Quantum no-signalling correlations and non-local games

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system, and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.


Introduction
Non-local games [18] have in the past decade acquired significant prominence, demonstrating both the power and limitations of quantum entanglement. 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 input a 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 by a predicate function λ : X × Y × A × B → {0, 1}. A probabilistic strategy is a family p = {(p(a, b|x, y)) (a,b)∈A×B : (x, y) ∈ X × Y } of probability distributions, one for each input pair (x, y), where the value p(a, b|x, y) denotes the probability that the players spit out the output (a, b) given they have received the input (x, y). Such families p are in addition required to satisfy a no-signalling condition, which ensures no communication between the 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, winning) strategies exist, while shared entanglement can produce perfect quantum strategies. Such strategies consist of two parts: a unit vector ξ in the tensor product H A ⊗ H B of two finite dimensional Hilbert spaces (representing the joint physical system of the players), and local measurement operators (E x,a ) x,a (for Alice) and (F y,b ) y,b (for Bob), leading to the probabilities p(a, b|x, y) = (E x,a ⊗ F y,b )ξ, ξ . Employing the commuting model of Quantum Mechanics leads, on the other hand, to the broader set of quantum commuting strategies, whose underlying no-signalling correlations arise from mutually 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 chain of 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 the quantum class C q -known, due to the work of Slofstra [70] (see also [22]) to be strictly larger than C q -and C ns is the class of all no-signalling correlations, playing a fundamental role in generalised probabilistic theories [3,4].
The long-standing question of whether C qa coincides with the class C qc of all quantum commuting correlations, known as Tsirelson's problem, was recently settled in the negative in [37]. Due to the works [38] and [56], this also 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 Embedding Problem in [37] follows complexity theory routes, and it remains of great interest if an operator algebraic approach is within reach. The classes of correlations we introduce are wider and hence may offer more flexibility in looking for counterexamples.
Our second source of motivation is the development of non-local games with quantum inputs and quantum outputs. A number of versions of quantum games have already been examined. In [19], the authors studied the computability and the parallel repetition behaviour of the entangled value of a rank one quantum game, where the players receive quantum inputs from the verifier, but a measurement is taken against a rank one projection to determine the likelihood of winning. In [30], the focus is on multiple round quantum strategies that are available to players with quantum memory, while the quantum-classical and extended non-local games considered in [67] both have classical outputs (see also [14]). Here, we propose a framework for quantum-to-quantum non-local games, which generalises directly (classical) non-local games. This allows us to define a quantum version of the graph homomorphism game (see [22,51,52,62]), and leads to notions 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 correlations given by Duan and Winter in [21]. Note that no-signalling (NS) correlations correspond 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 marginal channels are well-defined. In Section 4, we define the quantum versions of the 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 valued measures, which we develop in Section 3. The stochastic operator matrices defined therein replace the families (E x,a ) x∈X,a∈A of measurement operators that play a crucial role in the definitions of the classical classes (1.1). In Section 5, we define a universal operator system T X,A , whose concrete representations on Hilbert spaces are precisely determined by stochastic operator matrices. Our route passes through the definition of a universal ternary ring of operators V X,A of a given A × X-block operator isometry, which is a 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); this is 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 the elements of these classes and states on operator system tensor products. We see that, similarly to the case of classical NS correlations [49], each QNS correlation of the class Q qc arises from a state on the commuting tensor product T X,A ⊗ c T Y,B , and that similar descriptions hold for the rest of the aforementioned classes. Along with the hierarchy (1.2), we introduce an intermediate 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-quantum no-signalling (CQNS) correlations. We define their universal operator system, and provide analogous characterisations in terms of states on its tensor products; this is achieved in Section 7. In Section 8, we point out the canonical surjections Q x → CQ x → C x (where x denotes any specific correlation class from the set {loc, q, qa, qc, ns}). Combined with the separation results at 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 wellknown: its elements are precisely the local operations and shared randomness (LOSR) channels (see e.g. [73, p. 358]). Thus, the channels from Q q can be thought of as entanglement assisted LOSR transformations, and a similar interpretation can be adopted for the higher terms of (1.2).
The notion of a synchronous NS correlation [61] is of crucial importance when correlations are employed as strategies of non-local games. Here, we assume 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 product of |X| copies of the |A|-dimensional abelian C*-algebra. In Section 9, we propose two quantum versions of synchronicity. Fair correlations are defined in operational terms, but display a lower level of relevance than tracial correlations, which are defined operator algebraically, via traces on the universal C*-algebra of a stochastic operator matrix. Tracial QNS correlations are closely related to factorisable channels [1] which have been used to produce counterexamples to the asymptotic Birkhoff conjecture [31]. More precisely, if one restricts attention to QNS correlations that arise from the Brown algebra as opposed to the ternary ring of operators V X,A , then the tracial QNS correlations are precisely the couplings of a pair of factorisable channels with equal terms.
Restricted to CQNS and NS correlations, traciality produces classes of correlations that strictly contain synchronous NS correlations. The difference between synchronous and tracial NS correlations can be heuristically compared to that between projection and positive operator valued measures. The operational significance of tracial QNS, tracial CQNS and tracial NS correlations arises from the preservation of appropriate classes of states, which quantise the symmetry possessed by the classical pure states supported on the diagonal of a matrix algebra. The ground class, of locally reciprocal states, turns out to be a twisted version of de Finetti states [17]. Thus, the higher classes of quantum reciprocal and C*-reciprocal states can be 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 be used as strategies for quantum-to-quantum and classical-to-quantum nonlocal games. This is not an exhaustive treatment, and is rather intended to summarise several directions and provide a general context that we hope to investigate subsequently. In Subsection 10.1, we show that, when compared to NS correlations, CQNS correlations provide a significant advantage in the graph colouring game [15]. Employing the CQNS classes, we define new versions 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 chromatic 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 versions of this classical graph parameter. We show that ξ qc (G) does not degenerate, in that it is always lower bounded by d/θ(G), where d is the number of vertices of G and θ(G) is its Lovász number.
In Subsection 10.2, we define a non-commutative version of the graph homomorphism game [51]. We show that its perfect strategies from the class Q loc correspond precisely to non-commutative graph homomorphisms in the sense of Stahlke [71]. Thus, the perfect strategies from the larger classes in (1.2) can be thought of as quantum non-commutative graph homomorphisms. We note that special cases have been previously considered in [8] and [55]. The treatment in the latter papers was restricted to non-commutative graph isomorphisms, and the suggested approach was operator-algebraic. We remedy this by suggesting, up to our knowledge, the first operational approach to non-commutative graph homomorphisms, thus aligning the noncommutative case with the case of quantum homomorphisms between classical graphs [51].
Finally, in Subsection 10.3, we introduce a quantum version of non-local games that contains as a special case the games considered in the previous subsections. To this end, we view the rule predicate as a map between the projection lattices of algebras of diagonal matrices. We define game composition, 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 preserves traciality. Some of these results extend results previously proved in [57] in the case of classical no-signalling strategies.

