤ | A | θ ( G ) . Taking the minimum over all | A | completes the proof of the inequality.Realise A = Z d = { , , . . . , d − } and let X = A × A . Let ζ be a primitive | A | -th root of unity. For x = ( a ′ , b ′ ) and y = ( a ′′ , b ′′ ) ∈ X , let ξ x,y = 1 √ d ζ b ′′ ( a ′′ − a ′ ) d − X l =0 ζ ( b ′′ − b ′ ) l e l ⊗ e l − a ′ + a ′′ and write σ x,y = ξ x,y ξ ∗ x,y . We have σ x,y = 1 d d − X l,n =0 ζ ( b ′′ − b ′ )( l − n ) e l e ∗ n ⊗ e l − a ′ + a ′′ e ∗ n − a ′ + a ′′ = 1 d d − X l,n =0 ζ ( b ′′ − b ′ )( l − n ) e l + a ′ e ∗ n + a ′ ⊗ e l + a ′′ e ∗ n + a ′′ . Note that Θ = ( σ x,y ) x,y is a CQNS correlation; indeed,Tr A σ x,y = 1 d d − X l =0 e l + a ′′ e ∗ l + a ′′ = 1 d I A = 1 d d − X l =0 e l + a ′ e ∗ l + a ′ = Tr B σ x,y for all x, y ∈ X . Since X a,b ∈ A h σ x,y , e a e ∗ b ⊗ e a e ∗ b i = 1 d X a,b ∈ A δ a ′ ,a ′′ ζ ( b ′′ − b ′ )( a − b ) = dδ a ′ ,a ′′ δ b ′ ,b ′′ , we have that Θ is K d -proper.We claim that Θ is tracial. To see this, let E x,z,z ′ = ζ ( z ′ − z ) b ′ e z − a ′ e ∗ z ′ − a ′ ∈L ( C A ), x = ( a ′ , b ′ ) ∈ X , z, z ′ ∈ A , and set E x = ( E x,z,z ′ ) z,z ′ ∈ A , x ∈ X . Fix x ∈ X ; then P z ∈ A E x,z,z = I A . Furthermore, if ξ = ( ξ z ) z ∈ A , ξ z ∈ C A , then h E x ξ, ξ i = X z,z ′ ∈ A ζ z ′ b ′ h ξ z ′ , e z ′ − a ′ ih e z − a ′ , ξ z i ζ − zb ′ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X z ∈ A ζ zb ′ h ξ z , e z − a ′ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ . Thus, E = ( E x ) x ∈ X ∈ D X ⊗ M A ⊗ L ( C A ) is a semi-classical stochasticmatrix. Moreover, for x = ( a ′ , b ′ ) , y = ( a ′′ , b ′′ ) ∈ X and z, z ′ , w, w ′ ∈ A wehave Tr (cid:0) E x,z,z ′ E y,w ′ ,w (cid:1) = Tr (cid:16) ζ ( z ′ − z ) b ′ ζ ( w − w ′ ) b ′′ ( e z − a ′ e ∗ z ′ − a ′ )( e w ′ − a ′′ e ∗ w − a ′′ ) (cid:17) = δ z ′ − a ′ ,w ′ − a ′′ δ z − a ′ ,w − a ′′ ζ ( z ′ − z )( b ′ − b ′′ ) = d − X l,n =0 ζ ( b ′′ − b ′ )( l − n ) h e z ′ , e n + a ′ i h e w ′ , e n + a ′′ i h e l + a ′ , e z i h e l + a ′′ , e w i = d − X l,n =0 ζ ( b ′′ − b ′ )( l − n ) (cid:10) ( e l + a ′ e ∗ n + a ′ ) e z ′ ⊗ ( e l + a ′′ e ∗ n + a ′′ ) e w ′ , e z ⊗ e w (cid:11) = d h σ x,y ( e z ′ ⊗ e w ′ ) , e z ⊗ e w i . Therefore Θ is quantum tracial. It follows that ξ q ( K d ) ≤ d ; On the otherhand, θ ( K d ) = 1 and hence ξ qc ( K d ) ≥ d . Proposition 10.4 now impliesthat ξ qc ( K d ) = ξ q ( K d ) = d . (cid:3) Graph homomorphisms.
In this subsection, we consider a quan-tum version of the graph homomorphism game first studied in [51]. Let G and H be graphs with vertex sets X and A , respectively. Recall that thehomomorphism game G → H has Y = X , B = A , and λ ( x, y, a, b ) = 0 ifand only if, either x = y and a = b , or x ∼ y and a b . A synchronousNS correlation p = n ( p ( a, b | x, y )) a,b ∈ A : x, y ∈ X o is thus called a perfectx-strategy for the game G → H if p ∈ C x and x ∼ y, a b = ⇒ p ( a, b | x, y ) = 0 . For a subset κ ⊆ X × X , let P κ : M X → M X be the map given by P κ ( T ) = X ( x,y ) ∈ κ ( e x e ∗ x ) T ( e y e ∗ y ) , T ∈ M X . Thus, P κ is the Schur projection onto S κ ; it can be canonically identified withthe (positive) element P ( x,y ) ∈ κ ( e x e ∗ x ) ⊗ ( e y e ∗ y ) of D XX . We set ( P κ ) ⊥ = P κ c .For a graph G , we write for brevity P G = P E ( G ) . Proposition 10.6.
Let G (resp. H ) be a graph with vertex set X (resp. A ),and p = n ( p ( a, b | x, y )) a,b ∈ A : x, y ∈ X o be a synchronous NS correlation.The following are equivalent: (i) p is a perfect strategy for the homomorphism game G → H ; UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 59 (ii) hN p ( P G ) , ( P H ) ⊥ i = 0 .Proof. (i) ⇒ (ii) We have hN p ( P G ) , ( P H ) ⊥ i = * N p X x ∼ y e x e ∗ x ⊗ e y e ∗ y ! , X a ′ b ′ e a ′ e ∗ a ′ ⊗ e b ′ e ∗ b ′ + = X x ∼ y X a,b ∈ A X a ′ b ′ p ( a, b | x, y ) h e a e ∗ a ⊗ e b e ∗ b , e a ′ e ∗ a ′ ⊗ e b ′ e ∗ b ′ i = X x ∼ y X a ∼ b X a ′ b ′ p ( a, b | x, y ) h e a e ∗ a ⊗ e b e ∗ b , e a ′ e ∗ a ′ ⊗ e b ′ e ∗ b ′ i = 0 . (ii) ⇒ (i) If x ∼ y and a b then e x e ∗ x ⊗ e y e ∗ y ≤ P G and e a e ∗ a ⊗ e b e ∗ b ≤ ( P H ) ⊥ .By the monotonicity of the pairing, p ( a, b | x, y ) = (cid:10) N p (cid:0) e x e ∗ x ⊗ e y e ∗ y (cid:1) , e a e ∗ a ⊗ e b e ∗ b (cid:11) ≤ hN p ( P G ) , ( P H ) ⊥ i = 0 . (cid:3) General operator systems in M X were considered in [20] as a quantumversions of graphs (noting that S G is an operator system), while operatoranti-systems (that is, selfadjoint subspaces of M X each of whose elementshas trace zero [7]) were proposed as such a quantum version in [71] (notingthat S G is an operator anti-system). Note that one can pass from any ofthe two notions to the other by taking orthogonal complements. Due to thespecific definition of QNS correlations in [21], employed also here, it will beconvenient to use a slightly different (but equivalent) perspective on non-commutative graphs, which we now describe. Let Z be a finite set, H = C Z , H d be its dual space and d : H → H d be the map given by d( ξ )( η ) = h η, ξ i ;we write ξ d = d( ξ ). Note that, if T ∈ L ( H ) then(10.6) T d ξ d = ( T ∗ ξ ) d , T ∈ L ( H ) . Let θ : H ⊗ H → L ( H d , H ) be the linear map given by θ ( ξ ⊗ η )( ζ d ) = h ξ, ζ i η, ζ ∈ H. By (10.6),(10.7) θ (( S ⊗ T ) ζ ) = T θ ( ζ ) S d , ζ ∈ H ⊗ H, S, T ∈ L ( H ) . We denote by m : H ⊗ H → C the map given bym( ζ ) = * ζ, X z ∈ Z e z ⊗ e z + , ζ ∈ H ⊗ H. Let also f : H ⊗ H → H ⊗ H be the flip operator given by f ( ξ ⊗ η ) = η ⊗ ξ .Note that, if ξ, η, ζ , ζ ∈ H then h θ ( ξ ⊗ η ) ∗ ( ζ ) , ζ d2 i = h ζ , θ ( ξ ⊗ η ) ζ d2 i = h ζ , h ξ, ζ i η i = h ζ , ξ ih ζ , η i = h ξ d , ζ d2 ih ζ , η i = hh ζ , η i ξ d , ζ d2 i and henced − ( θ ( ξ ⊗ η ) ∗ (d − ( ζ d1 )) = d − ( h ζ , η i ξ d ) = h η, ζ i ξ = θ ( η ⊗ ξ )( ζ d1 );thus,(10.8) d − ◦ θ ( ζ ) ∗ ◦ d − = ( θ ◦ f )( ζ ) , ζ ∈ H ⊗ H. In addition, X z ∈ Z h θ ( ξ ⊗ η )(d( e z )) , e z i = X z ∈ Z h ξ, e z ih η, e z i = m( ξ ⊗ η ) , and hence(10.9) m( ζ ) = X z ∈ Z h ( θ ( ζ ) ◦ d)( e z ) , e z i , ζ ∈ H ⊗ H. Definition 10.7.