Preliminaries
All inner products appearing in the paper will be assumed linear in the first variable. Let H be a Hilbert space. We denote by B(H) the space of all bounded linear operators on H and often write L(H) if H is finite dimensional. If ξ, η ∈ H, we write ξη * for the rank one operator given by (ξη * )(ζ) = ζ, η ξ. In addition to inner products, ·, · will denote bilinear dualities between a vector space and its dual. We write B(H) + for the cone of positive operators in B(H), denote by T (H) its ideal of trace class operators, and by Tr -the trace functional on T (H).
An operator system is a self-adjoint subspace S of B(H) for some Hilbert space H, containing the identity operator I H . The linear space M n (S) of all n by n matrices with entries in S can be canonically identified with a subspace 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 all hermitian elements of S. If K is a Hilbert space, T ⊆ B(K) is an operator system and φ : S → T is a linear map, we let φ (n) : M n (S) → M n (T ) be the (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 completely positive if φ (n) is positive for every n ∈ N. We call φ a complete order embedding if it is injective and φ −1 | 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 canonical way; a state of S is a unital positive (linear) map φ : S → C. We denote by S(S) the (convex) set of all states of S. We note that every operator system is an operator space in a canonical fashion, and denote by S d the dual Banach space of S, equipped with its canonical matrix order structure. Operator systems can be described abstractly via a set of axioms [58]; we refer the reader to [23], [58] and [64] for details and for further background on operator space theory.
We denote by |X| the cardinality of a finite set X, let H X = ⊕ x∈X H and denote 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] = {1, . . . , 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 to the basis (e x ) x∈X , matrices, and let ∆ X : M X → D X be the corresponding conditional expectation.
When ω is a linear functional on M X , we often write ω = ω X . The canonical 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 canonical basis); see e.g. [63,Theorem 6.2]. We will thus consider M X as self-dual space with pairing On the other hand, note that the Banach space predual B(H) * can be canonically identified with T (H); every normal functional φ : B(H) → C thus corresponds 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 sometimes come in the form of a direct product), we will use a mixture of the two dualities just discussed: if ω, ρ ∈ M X , S ∈ T (H) and T ∈ B(H), it will be convenient to continue writing ρ ⊗ T, ω ⊗ S = 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 the sequel, we use the symbol ⊗ to denote the algebraic tensor product of vector spaces. For an element ω X ∈ M X and a Hilbert space H, we let L ω X : 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 probability distribution 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 trace preserving map Φ : M X → M A ; such a Φ will be called ( Tr A Γ(ρ ′ ) = 0 and Tr B Γ(ρ ′′ ) = 0 provided 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 x ′ ⊗ e y e * y ′ , we have that x∈X ρ ′ x,x,y,y ′ = 0 for all y, y ′ ∈ Y . Thus Tr x∈X ρ ′ x,x,y,y ′ e x e * x = 0, and hence 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 ′ , for all y, y ′ ∈ Y . It follows that Tr A Γ(ρ ′ ) = 0. The second property is verified similarly, while the converse direction of the statement is trivial.
A classical correlation over (X, Y, A, B) is a family p = (p(a, b|x, y)) (a,b)∈A×B : (x, y) ∈ X × Y , 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 nosignalling (NS) correlation) is a correlation p = ((p(a, b|x, y)) a,b ) x,y that satisfies the conditions We denote by C ns the set of all NS correlations and identify its elements with classical channels from D XY to D AB . Given a classical correlation p, we write Γ p = Φ Np ; thus, Γ p : M XY → M AB is the (X × Y, A × B)-classical channel given by (2.6) Γ p (ρ) = x∈X,y∈Y a∈A,b∈B p(a, b|x, y) ρ(e x ⊗ e y ), e x ⊗ e y 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 an NS correlation precisely when Γ p is a QNS correlation. Indeed, if Tr ρ X = 0 and p satisfies (2.4) and (2.5) then Let H 1 , . . . , H k be Hilbert spaces, at most one of which is infinite dimensional, T ∈ B(H 1 ⊗ · · · ⊗ H k ) and f be a bounded functional on B(H i 1 ⊗ · · · ⊗ H i k ), where k ≤ n and i 1 , . . . , i k are distinct elements of [n] (not necessarily in increasing order). We will use the expression L f (T ), or T, f (in the case k = n), without mentioning explicitly that a suitable permutation of the tensor factors has been applied before the action of f . We note that, if g is a bounded functional on B(H j 1 ⊗ · · · ⊗ H j l ), where l ≤ n and the subset {j 1 , . . . , j l } does not intersect {i 1 , . . . , i k }, then Considering an element ω ∈ M X as a functional on M X via (2.1), we have

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 such that We say that E acts on H. This terminology becomes natural after noting that the operator stochastic matrices E ∈ D X ⊗ D A ⊗ B(C) coincide, after the 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 we write E = (E x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ . Note that thus, E a,a ′ = L eae * 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 Choi matrices of unital completely positive maps Φ E : 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 a∈A E a = I H . If E a is a projection for each a ∈ A, the family (E a ) a∈A is called a projection valued measure (PVM).
The following are equivalent: (i) E is a stochastic operator matrix; In particular, if E is a stochastic operator matrix then (E x,x,a,a ) a∈A is a POVM for every x ∈ X.
where ω i is a state in M X and λ i ∈ C, i = 1, 2, 3, 4. Then (iii)⇒(i) By (2.7), for all ω X ∈ S(M X ) and all normal states τ on B(H), we have By polarisation and linearity, 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 spaceK, an isometry V : C X ⊗ H →K and a unital *-homomorphism π : M A → B(K) 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 as a block operator matrix. For ξ, η ∈ H, x, x ′ ∈ X and a, a ′ ∈ A, we have and thus E is positive. Since V is an isometry, we have Let (E x,a ) a∈A be a POVM on a Hilbert space H for every x ∈ X. A stochastic operator matrix of the form will be called classical. A general stochastic operator matrix can thus be thought of as a coordinate-free version of a finite family of POVM's. (ii) The following generalisation of Naimark's Dilation Theorem was proved in [59]: if (E x,a ) a∈A ⊆ B(H), x ∈ X, are POVM's then there exist a Hilbert spaceH, a PVM (Ẽ a ) a∈A ⊆ B(H) and isometries V x : H →H, x ∈ X, with orthogonal ranges such that

Remarks. (i)
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) and let V = (V a,x ) a,x be the isometry from Theorem 3.1. SetẼ 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 (Ẽ 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 be given by (3.2). Recall that the predual Φ * : , e a e * a ′ ; (3.6) now follows by linearity. By Choi's Theorem, every quantum channel Φ : M X → M A has the form Γ E,1 for some stochastic operator matrix Proof. The channel Γ E,σ is (X, A)-classical if and only if Γ E,σ (e x e * x ′ ) = 0 whenever x = x ′ and Γ E,σ (e x e * x ), e a e * a ′ = 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 is classical.

Three subclasses of QNS correlations
In this section, we introduce several classes of QNS correlations, which generalise corresponding classes of NS correlations studied in the literature (see e.g. [49]).