A linear subspace
U ⊆ H ⊗ H is called skew if m( U ) = { } and symmetric if f ( U ) = U . Suppose that U is a symmetric skew subspace of H ⊗ H . Let S U = θ ( U );by (10.8) and (10.9), the subspace S U of L ( H d , H ) satisfies • T ∈ S U = ⇒ d − ◦ T ∗ ◦ d − ∈ S U , and • T ∈ S U = ⇒ P z ∈ Z h ( T ◦ d)( e z ) , e z i = 0.We call a subspace of L ( H d , H ) satisfying these properties a twisted op-erator anti-system . Conversely, given a twisted operator anti-system S ⊆L ( H d , H ), (10.8) and (10.9) imply that the subspace U S = θ − ( S ) of H ⊗ H is symmetric and skew. Given a graph G , let U G = span { e x ⊗ e y : x ∼ y } ;it is clear that U G is a symmetric skew subspace of C X ⊗ C X . We thus con-sider symmetric skew subspaces of C X ⊗ C X as a non-commutative versionof graphs.We write P U for the orthogonal projection from C X ⊗ C X onto U . Let U ⊥ ⊂ (cid:0) C X ⊗ C X (cid:1) d be the annihilator of U and write P U ⊥ ∈ L (cid:0) ( C X ⊗ C X ) d (cid:1) for the orthogonal projection onto U ⊥ . Observe that ζ d ∈ U ⊥ if and only if ζ belongs to the orthogonal complement U ⊥ of U in C X ⊗ C X . Thus, for ζ ∈ H ⊗ H we have P U ⊥ ζ d = ζ d ⇔ P ⊥U ( ζ ) = ζ ⇔ h P ⊥U ( ζ ) , ζ d i = 1 ⇔ h ζ, ( P ⊥U ) d ( ζ d ) i = 1 ⇔ ( P ⊥U ) d ( ζ d ) = ζ d , and hence(10.10) P U ⊥ = ( P ⊥U ) d . Let A be a finite set and ω ∈ M A . Writing f ω for the functional on M A given by f ω ( ρ ) = Tr( ρω t ), we have that the map ω → f ω is a complete orderisomorphism from M A onto M d A (see e.g. [63, Theorem 6.2]). On the otherhand, the map ω d ω t is a *-isomorphism from L (cid:0) ( C A ) d (cid:1) onto M A . Thecomposition of these maps, ω d f w t , is thus a complete order isomorphism UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 61 from L (cid:0) ( C A ) d (cid:1) onto M d A . In the sequel, we identify these two spaces; notethat, via this identification,(10.11) h ρ, ω d i = h ρ, ω t i = Tr( ρω ) , ρ, ω ∈ M A . Definition 10.8.
Let X and A be finite sets and U ⊆ C X ⊗ C X , V ⊆ C A ⊗ C A be symmetric skew subspaces. A QNS correlation Γ : M XX → M AA is called (i) a quantum commuting homomorphism from U to V (denoted U qc →V ) if Γ is tracial and (10.12) h Γ ( P U ) , P V ⊥ i = 0 . (ii) a quantum homomorphism from U to V (denoted U q → V ) if Γ isquantum tracial and (10.12) holds; (iii) a local homomorphism from U to V (denoted U loc → V ) if Γ is locallytracial and (10.12) holds. Given operator anti-systems
S ⊆ M X and T ⊆ M A , Stahlke [71] definesa non-commutative graph homomorphism from S to T to be a quantumchannel Φ : M X → M A with family { M i } mi =1 of Kraus operators, such that M i S M ∗ j ⊆ T , i, j = 1 , . . . , m ; if such Φ exists, he writes S → T . Theappropriate version of this notion for twisted operator anti-systems – directlymodelled on Stahlke’s definition – is as follows. For T ∈ M Z , we write T = T ∗ t for the conjugated matrix of T . Definition 10.9.
Let X and A be finite sets, and S ⊆ L (cid:0) ( C X ) d , C X (cid:1) and T ⊆ L (cid:0) ( C A ) d , C A (cid:1) be twisted operator anti-systems. A homomorphism from S into T is a quantum channel Φ : M X → M A , Φ( T ) = m X i =1 M i T M ∗ i , such that M j S M d i ⊆ T , i, j = 1 , . . . , m. If S and T are twisted operator anti-systems, we write S → T as in [71]to denote the existence of a homomorphism from S to T . Proposition 10.10.
Let X and A be finite sets and U ⊆ C X ⊗ C X , V ⊆ C A ⊗ C A be symmetric skew spaces. Then U loc → V if and only if S U → S V .Proof. Suppose that U loc → V and let Γ be a locally tracial QNS correlationfor which (10.12) holds. By Theorem 9.9, there exist quantum channelsΦ j : M X → M A , j = 1 , . . . , k , such that Γ = P kj =1 λ j Φ j ⊗ Φ ♯j as a convexcombination. We have k X j =1 λ j D(cid:16) Φ j ⊗ Φ ♯j (cid:17) ( P U ) , P V ⊥ E = h Γ ( P U ) , P V ⊥ i = 0; since each of the terms in the sum on the left hand side is non-negative,selecting j with λ j > j , we have(10.13) D(cid:16) Φ ⊗ Φ ♯ (cid:17) ( P U ) , P V ⊥ E = 0 . Let Φ( ω ) = P mi =1 M i ωM ∗ i , ω ∈ M X , be a Kraus representation of Φ. For ω ∈ M X , we haveΦ ♯ ( ω ) = m X i =1 (cid:0) M i ω t M ∗ i (cid:1) t = m X i =1 ( M ∗ i ) t ωM t i = m X i =1 M i ωM ∗ i . It follows that(10.14) (cid:16) Φ ⊗ Φ ♯ (cid:17) ( ρ ) = m X i,j =1 ( M i ⊗ M j ) ρ ( M i ⊗ M j ) ∗ , ρ ∈ M XX . Let ξ ∈ U and η ∈ V ⊥ be unit vectors; then ξξ ∗ ≤ P U . In addition, η d = P V ⊥ η d and hence ( ηη ∗ ) d = η d ( η d ) ∗ ≤ P V ⊥ ; thus, (10.13) implies D(cid:16) Φ ⊗ Φ ♯ (cid:17) ( ξξ ∗ ) , ( ηη ∗ ) d E = 0 . By (10.14) and positivity, (cid:10) ( M i ⊗ M j )( ξξ ∗ )( M i ⊗ M j ) ∗ , ( ηη ∗ ) d (cid:11) = 0 , i, j = 1 , . . . , m, which, by (10.11), means that (cid:10) ( M i ⊗ M j ) ξ, η (cid:11) = 0 , i, j = 1 , . . . , m. Thus, ( M i ⊗ M j ) ξ ∈ V for every ξ ∈ U and, by (10.7), M j θ ( ξ ) M d i = θ (( M i ⊗ M j ) ξ ) ∈ S V , ξ ∈ U , that is, S U → S V .Conversely, suppose that Φ : M X → M A is a quantum channel with afamily of Kraus operators ( M i ) mi =1 ⊆ L ( C X , C A ) such that M j S U M d i ⊆ S V , i, j = 1 , . . . , m . The previous paragraphs show that D (Φ ⊗ Φ ♯ )( ξξ ∗ ) , ( ηη ∗ ) d E = Tr((Φ ⊗ Φ ♯ )( ξξ ∗ )( ηη ∗ )) = 0for all unit vectors ξ ∈ U , η ∈ V ⊥ . It follows that(Φ ⊗ Φ ♯ )( ξξ ∗ ) = ( ηη ∗ ) ⊥ (Φ ⊗ Φ ♯ )( ξξ ∗ )( ηη ∗ ) ⊥ , for all unit vectors η ∈ V ⊥ . Taking infimum over all such η , we obtain(Φ ⊗ Φ ♯ )( ξξ ∗ ) = P V (Φ ⊗ Φ ♯ )( ξξ ∗ ) P V , for all unit vectors ξ ∈ U . Thus, by (10.11) and (10.10),Tr (cid:16)(cid:16) Φ ⊗ Φ ♯ (cid:17) ( ξξ ∗ ) P ⊥V (cid:17) = D (Φ ⊗ Φ ♯ )( ξξ ∗ ) , P V ⊥ E = 0 , for all unit vectors ξ ∈ U . Writing P U = P li =1 ξ i ξ ∗ i , where ( ξ i ) li =1 is anorthonormal basis of U , we obtain (cid:10) (Φ ⊗ Φ ♯ )( P U ) , P V ⊥ (cid:11) = 0. (cid:3) UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 63
For graphs G and H , write G → H if there exists a homomorphism from G to H . The next corollary justifies viewing the symmetric skew spaces asnon-commutative graphs. Corollary 10.11.