Proof. Let Denote by A (resp. B) the C*-algebra, generated by E 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 ) gives rise to a *-representation π : Inequality (ii) now follows from the contractivity of *-representations. In addition, 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 and (4.1) follows by linearity.
To show (iii), let σ ∈ T (H) be a state. Suppose that ρ X ∈ M X is traceless and ρ Y ∈ M Y . For every τ B ∈ M B , by (4.1) and Theorem 3.1, we have Thus, (2.2) is satisfied; by symmetry, so is (2.3).
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 a commuting 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.
Remark 4.4. Recall that a classical NS correlation p over (X, Y, A, B) is called 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 for all x, y, a, b, and a unit vector ξ ∈ H, such that p(a, b|x, y) = E x,a F y,b ξ, ξ , 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 Reshuffling the terms of the tensor product, we consider E ⊗F as an element of M XY ⊗M AB ⊗B(H A ⊗H B ); to underline this distinction, the latter element will henceforth be denoted by E ⊙ F . Note that, if  (ii) A QNS correlation will be called approximately quantum if it is the limit of a sequence of quantum QNS correlations.
We denote by Q q (resp. Q qa ) the set of all quantum (resp. approximately quantum) 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 correlations can equivalently be defined using arbitrary, as opposed to pure, states.  p(a, b|x, y) = (E x,a ⊗ F y,b )ξ, ξ , 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 the families (E x,a ) a∈A , x ∈ X, and (F y,b ) b∈B , y ∈ Y , then E ⊙ F is classical and determined by the family {(E x,a ⊗ F y,b ) (a,b)∈A×B : (x, y) ∈ X × Y }. As in Remark 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.
thus, Q q is convex, and the convexity of Q qa follows from the fact that Q qa = Q q .
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}.

The operator system of a stochastic operator matrix
Recall [36,74] that a ternary ring is a complex vector space V, equipped with a ternary operation [·, ·, ·] : V × V × V → V, linear on the outer variables and conjugate linear in the middle variable, such that A ternary representation of V is a linear map θ : V → B(H, K), for some Hilbert spaces H and K, such that 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 0 X,A be the ternary ring, generated by elements v a,x , x ∈ X, a ∈ A, satisfying the relations Note that (5.1) implies Indeed, suppose that (5.2) holds for u = u i , i = 1, 2, 3. Then (5.2) now follows by induction.
conversely, a family {V a,x : x ∈ X, a ∈ A} ⊆ B(H, K) satisfying (5.3) clearly gives rise to a non-degenerate ternary representation θ : V 0 X,A → B(H, K). We therefore call such a family a representation of the relations (5.1). We note that the set of representations of (5.1) is non-empty. Indeed, consider isometries V x , x ∈ X, with orthogonal ranges on some Hilbert space H, where the direct sum is taken over all equivalence classes of representations of the relations (5.1). For u ∈ V 0 X,A , let u 0 := θ (u) .
The ternary product of V 0 X,A thus induces a ternary product on V 0 X,A /N , andθ induces a ternary representation of V 0 X,A /N that will be denoted in the same way. Letting u := θ (u) , u ∈ V 0 X,A /N , we have that · is a norm on V 0 X,A /N , and hence V 0 X,A /N is a ternary pre-C*-ring (see [74]). We let V X,A be the completion of V 0 X,A /N ; thus, V X,A is a ternary C*-ring [74]. Note thatθ extends to a ternary representation of V X,A (denoted in the same way) onto a concrete TRO, and the equality u = θ (u) continues to hold for every u ∈ V X,A . We thus have that V X,A is a TRO in a canonical fashion. It is clear 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 faithfully as a concrete ternary ring of operators in B(H, K) for some Hilbert spaces , the C*-algebra C X,A may be defined by letting Each representation V = (V a,x ) a,x of the relations (5.1) gives rise [34] to a unital *-representation π V of C X,A on H V by letting 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)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 some representation V of the relations (5.1).
(ii) follows from the fact that θ (u) = u , u ∈ V X,A .
of C X,A the Brown-Cuntz operator system. Note that relations (5.1) imply Theorem 5.2. Let H be a Hilbert space and φ : T X,A → B(H) be a linear map. The following are equivalent: Proof. (i)⇒(ii) By Arveson's Extension Theorem and Stinespring's Theorem, there exist a Hilbert space K, a *-representation π : By (5.4) and Theorem 3.1, φ(e x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ is a stochastic operator matrix.
(ii)⇒(iii) By Theorem 3.1, there exist a Hilbert space K and an isometry a,a ′ is a stochastic operator matrix acting on H. Letting V be the isometry, associated with E via Theorem 3.1, we have that φ := π V | T X,A satisfies the required conditions. Let S be an operator system. Recall that the pair (C * u (S), ι) is called a 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 space and 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 the universal C*-cover is unique up to a *-isomorphism. The following corollary is immediate from Theorem 5.2.
We will need the following slight extension of the equivalence (i)⇔(ii) of Theorem 5.2.
Proposition 5.4. Let H be a Hilbert space and φ : T X,A → B(H) be a linear map. The following are equivalent: (i) φ is a completely positive map; Proof. (i)⇒(ii) It follows from Theorem 3.1 and Lemma 5.
Since T X,A ⊆ C X,A as an operator subsystem, we have Assume first that T is invertible. Let ψ : T X,A → B(H) be the map given by Let ω = (ω x,x ′ ) ∈ M X and σ be a state in T (H). Using (5.6), we have By Theorem 3.1, F is a stochastic operator matrix; by Theorem 5.2, ψ is completely positive. Since φ(·) = T 1/2 ψ(·)T 1/2 , so is φ.
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 previous paragraph, φ ǫ is completely positive. Since φ = lim ǫ→0 φ ǫ in the point-norm topology, φ is completely positive. Let we consider L X,A as an operator subsystem of M XA . For the next proposition, note that, by [16,Corollary 4.5], if T is a finite dimensional operator system than its (matrix ordered) dual T d is an operator system, when equipped with any faithful state of T as an Archimedean order unit. It is straightforward to verify that, in this case, is well-defined. It is clear that Λ + is additive and where φ 1 , φ 2 , ψ 1 and ψ 2 are positive functionals then, by the additivity of Λ + , we have that Λ . It is straightforward that the map Λ h is R-linear, and thus it extends to a (C-)linear map Λ : This shows that Λ is completely positive.
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 follows that Λ is surjective.
Thus, Λ −1 is completely positive, and the proof is complete.
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 completely positive map ψ : S → T such that J = ker(ψ). If J is a kernel in S, the quotient space S/J can be equipped with a unique operator system structure with the property that, whenever T is an operator system and φ : S → T is a completely positive map annihilating J, the induced mapφ : S/J → T is completely positive. If T is an operator system, a surjective map φ : S → T is called complete quotient, if the mapφ is a complete order isomorphism. We refer the reader to [43] for further details. Let and 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 order embedding. By [28,Proposition 1.8 Thus, ker(Λ * ) = J X,A .

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 NS correlations given in [49] (see also [26] and [62]). We will use the tensor theory of operator systems developed in [42]. If S and T are operator systems, 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 The commuting tensor product S ⊗ c T sits completely order isomorphically in the maximal tensor 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 property that it linearises jointly completely positive maps θ : S × T → B(H). We refer 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 ′ , Given a linear functional s : T X,A ⊗ T Y,B → C (or a linear functional s : Theorem 6.2. Let X, Y, A, B be finite sets and Γ : M XY → M AB be a linear map. The following are equivalent: It follows from (2.2) and (2.3) that the Choi matrix C := C x,x ′ ,y,y ′ a,a ′ ,b,b ′ of Γ satisfies the following conditions (see also [21]): 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, via the identification Λ given by (5.8). The state s of T X,A ⊗ max T Y,B corresponding to C via (6.2) satisfies ; thus, C is the Choi matrix of Γ s . By (5.5) and and the definition of the maximal tensor procuct, that is, (2.2) holds; similarly, (c) implies (2.3). (ii) there exists a state s : 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 (iii)⇒(i) Let s be a state on C X,A ⊗ max C Y,B and write π s and ξ s for the corresponding GNS representation of C X,A ⊗ max C Y,B and for its cyclic vector, respectively. Then E := (π s (e x,x ′ ,a,a ′ ⊗ 1)) x,x ′ ,a,a ′ and F := (π s (1 ⊗ f y,y ′ ,b,b ′ ) y,y ′ ,b,b ′ form a commuting pair of stochastic operator matrices; moreover, for x, x ′ ∈ X, y, y ′ ∈ Y , a, a ′ ∈ A and b, b ′ ∈ B, we have 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 bijection from the state space of T X,A ⊗ c T Y,B onto Q qc . It is straightforward that it 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 range is (convex and) closed. (i) Γ is an approximately quantum QNS correlation; (ii) there exists a state s : Proof. The proof is along the lines of the proof of [62, Theorem 2.8]; we include the details for the convenience of the reader.
Let s be a state satisfying (iii). By [39,Corollary 4.3.10], s can be approximated in the weak* topology by elements 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 1 , . . . , ξ n ∈ H X ⊗ H Y and positive scalars λ 1 , . . . , λ n with n i=1 λ i = 1 such that It follows that Γ s is in the closure of the set of correlations of the form Γ 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.
(i)⇒(iii) Given ε > 0, let E and F be stochastic operator matrices acting on finite dimensional Hilbert spaces H X and H Y , respectively, and ξ ∈ H X ⊗ H Y be a unit vector, such that and s be a cluster point of the sequence {s 1/n } n in the weak* topology. Then Recall [63] that, given any Archimedean ordered unit (AOU) space V , there exists a (unique) operator system OMIN(V ) (resp. OMAX(V )) with underlying space V , called the minimal operator system (resp. the maximal operator system) of V that has the property that every positive map from an operator system T into V (resp. from V into an operator system T ) is automatically 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 denote by OMIN(V ) (resp. OMAX(V )) the minimal (resp. maximal) operator system of the AOU space, underlying V . Lemma 6.6. Let V and W be finite dimensional AOU spaces with units e and f , respectively.
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 u ǫ → ǫ→0 0 and u + u ǫ ∈ D for every ǫ > 0, then u ∈ D. Assume, without loss of generality, that u ǫ ≤ 1 for all ǫ > 0. Set L = 2 dim(V ) dim(W ) + 1 and, using Carathéodory's Theorem, write Suppose that S ∈ M n (OMAX(V )) + and α ∈ M 1,nm . By the definition of the maximal tensor product [42], if ǫ > 0 then S + ǫ1 n has the form of S 0 in (6.4). Hence Since α (1 n ⊗ T 0 ) α * ∈ D, the previous paragraph shows that Now let T ∈ M m (OMAX(W )) + , and write T + ǫ1 m in the form of T 0 in (6.4). Then By the previous paragraph, α (S ⊗ 1 m ) α * ∈ D; by the first paragraph, Let u ∈ (OMAX(V ) ⊗ max OMAX(W )) + . By the definition of the maximal 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 1,nm , such that u + ǫ1 = α (S ⊗ T ) α * . By the previous and the first paragraph, u ∈ D.
Theorem 6.7. Let X, Y, A, B be finite sets and Γ : M XY → M AB be a linear map. The following are equivalent: (i) Γ is a local QNS correlation; (ii) there exists a state s : Using [40,Theorem 9.9] and [28, Proposition 1.9], we can identify s with , and non-negative scalars λ 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 : It is now straightforward to see that s is the functional corresponding to m i=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 ).

Classical-to-quantum no-signalling correlations
In this section, we consider the set of classical-to-quantum no-signalling correlations, and provide descriptions of its various subclasses in terms of canonical operator systems. for all x, x ′ , x ′′ ∈ X and all y, y ′ , y ′′ ∈ Y .
We write E = (E x ) x∈X ; note that, in its own right, E x is a stochastic operator matrix in Suppose that E = (E x ) x∈X and F = (F y ) y∈Y form a commuting pair of semi-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 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 quantum CQNS 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 A i,x (resp. σ B i,y ) in M A (resp. M B ) and scalars λ i > 0, i = 1, . . . , m, such that x ⊗ e y e * y = 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 correlations of class x; thus, the elements of CQ x will often be considered as channels from D XY to M AB . Similarly to the proof of Proposition 4.3, it can be shown that quantum and quantum commuting CQNS correlations can be defined using normal (not necessarily vector) states.
In the next lemma, for (finite) sets X and A and a Hilbert space H, we let for brevitỹ Lemma 7.2. Let H be a Hilbert space, E ∈ M X ⊗M A ⊗B(H) be a stochastic operator 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) stochastic operator matrix, is a stochastic operator matrix that forms a commuting pair with E then We now have for all x, x ′ ∈ X and all a, a ′ ∈ A. The second identity in (7.3) is equally straightforward. Finally, for (7.4), notice that, if E = E x,x ′ ,a,a ′ and F = F y,y ′ ,b,b ′ then both sides of the identity are equal to Proof. It is trivial that if Γ ∈ Q ns then Γ| D XY ∈ CQ ns . Conversely, suppose that E ∈ CQ ns , and let ρ X ∈ M X and ρ Y ∈ M Y be states, with Tr(ρ X ) = 0. By (7.1), Conversely, suppose that E Θ ∈ CQ qc , where Θ = (σ x,y ) x∈X,y∈Y is a CQNS correlation. 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 follows 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.