Let G and H be graphs. We have that G → H if andonly if U G loc → U H .Proof. Write X and A for the vertex sets of G and H , respectively. Assumethat G → H . By [71], S G → S H . Write { M i } mi =1 for the set of Krausoperators such that M i S G M ∗ j ⊆ S H , i, j = 1 , . . . , m . Let J X : C X → C X bethe map given by J X ( η ) = ¯ η . Then θ ( e x ⊗ e y ) = J X ◦ e y e ∗ x ◦ d − , x, y ∈ X .Therefore,( J A ◦ M i ◦ J X )( J X ◦ S G ◦ d − )(d ◦ M ∗ j ◦ d − ) ⊆ J A ◦ S H ◦ d − , implying M i S U G M d j ⊆ S U H ; by Proposition 10.10, U G loc → U H . The conversefollows after reversing the arguments. (cid:3) General quantum non-local games.
We write P M for the projec-tion lattice of a von Neumann algebra M , and denote as usual by ∨ (resp. ∧ ) the join (resp. the wedge) operation in P M ; thus, for P , P ∈ P M , theprojection P ∨ P (resp. P ∧ P ) has range the closed span (resp. the inter-section) of the ranges of P and P . If M and N are von Neumann algebras,a map ϕ : P M → P N is called join continuous if ϕ ( ∨ i ∈ I P i ) = ∨ i ∈ I ϕ ( P i ) forany family { P i } i ∈ I ⊆ P M . Note that if M is finite dimensional, then joincontinuity is equivalent to the preservation of finite joins.Let H be a Hilbert space and P be an orthogonal projection on H withrange U . As in Subsection 10.2, we denote by U ⊥ the annihilator of U in thespace H d , and by P ⊥ – the orthogonal projection on H d with range U ⊥ . Definition 10.12.
Let X , Y , A and B be finite sets. (i) A map ϕ : P M XY → P M AB (resp. ϕ : P D XY → P M AB , ϕ : P D XY →P D AB ) is called a quantum non-local game (resp. a classical-to-quantum non-local game , a classical non-local game ) if ϕ is joincontinuous and ϕ (0) = 0 . We say that such ϕ is a game from XY to AB . (ii) A QNS (resp. CQNS, NS) correlation Λ is called a perfect strat-egy for the quantum (resp. classical-to-quantum, classical) non-localgame ϕ if (10.15) h Λ( P ) , ϕ ( P ) ⊥ i = 0 , P ∈ P M XY ( resp. P ∈ P D XY ) . Remark 10.13. (i)
Join continuous zero-preserving maps ϕ : P B ( H ) →P B ( K ) , where H and K are Hilbert spaces, were first considered by J. A.Erdos in [25]. They are equivalent to bilattices introduced in [69] – that is,subsets B ⊆ P B ( H ) × P B ( K ) such that ( P, , (0 , Q ) ∈ B for all P ∈ P B ( H ) , Q ∈ P B ( K ) , and ( P , Q ) , ( P , Q ) ∈ B ⇒ ( P ∨ P , Q ∧ Q ) ∈ B and ( P ∧ P , Q ∨ Q ) ∈ B . Thus, quantum non-local games (resp. classical-to-quantum non-local games, classical non-local games) can be alternativelydefined as bilattices; we have chosen to use maps instead because they aremore convenient to work with when compositions are considered (see Defi-nition 10.15).Conditions (10.15) are reminiscent of J. A. Erdos’ characterisation [25] ofreflexive spaces of operators, introduced by L. N. Loginov and V. S. Shulmanin [47]. As shown in [25], a subspace S ⊆ B ( H, K ) ( H and K being Hilbertspaces) is reflexive in the sense of [47] if and only if there exists a joincontinuous zero-preserving map ϕ : P B ( H ) → P B ( K ) such that S coincideswith the the spaceOp( ϕ ) = n T ∈ B ( H, K ) : ϕ ( P ) ⊥ T P = 0 , for all P ∈ P B ( H ) o . (ii) The quantum (resp. classical-to-quantum, classical) non-local game ϕ with ϕ ( P ) = I AB for every non-zero P ∈ P M XY (resp. P ∈ P D XY ) will bereferred to as the empty game . It is clear that the set of perfect strategiesfor the empty game coincides with the class of all no-signalling correlations. (iii) Let G be a graph with vertex set X and A be a finite set. Thequantum graph colouring game considered in Subsection 10.1 is the classical-to-quantum non-local game ϕ : P D XX → P M AA , given by ϕ ( e x e ∗ x ⊗ e y e ∗ y ) = ( | A | Ω ⊥ A if x ∼ yI otherwise.Similarly, letting U ⊆ C X ⊗ C X and V ⊆ C A ⊗ C A be symmetric skewspaces, we define the homomorphism game U → V to be the quantum non-local game ψ , given by ψ ( P ) = P V if 0 = P ≤ P U P = 0 I otherwise.For x ∈ { loc , q , qc } , we have that U x → V if and only if the game U → V hasa perfect strategy of class Q x .Let ( X, Y, A, B, λ ) be a non-local game. For a subset α ⊆ X × Y , let P α ∈ P D XY be the projection with range span { e x ⊗ e y : ( x, y ) ∈ α } . For( x, y ) ∈ X × Y , let β x,y ( λ ) = { ( a, b ) ∈ A × B : λ ( x, y, a, b ) = 1 } . We associate with λ the (unique) classical non-local game ϕ λ : P D XY →P D AB determined by the requirement ϕ λ (cid:0) P { ( x,y ) } (cid:1) = P β x,y ( λ ) , ( x, y ) ∈ X × Y. Proposition 10.14.