7.2. Description in terms of states. We next introduce an operator system, universal for classical-to-quantum no-signalling correlations in a similar manner that T X,A is universal for the (fully) quantum correlations, and describe the subclasses of CQNS correlations via states on tensor products of its copies. Let 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 , and let 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 1 , . . . , S n , their coproduct S = S 1 ⊕ 1 · · · ⊕ 1 S n is an operator system, equipped with complete order embeddings ι i : S i → S, characterised by the universal property that, whenever R is an operator system 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, 2, . . . , n. We refer the reader to [41, Section 8] for a detailed account 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 free product A 1 * 1 · · · * 1 A n , amalgamated over the unit. It was shown in [27, Theorem 5.2] that S ∼ = c.o.i. A 1 ⊕ 1 · · · ⊕ 1 A n . In particular, we have An application of [62, Lemma 2.8] now shows that Theorem 7.5. Let H be a Hilbert space and φ : R X,A → B(H) be a linear map. The following are equivalent: (i) φ is a unital completely positive map; (ii) φ(e x,a,a ′ ) a,a ′ ∈A x∈X is a semi-classical stochastic operator matrix.
Proof. (i)⇒(ii) The restriction φ x of φ to the x-th copy of M A is a unital completely positive map. By Choi's Theorem, φ x (e x,a,a ′ ) a,a ′ is a stochastic operator matrix in M A ⊗B(H) for every x ∈ X; thus, φ(e x,a,a ′ ) a,a ′ ∈A 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 by letting φ x (e a e * a ′ ) = φ(e x,a,a ′ ). By Choi's Theorem, φ x is a (unital) completely positive map. By the universal property of the coproduct, there exists a (unique) unital completely positive map ψ : R X,A → B(H) whose restriction to the x-th copy of M A coincides with φ x . It follows that ψ = φ, and hence φ is completely positive.
Remark 7.6. By [27, Theorem 5.1], R X,A is an operator system quotient of M XA . Now [28,Proposition 1.8] shows that, if is a well-defined unital complete order isomorphism.
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 existence of a unital completely positive map β X,A : R X,A → T X,A such that Similarly, the matrix (δ x,x ′ e x,a,a ′ ) x,x ′ ,a,a ′ is stochastic, and Theorem 5.2 implies the existence of a unital completely positive map β ′ X,A : T X,A → R X,A such that Proof. It is clear that the map t → E t is bijective. It is also straightforward to see that, for a linear functional s : The claims now follow from Theorems 6.2, 6.3, 6.5, 6.7, 7.3 and the functoriality of the involved tensor products.
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 can be 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 a direct consequence of the injectivity of the minimal tensor product in the operator system category, while for the class CQ qc , this is a consequence of Remark 7.4.

Classical reduction and separation
Let X and A be finite sets. We let where the free product is amalgamated over the unit, and 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 . We let (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 the relations a∈A e x,a = 1, x ∈ X, are satisfied. Note that, by the universal property of the operator system coproduct, S X,A is characterised by the following property: whenever H is a Hilbert 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 reduction 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 after an 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: Moreover, the maps E : Q x → CQ x and N : CQ x → C x are well-defined and surjective.
We identify an element N of C x with the corresponding classical-toquantum channel from D XY into M AB , and an element E of CQ x with the corresponding quantum channel from M XY into M AB . The subsequent table summarises the inclusions between the various classes of correlations: By Bell's Theorem, C loc = C q for all subsets X, Y, A, B of cardinality at least 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]), implying that CQ qc = CQ ns and Q qc = Q ns .
It was recently shown [37] that the inequality C qa = C qc also holds true for suitable sets X, Y , A and B, thus resolving the long-standing Tsirelson Problem and, by [38] and [56], the Connes Embedding Problem, in the negative. 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, 2, with X 1 ⊆ X 2 and A 1 ⊆ A 2 . There exist unital completely positive maps ι 1 : S X 1 ,A 1 → S X 2 ,A 2 and ι 2 : Proof. Denote the canonical generators of S X 1 ,A 1 by e x,a , and of S X 2 ,A 2by f x,a . By induction, it suffices to prove the claim in two cases. Case 1. X 1 = X 2 and A 2 = A 1 ∪ {a 2 }, where a 2 ∈ A 1 .
Let a 1 ∈ A 1 . Define the maps ι 1 and ι 2 by setting and By the universal property of the operator systems S X,A , ι 1 and ι 2 are unital completely positive maps, and the condition ι 2 • ι 1 = id is readily verified.
Proof. By [37], there exist (finite) sets X 0 , Y 0 , A 0 and B 0 and an NS correlation for the maps arising from Lemma 8.2 for the operator systems S X 0 ,A 0 and S X,A (resp. S Y 0 ,B 0 and S Y,B ). By the functoriality of the commuting tensor product, the map t : The NS correlation q ∈ C qc (X, Y, A, B) arising from t as in (8.1) does not belong to the 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 [49,Corollary 3.3], contradicts the fact that 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 Remark 8.1. Claim (iii) follows from Theorems 6.3 and 6.5, while (iv) -from (ii) and Theorem 7. Recall that an operator system S is said to possess the operator system local 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 subsystem and ϕ : T → A/I is a unital completely positive map, there exists a unital completely 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 the operator systems we introduced possess OSLPP.
be an element such that 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 functoriality of the maximal tensor product, w ∈ (S ⊗ max M n (B(H))) + . We have 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 possesses OSLLP. Proposition 8.4, combined with Corollary 5.6 and Remark 7.6, yield the following corollary.
Corollary 8.5. Let X and A be finite sets. Then T X,A and R X,A possess OSLPP.
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 special case where |X| = 1, when A X,A ∼ = D A , B X,A ∼ = M A and C X,A ∼ = C * u (M A ).

Quantum versions of synchronicity
Let X and A be finite sets, Y = X and B = A. We will often distinguish the notation for X vs. Y (resp. A vs. B) although they coincide, in order to make clear with respect to which term in a tensor product a partial trace is taken. An NS correlation p = (p(a, b|x, y)) a,b∈A : x, y ∈ X 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 synchronicity. Our main motivation is the following result, which was proved in [61]. (ii) p is synchronous and quantum if and only if there exist a finite dimensional 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 the opposite C*-algebra of A; recall that A op has the same underlying set (whose elements will be denoted by u op , for u ∈ A), the same involution, linear structure and norm, and multiplication given by 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 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 transposition map u → (u t ) op is a *-isomorphism between M X and M op X . It was shown in [44] that there exists a *-isomorphism ∂ A : A X,A → A op X,A such that ∂ 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 ∂ : 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 x ∈ X, a ∈ A. Using (9.2), we have 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 F x,x ′ ,a,a ′ x,x ′ ,a,a ′ is a stochastic operator matrix. Note that x,a ′ ,a . By the universal property of C X,A , there exists a *-homomorphism π ′ : π (C X,A ) → B H d such that x, x ′ ∈ X, a, a ′ ∈ A. By the paragraph before Lemma 9.2, π ′ • π can be regarded as a *-homomorphism from C X,A into C op X,A , which maps e x,x ′ ,a,a ′ to e op x ′ ,x,a ′ ,a . The claim follows by symmetry.
(ii) The words of the form e x 1 ,a 1 ,a ′ 1 . . . e x k ,a k ,a ′ k span a dense * -subalgebra of B X,A . As u → (u t ) op is a *-isomorphism from M A to M op A that maps the matrix unit e a e * a ′ to (e a ′ e * a ) op , the universal property of the free product implies that the map ∂ B given by extends to the desired *-isomorphism.
If U is a subspace of a C * -algebra A, we call a linear functional s : 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)). We write 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 y,y ′ ∈Y ρ x,x,y,y ′ e y ′ e * y = x,x ′ ∈X y∈Y ρ x,x ′ ,y,y e x e * x ′ , that is, when 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 precisely when 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 CQNS correlation E : D XY → M AB , an NS correlation N : Proof. We only show (i); the proofs of (ii)-(iv) are similar. Let Γ be a QNS correlation. By Theorem 6.2, there exists a state s ∈ T X,A ⊗ max T X,A → C such that Γ = Γ s . The condition Thus, letting µ (1) y,y ′ = x∈X ρ x,x,y,y ′ , we have that the left hand side of (9.7) coincides with Similarly, letting µ (2) x,x ′ = y∈X ρ x,x ′ ,y,y , we have that the right hand side of (9.7) coincides with Conversely, assuming that Γ is fair, the previous paragraph shows that for any u of the form u = y,y ′ ∈X ( 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 y∈X e y,y, From this we deduce that (9.8) holds for any u = e y,y ′ ,b,b ′ , y, y ′ ∈ X, b, b ′ ∈ A.
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 mapφ : S op → B(K d ), given byφ(u d ) = φ(u) d , is also unital and completely isometric. We thus write u op = u d in the (abstract) operator system S op .
For a linear map Φ : 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 (ii) Suppose that ψ : (iii) The transposition is a (unital) complete order isomorphism from M X onto M op X . The statement follows after observing that, under the latter identification, Φ ♯ coincides with Φ op .

Corollary 9.6. A local QNS correlation Γ is fair if and only if
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 ⊗ ∂) −1 is fair. As in the proof of Theorem 6.7, identify s with a convex combination m i=1 λ i φ i ⊗ ψ i , where φ i and ψ i are states on T X,A , i = 1, . . . , m; then the fairness condition is equivalent to 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 , u ∈ T X,A . By Lemma 9.2,ψ i is a state. Moreover, , that is, the quantum channel Ψ ♯ i corresponds toψ i . Identity (9.9) now follows from (9.10). The converse implication follows by reversing the previous steps.
(i) A CQNS correlation E is fair if and only if there is a state t : R X,A ⊗ max R X,A → C such that t • (id ⊗ ∂ B ) −1 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 CQNS correlation. By (9.6), E • ∆ XY : M XY → M AB is a fair QNS correlation. By and s • (id ⊗ ∂) −1 is fair. By Theorem 9.4 (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.
Remark. It follows from Theorem 9.1, Theorem 9.4 and Corollary 9.7 that fair 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 paragraph before 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 Amatrix 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 belongs to D X ⊗ M A ⊗ A.
Let E = (g x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ be a stochastic A-matrix, and set 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 permutation of the tensor factors, we can consider E ⊗ E op as an element of (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, 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.
In the sequel, we write Γ E,τ = Γ f E,τ ; by Theorem 6.3, Γ E,τ ∈ Q qc . By Theorem 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 .
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 if there exists quantum channels Φ j : M X → M A , j = 1, . . . , k, such that as a convex combination. In particular, if Γ is a locally tracial QNS correlation 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 = E x,x ′ ,a,a ′ x,x ′ ,a,a ′ . By the proof of Lemma 9.2, E op := E d x ′ ,x,a ′ ,a x,x ′ ,a,a ′ is a stochastic operator matrix. Let σ be any positive functional on L(H ⊗ H d ) that extends the state s τ A which, by nuclearity, 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 channels and Γ is the convex combination (9.12). Letting λ x, x ′ ∈ X, we have that the matrix C j = λ (j) x,x ′ ,a,a ′ 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 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 (µ j ) k j=1 = k j=1 λ j µ j . We have 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 . Without loss of generality, assume that A = C(Ω), where Ω is a compact Hausdorff topological space, and µ is a Borel probability measure on Ω such that τ A (f ) = 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 by Φ(s) e x e * x ′ = h x,x ′ ,a,a ′ (s) a,a ′ . We have It follows that Γ is in the closed hull of the set of all correlations of the form (9.12). An argument using Carathéodory's Theorem, similar to the one in the proof of Remark 4.10, shows that Γ has the form (9.12).
Remark 9.10. (i) Every tracial QNS correlation Γ = Γ E,τ is fair. Indeed, writing E = (g x,x ′ ,a,a ′ ), we have x,x ′ ,a,a ′ ). It can be seen from Corollary 9.6 and Theorem 9.9 (see the closing remarks of this section) that the converse does not hold true.
(ii) The set of all tracial (resp. quantum tracial, locally tracial) QNS correlations over (X, A) is convex. Indeed, suppose that A (resp. B) is a unital 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, 1), C = A⊕B, τ : C → C be given by τ (u ⊕ v) = λτ A (u) + (1 − λ)τ B (v), and G = E ⊕ F , considered as an element of M X ⊗ M A ⊗ C. Then G is a stochastic C-matrix and (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, locally tracial) QNS correlation. By [22,Theorem 4.2], the set C s q of synchronous quantum NS correlations is not closed if |X| = 5 and |A| = 2. Let p ∈ C s q \C s q . Then p is a synchronous NS correlation and does not lie in C q . Assume that Γ p is quantum tracial. By Theorem 9.9, Γ p ∈ Q q and hence, by Remark 8.1, p ∈ C q , a contradiction. It follows that the set of quantum tracial NS correlations is not closed.
(iv) The set of all tracial QNS correlations is closed; this can be seen via a standard argument (see e.g. [54]): Assuming that (Γ n ) n∈N is a sequence of tracial QNS correlations converging to the QNS correlation Γ, let A n be a unital C*-algebra with a trace τ n , and E n = g (n) x,x ′ ,a,a ′ be a stochastic A n -matrix such that Γ n = Γ En,τn . Let A be the tracial ultraproduct of the family {(A n , τ n )} n∈N with respect to a non-trivial ultrafilter u [33, Section 4]. 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 Γ(e x e * x ′ ⊗ e y e * y ′ ), e a e * a ′ ⊗ e b e * b ′ = lim n→∞ τ n g (n) x,x ′ ,a,a ′ g We next show that the class of all tracial QNS correlations, as well as each of 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 : Equivalently, let E op be the stochastic A op -matrix g op u ′ ,u , and recall that s τ : A ⊗ max A op → C is the state given by s τ (u ⊗ v op ) = τ (uv). Then 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.
Proof. (i) Let τ be a trace on C X,A , A be a C*-algebra, τ A be a trace on A, thus, F := (h a,a ′ ) a,a ′ ∈ M A ⊗ B. Moreover, To see that F is positive, we assume that A and C X,A are faithfully represented and let V x and V a,x be operators such that (V x ) x is a row isometry, x 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 (ii) and (iii) follow from the fact that if the C*-algebra A is finite dimensional (resp. abelian) and τ factors through a finite-dimensional (resp. abelian) C*-algebra then so does τ B . (ii) Recall that a state ρ ∈ M XX is called de Finetti [17] if there exist states ω i ∈ M Z , i = 1, . . . , k, such that ρ = k j=1 λ j ω j ⊗ ω j as a convex combination. By (9.13) and Theorem 9.9, Υ loc (X) = conv ω ⊗ ω t : ω a state in M X .