An NS correlation p is a perfect strategy for the non-local game (with rule function) λ if and only if N p is a perfect strategy for ϕ λ . UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 65
Proof.
Note that, if ( x, y ) ∈ X × Y then (cid:0) P β x,y ( λ ) (cid:1) ⊥ has range span { e a e ∗ a ⊗ e b e ∗ b : λ ( x, y, a, b ) = 0 } . As in Proposition 10.6, it is thus easily seen that p is a perfect strategy for λ if and only if D N p (cid:0) P { ( x,y ) } (cid:1) , (cid:0) P β x,y ( λ ) (cid:1) ⊥ E = 0 , ( x, y ) ∈ X × Y. Assume that p is a perfect strategy for λ . For a projection P ∈ D XY ,write P = ∨{ P { ( x,y ) } : P ( e x ⊗ e y ) = e x ⊗ e y } ; then ϕ λ ( P ) = ∨{ P β x,y ( λ ) : P ( e x ⊗ e y ) = e x ⊗ e y } . Thus, (cid:10) N p ( P { ( x,y ) } ) , ϕ λ ( P ) ⊥ (cid:11) = 0 for all pairs ( x, y ) with P ( e x ⊗ e y ) = e x ⊗ e y . Taking the join over all those ( x, y ), we conclude that hN p ( P ) , ϕ λ ( P ) ⊥ i =0. The converse is direct from the first paragraph. (cid:3) Definition 10.15.
Let X , Y , A , B , Z and W be finite sets and ϕ (resp. ϕ ) be a game from XY to AB (resp. from AB to ZW ). The composition of ϕ and ϕ is the game ϕ ◦ ϕ from XY to ZW . It is clear that ϕ ◦ ϕ is well-defined in all cases except when ϕ is aquantum game, while ϕ is a classical-to-quantum game. Lemma 10.16.
Let X , A and Z be finite sets, H and K be Hilbert spacesand E ∈ M X ⊗ M A ⊗ B ( H ) and F ∈ M A ⊗ M Z ⊗ B ( K ) be stochastic operatormatrices. Set G x,x ′ ,z,z ′ = X a,a ′ ∈ A F a,a ′ ,z,z ′ ⊗ E x,x ′ ,a,a ′ , x, x ′ ∈ X, z, z ′ ∈ Z. Then G = ( G x,x ′ ,z,z ′ ) x,x ′ ,z,z ′ is a stochastic operator matrix in M X ⊗ M Z ⊗B ( K ⊗ H ) .Proof. Let V = ( V a,x ) a,x (resp. W = ( W z,a ) z,a ) be an isometry from H X (resp. K A ) to ˜ H A (resp. ˜ K Z ) for some Hilbert space ˜ H (resp. ˜ K ), suchthat E x,x ′ ,a,a ′ = V ∗ a,x V a ′ ,x ′ and F a,a ′ ,z,z ′ = W ∗ z,a W z ′ ,a ′ for all x, x ′ ∈ X , a, a ′ ∈ A and z, z ′ ∈ Z . Set U z,x = X a ∈ A W z,a ⊗ V a,x , x ∈ X, z ∈ Z. For x, x ′ ∈ X , we have X z ∈ Z U ∗ z,x U z,x ′ = X z ∈ Z X a ∈ A W ∗ z,a ⊗ V ∗ a,x ! X a ′ ∈ A W z,a ′ ⊗ V a ′ ,x ′ ! = X z ∈ Z X a,a ′ ∈ A W ∗ z,a W z,a ′ ⊗ V ∗ a,x V a ′ ,x ′ = X a,a ′ ∈ A X z ∈ Z W ∗ z,a W z,a ′ ! ⊗ V ∗ a,x V a ′ ,x ′ = X a,a ′ ∈ A δ a,a ′ I K ⊗ V ∗ a,x V a ′ ,x ′ = X a ∈ A I K ⊗ V ∗ a,x V a,x ′ = δ x,x ′ I K ⊗ I H ;thus, ( U z,x ) z,x is an isometry from ( K ⊗ H ) X into ( ˜ K ⊗ ˜ H ) Z . In addition,for x, x ′ ∈ X and z, z ′ ∈ Z , we have U ∗ z,x U z ′ ,x ′ = X a ∈ A W ∗ z,a ⊗ V ∗ a,x ! X a ′ ∈ A W z ′ ,a ′ ⊗ V a ′ ,x ′ ! = X a,a ′ ∈ A F a,a ′ ,z,z ′ ⊗ E x,x ′ ,a,a ′ = G x,x ′ ,z,z ′ . By Theorem 3.1, G is a stochastic operator matrcx acting on K ⊗ H . (cid:3) We call the stochastic operator matrix G from Lemma 10.16 the compo-sition of F and E and denote it by F ◦ E . Theorem 10.17.