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 the necessity to employ opposite C*-algebras. Thus, quantum reciprocal states can be viewed as an entanglement assisted version of (twisted) de Finetti states, while C*-reciprocal states -as their commuting model version.
(iii) C*-reciprocal states are closely related to factorisable channels introduced in [1] (see also [32,53], to which we refer the reader for the definition used 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 Neumann algebra A, and (g x,x ′ ) x,x ′ and (h y,y ′ ) y,y ′ are matrix unit systems -a special type of stochastic A-matrices (see [53,Proposition 3.1]). Equivalently, the Choi matrices of factorisable channels Φ : M X → M X can be described [32,Definition 3.1] as the matrices of the form τ (v * a,x v a ′ ,x ′ ) x,x ′ ,a,a ′ , where V = (v a,x ) a,x ∈ M X (A) is a unitary matrix. Note that, if E is the stochastic operator matrix corresponding to V , then the QNS correlation Γ = Γ E,τ has marginal channels Γ A (·) = Γ(· ⊗ I) and Γ B (·) = Γ(I ⊗ ·) that coincide with Φ. We can thus view tracial QNS correlations as generalised couplings of factorisable channels. Here, by a coupling of the pair (Φ, Ψ) of channels, we mean a channel Γ with Γ A = Φ and Γ B = Ψ -a generalisation of classical coupling of probability distributions in the sense of optimal transport [72]. 9.3. Tracial CQNS correlations. In this subsection, we define a tracial version of CQNS correlations. Let A be a unital C*-algebra, τ : A → C be a trace 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 a∈A g x,a,a = 1, for each x ∈ X. Set E op = (g op x,a ′ ,a ) x,a,a ′ ; thus, E op ∈ D X ⊗ M A ⊗ A op and Lemma 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 positive map 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 is completely positive by Lemma 9.5. Write In the sequel, we write E E,τ = E f E,τ ; by Theorem 7.7, E E,τ ∈ CQ qc .
A is a semi-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 as in (i) can be chosen to be finite dimensional; (iii) locally tracial if it it is tracial and the C*-algebra as in (i) can be chosen to be abelian.
(ii) E is locally tracial if and only if there exist channels E j : D X → M A , j = 1, . . . , k, such that 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 for some finite dimensional C*-algebra A and a trace τ : A → C. The matrix E = δ x,x ′ g x,a,a ′ x,x ′ ,a,a ′ is a stochastic matrix in M X ⊗ M A ⊗ A and hence gives rise, via Theorem 5.2, to a canonical *-homomorphism πẼ : C X,A → A. Lettingτ = τ • πẼ, we have thatτ is a trace on C X,A and Γ E = Γτ . 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 traceτ , defined in the proof of (i), now factors through an abelian C*-algebra, and hence Γ E is locally tracial. By Theorem 9.9, there exists quantum channels Φ j : M X → M A , j = 1, . . . , k, such that Γ E = k j=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, there exists an abelian C*-algebra A, a *-representation π : C X,A → A and a trace τ on A such that Γ E = Γ τ •π . The stochastic operator matrix E = π(e x,x,a,a ′ ) x,a,a ′ is semi-classical and E = E E,τ .
We now specialise Definition 9.11 to states in D XX , that is, bipartite probability distributions. A probability distribution q = (q(x, y)) x,y∈X on 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 quantum reciprocal (resp. locally reciprocal). We denote by Υ cl (X) (resp. Υ cl q (X), Υ cl loc (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 probability distributions coincides with the well-known class of exchangeable probability distributions, that is, the convex combinations of the form where q i is a probability distribution on X, i = 1, . . . , n. Thus, C*-reciprocal and quantum reciprocal probability distributions can be viewed as quantum versions of exchangeable distributions.
loc (X) ⊆ Υ loc (A). 9.4. Tracial NS correlations. The correlation classes introduced in Sections 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 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 be chosen to be finite dimensional; (iii) locally tracial if it it is tracial and the C*-algebra A in (i) can be chosen to be abelian.
The next two propositions are analogous to Theorem 9.9 and 9.12, respectively, 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 = k j=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.
If N is locally tracial then N Υ cl loc (X) ⊆ Υ cl loc (A). 9.5. Reduction for tracial correlations. We next specialise the statements 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 only if Γ p is tracial (resp. quantum tracial, locally tracial, fair); (ii) E is tracial (resp. quantum tracial, locally tracial, fair) if and only if 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. quantum tracial, locally tracial, fair) CQNS correlations onto the class of all tracial (resp. quantum tracial, locally tracial) NS correlations; (iv) the map C is a surjection from the class of all tracial (resp. quantum tracial, locally tracial, fair) QNS correlations onto the class of all tracial (resp. quantum tracial, locally tracial, fair) CQNS correlations.
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 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-classical stochastic A-matrix and, trivially, Conversely, suppose that Γ p = Γ E,τ , where E = (g x,x ′ ,a,a ′ ) x,x ′ ,a,a ′ is a stochastic 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 a trace on A such that Γ = Γ E,τ then C(Γ) = E E ′ ,τ , where E ′ is given as in the second paragraph of the proof.
(iv) is similar to (iii). All remaining statements about quantum tracial and locally tracial correlations are analogous.
Turning to the case of fair correlations, (ii) follows from the equivalence showing that p is fair if and only if so is E p . As Γ p = Γ Ep , the equivalence with fairness of Γ p follows from (ii).
We conclude this section with a comparison between the different classes of correlations of synchronous type. Note first that, if p is a synchronous quantum commuting NS correlations then, by Theorem 9.1, N p is a tracial NS correlation. In fact, the synchronous quantum commuting NS correlations arise precisely from classical stochastic A-matrices (g x,a ) x,a , where each (g x,a ) a∈A is a PVM, as opposed to POVM. Theorem 9.4 implies that tracial QNS correlations are necessarily fair. We summarise these inclusions below: The inclusions in the table are all strict. Indeed, for the first column this follows from [22]. It can be shown, using results on the completely positive semidefinite cone of matrices [46,13] that Υ cl loc = Υ cl q [2]. The properness of the first inclusion in the second column now follows from Remark 9.13. The properness of the second inclusion in the second column was pointed out in Remark 9.10 (iii), and Theorem 9.20 implies that the first and the second inclusions in the third and the fourth column are proper.