Let ϕ (resp. ϕ ) be a quantum game from XY to AB (resp. from AB to ZW ) and x ∈ { loc , q , qa , qc , ns } . (i) If Γ i is a perfect strategy for ϕ i from the class Q x , i = 1 , , then Γ ◦ Γ is a perfect strategy for ϕ ◦ ϕ from the class Q x . (ii) Asume that X = Y , A = B and Z = W . If Γ i is a perfect tracial(resp. quantum tracial, locally tracial) strategy for ϕ i , i = 1 , , then Γ ◦ Γ is a perfect tracial (resp. quantum tracial, locally tracial)strategy for ϕ ◦ ϕ .Proof. First note that if Γ i is a QNS correlation then so is Γ ◦ Γ . Indeed,suppose that ρ ∈ M XY is such that Tr X ρ = 0. By Remark 2.1, Tr A Γ ( ρ ) =0, and hence, again by Remark 2.1, Tr Z (Γ (Γ ( ρ )) = 0.Suppose that Γ i ∈ Q qc , i = 1 ,
2. Let ( E ( i ) , F ( i ) ) be a commuting pairof stochastic operator matrices acting on a Hilbert space H i , and σ i be anormal state on B ( H i ), such that Γ i = Γ E ( i ) · F ( i ) ,σ i , i = 1 ,
2. Write E (1) = (cid:16) E (1) x,x ′ ,a,a ′ (cid:17) , F (1) = (cid:16) F (1) y,y ′ ,b,b ′ (cid:17) , E (2) = (cid:16) E (2) a,a ′ ,z,z ′ (cid:17) and F (2) = (cid:16) F (2) b,b ′ ,w,w ′ (cid:17) .Set H = H ⊗ H , σ = σ ⊗ σ , E = E (2) ◦ E (1) and F = F (2) ◦ F (1) ;note that, by Lemma 10.16, E and F are stochastic operator matrices. It is UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 67 straightforward that (
E, F ) is a commuting pair. Write E = ( E x,x ′ ,z,z ′ ) and F = ( F y,y ′ ,w,w ′ ). Note that X a,a ′ ,b,b ′ D E (1) x,x ′ ,a,a ′ F (1) y,y ′ ,b,b ′ , σ E D E (2) a,a ′ ,z,z ′ F (2) b,b ′ ,w,w ′ , σ E = X a,a ′ ,b,b ′ D(cid:16) E (2) a,a ′ ,z,z ′ ⊗ E (1) x,x ′ ,a,a ′ (cid:17) (cid:16) F (2) b,b ′ ,w,w ′ ⊗ F (1) y,y ′ ,b,b ′ (cid:17) , σ ⊗ σ E = (cid:10) E x,x ′ ,z,z ′ F y,y ′ ,w,w ′ , σ (cid:11) , and hence(Γ ◦ Γ ) (cid:0) e x e ∗ x ′ ⊗ e y e ∗ y ′ (cid:1) = X a,a ′ ,b,b ′ D E (1) x,x ′ ,a,a ′ F (1) y,y ′ ,b,b ′ , σ E Γ ( e a e ∗ a ′ ⊗ e b e ∗ b ′ )= X z,z ′ ,w,w ′ X a,a ′ ,b,b ′ h E (1) x,x ′ ,a,a ′ F (1) y,y ′ ,b,b ′ , σ ih E (2) a,a ′ ,z,z ′ F (2) b,b ′ ,w,w ′ , σ i e z e ∗ z ′ ⊗ e w e ∗ w ′ = X z,z ′ ,w,w ′ (cid:10) E x,x ′ ,z,z ′ F y,y ′ ,w,w ′ , σ (cid:11) e z e ∗ z ′ ⊗ e w e ∗ w ′ ;thus, Γ ◦ Γ = Γ E · F,σ .If Γ i ∈ Q q , i = 1 ,
2, then the arguments in the previous paragraph –replacing operator products by tensor products as necessary – show thatΓ ◦ Γ ∈ Q q . By the continuity of the composition, the assumptions Γ i ∈Q qa , i = 1 ,
2, imply that Γ ◦ Γ ∈ Q qa . Finally, assume that Γ i ∈ Q loc , i = 1 ,
2, and write Γ i = P m i k =1 λ ( i ) k Φ ( i ) k ⊗ Ψ ( i ) k as a convex combination, whereΦ ( i ) k : M X → M A and Ψ ( i ) k : M Y → M B are quantum channels, i = 1 , ◦ Γ = m X k =1 m X l =1 λ (1) k λ (2) l (cid:16) Φ (2) l ◦ Φ (1) k (cid:17) ⊗ (cid:16) Ψ (2) l ◦ Ψ (1) k (cid:17) as a convex combination, and hence Γ ◦ Γ ∈ Q loc .Suppose that Γ i is a tracial QNS correlation; thus, there exist unitalC*-algebras A and A , traces τ and τ on A and A , respectively, andstochastic matrices E (1) ∈ M X ⊗ M A ⊗ A and E (2) ∈ M A ⊗ M Z ⊗ A , suchthat Γ i = Γ E ( i ) ,τ i , i = 1 ,
2. The arguments given for (i) show thatΓ ◦ Γ = Γ E (2) ◦ E (1) ,τ ⊗ τ , where τ ⊗ τ is the product trace on A ⊗ min A ; E (2) ◦ E (1) is considered asa stochastic A ⊗ min A -matrix (note that we identify (cid:0) E (2) ◦ E (1) (cid:1) op with E (2)op ◦ E (1)op in the natural way).It remains to show that if Γ i is a perfect strategy for ϕ i , i = 1 , ◦ Γ is a perfect strategy for ϕ ◦ ϕ . Let P ∈ P M XY and ω bea pure state with ω ≤ P . Then Γ ( ω ) = ϕ ( P )Γ ( ω ) ϕ ( P ) and henceΓ ( ω ) ≤ ϕ ( P ). Similarly, for any pure state σ with σ ≤ ϕ ( P ) we have Γ ( σ ) = ϕ ( ϕ ( P ))Γ ( σ ) ϕ ( ϕ ( P )), giving h (Γ ( σ ) , ( ϕ ◦ ϕ )( P ) ⊥ i = 0. Inparticular, h (Γ ◦ Γ )( ω ) , ( ϕ ◦ ϕ )( P ) ⊥ i = 0 . As in the proof of Proposition 10.10, this yields h (Γ ◦ Γ )( P ) , ( ϕ ◦ ϕ )( P ) ⊥ i = 0 , establishing the claim. (cid:3) Suppose that p (resp. p ) is an NS correlation from XY to AB (resp.from AB to ZW ). It is straightforward to verify that the correlation p with N p = N p ◦ N p is given by p ( z, w | x, y ) = X a ∈ A X b ∈ B p ( z, w | a, b ) p ( a, b | x, y );we write p = p ◦ p . Such compositions were first studied in [57]. For anon-local game from XY to AB (resp. from AB to ZW ) with rule function λ (resp. λ ), let λ ◦ λ : X × Y × Z × W → { , } be given by( λ ◦ λ )( x, y, z, w ) = 1 ⇔ ∃ ( a, b ) s.t. λ ( x, y, a, b ) = λ ( a, b, z, w ) = 1 . Combining Theorem 10.17 with classical reduction and Proposition 10.14,we obtain the following perfect strategy version of [57, Proposition 3.5],which simultaneously extends the graph homomorphism transitivity resultscontained in [57, Theorem 3.7].
Corollary 10.18.
Let λ (resp. λ ) be the rule functions of non-local gamesfrom XY to AB (resp. from AB to ZW ) and x ∈ { loc , q , qa , qc , ns } . If p i is a perfect strategy for λ i from the class C x , i = 1 , , then p ◦ p is a perfectstrategy for λ ◦ λ from the class C x . Combining Theorem 10.17 with Remark 10.13 (iii) yields the followingtransitivity result; in view of Proposition 10.10, it extends [71, Proposition9].
Corollary 10.19.
Let X , A and Z be finite sets, U ⊆ C X ⊗ C X , V ⊆ C A ⊗ C A and W ⊆ C Z ⊗ C Z be symmetric skew spaces, and x ∈ { loc , q , qc } .If U x → V and V x → W then U x → W . Acknowledgement.
It is our pleasure to thank Michael Brannan, Li Gao,Marius Junge, Dan Stahlke and Andreas Winter for fruitful discussions onthe topic of this paper.
Note.
After the paper was completed, we became aware of the work [9],in which the authors define quantum-to-classical no-signalling correlationsand study a version of the homomorphism game from a non-commutative toa classical graph. Although there are similarities between our approaches,there is no duplication of results in the current paper with those in [9].
UANTUM NO-SIGNALLING CORRELATIONS AND NON-LOCAL GAMES 69
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