Let p = {p(·|x) : x ∈ X} and q = {q(·|x) : x ∈ X} be families of distributions so that, for some x ∈ X, we have that supp p(·|x)∩supp q(·|x) = ∅. Thenp = 1/2(p ⊗ q + q ⊗ p) is a fair NS correlation. However,p is not tracial; indeed, assuming the contrary, we have thatp = m j=1 λ j p j ⊗ p j as a convex combination, where {p j } m j=1 consists of families of probability distributions indexed by X. Sincẽ p(a, a|x, x) = 1 2 (p(a|x)q(a|x) + q(a|x)p(a|x)) = 0, a ∈ A, we have m j=1 λ j p j (a|x) 2 = 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 fourth column.
Using Theorem 9.9 and Proposition 9.15, one can easily see that the second and third inclusion on the first row are strict, and hence these inclusions are 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 with at least one x having | supp q(·|x)| > 1, is not synchronous, but is locally tracial; thus, the first inclusion in the first, second and third rows are strict.

Correlations as strategies for non-local games
In this section, we discuss how QNS correlations can be viewed as perfect strategies for quantum non-local games, extending the analogous viewpoint on NS correlations to the quantum case. Let X, Y , A and B be finite sets. A non-local game on (X, Y, A, B) is a cooperative game, played by two players against a verifier, determined by a rule function (which will often be identified with the game) λ : X × Y × A × B → {0, 1}. The set X (resp. Y ) is interpreted as a set of questions to, while the set A (resp. B) as a set 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 NS correlation 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) of questions, the players choose their answers according to the probability distribution p(·, ·|x, y), they will win every round of the game.
10.1. Quantum graph colourings. Let G be a simple graph on a finite set 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 colouring of G is a map f : X → A, where A is a finite set, such that 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 Thus, an NS correlation p = {(p(a, b|x, y)) a,b∈A : x, y ∈ X} is a perfect strategy of the G-colouring game if (S) p is synchronous; (P) x ∼ y ⇒ p(a, a|x, y) = 0 for all a.
It is easy to see that if p is a perfect strategy of the G-colouring game from the class C loc then G possesses a classical colouring from the set A. Thus, the perfect 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 number of 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 finite set A, we let Ω A be the non-normalised maximally entangled matrix in M AA , namely, 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 =⇒ E p e x e * x ⊗ e y e * y , Ω A = 0. Proof. The claim is immediate from the fact that p (a, a|x, y) .
Remark 10.1 allows to generalise the classical x-colourings of a graph G to the quantum setting as follows. 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 The orthogonal rank ξ(G) of G is given by ξ(G) = min k : ∃ an orthogonal representation of G in C k .
Proposition 10.3. Let G be a graph with vertex set X. The following are equivalent: (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 representation of G. Let E 0 : D X → M A be the quantum channel given by and set E = E 0 ⊗ E ♯ 0 ; by Proposition 9.15, E is locally tracial. If x ∼ y then 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 = k j=1 λ j E j ⊗ E ♯ j as a convex combination with positive coefficients, where E j : D X → M A is a quantum channel, j = 1, . . . , k.
Suppose that x ∼ y. Then x ⊗ e y e * y , Ω A = 0 and hence, by the non-negativity of each of the terms of the sum, , Ω A = 0.
Let ξ x be a unit eigenvector of E 1 (e x e * x ), corresponding to a positive eigenvalue, x ∈ X. Condition (10.1) implies that ξ x ξ * x ⊗ ξ y ξ * y t , Ω A = 0, which in turn means, by the arguments in the previous paragraph, that ξ x , ξ y = 0.
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 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 by a set A. By Theorem 9.20, E p ∈ CQ x . By Remark 10.1, E p is G-proper.
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 Proposition 10.4 (ii) in the case x = loc. In [52], an example of a graph G on 13 vertices was 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) in the case x = q. We do not know if a strict inequality can occur in the case x = 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 not whether 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 Lovász number θ(G) of G. We refer the reader to [48] for the definition and properties of the 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 e x e * y : (x, y) ∈ κ ; thus, S κ is a linear subspace of M X which is a bimodule over the diagonal algebra D X . We write and let S G := S E(G) be the graph operator system of G [20], and S 0 G := S E 0 (G) be the graph operator anti-system of G [71] (here we use the terminology of [7]).
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 quantum qc-colouring of G, such that Assume, without loss of generality, that A ⊆ B(H) as a unital C*-subalgebra and that ξ ∈ H is a unit vector with τ (u) = uξ, ξ , u ∈ A. Set ξ x,a,a ′ = E x,a,a ′ ξ, x ∈ X, a, a ′ ∈ A; then We note that (10.2) a∈A ξ x,a,a = ξ, x ∈ X.
In addition, if x ∼ y then Note that, up to an application of the canonical shuffle, x ⊗ e y e * y ) x,y , and hence, after another application of the canonical shuffle, Choi's Theorem implies that the linear map Ψ : M AA → M X , given by By Theorem 3.1, there exist operators V a,x such that (V a,x ) a,x is an isometry and E x,a,a ′ = V * a,x V a ′ ,x , x ∈ X, a, a ′ ∈ A. Thus, if x ∈ X then Write Ψ(I AA ) = D + T , where D is diagonal and T ⊥ S G . We have shown that D ≤ |A|I X ; thus |A|I X + T ∈ M + X . It follows that Ψ(I AA ) ≤ |A|I X + T ≤ max |A|I X + S : S ∈ S ⊥ G , |A|I X + S ∈ M + X = |A|θ(G). (10.4) Let J X be the matrix in M X all of whose entries are equal to one. By Taking the minimum over all |A| completes the proof of the inequality.

Graph homomorphisms.
In this subsection, we consider a quantum 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 the homomorphism game G → H has Y = X, B = A, and λ(x, y, a, b) = 0 if and only if, either x = y and a = b, or x ∼ y and a ∼ b. A synchronous NS correlation p = (p(a, b|x, y)) a,b∈A : x, y ∈ X is thus called a perfect x-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 Thus, P κ is the Schur projection onto S κ ; it can be canonically identified with the (positive) element (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 0 (G) . Proposition 10.6. Let G (resp. H) be a graph with vertex set X (resp. A), and p = (p(a, b|x, y)) a,b∈A : x, y ∈ X be a synchronous NS correlation. The following are equivalent: (i) p is a perfect strategy for the homomorphism game G → H; (ii) N p (P G ), (P H ) ⊥ = 0.
Proof. (i)⇒(ii) We have (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) = N p e x e * x ⊗ e y e * y , e a e * a ⊗ e b e * b ≤ N p (P G ), (P H ) ⊥ = 0.
General operator systems in M X were considered in [20] as a quantum versions of graphs (noting that S G is an operator system), while operator anti-systems (that is, selfadjoint subspaces of M X each of whose elements has trace zero [7]) were proposed as such a quantum version in [71] (noting that S 0 G is an operator anti-system). Note that one can pass from any of the two notions to the other by taking orthogonal complements. Due to the specific definition of QNS correlations in [21], employed also here, it will be convenient to use a slightly different (but equivalent) perspective on noncommutative 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) = η, ξ ; we write ξ d = d (ξ). Note that, if T ∈ L(H) then Note that, if ξ, η, ζ 1 , ζ 2 ∈ H then θ(ξ ⊗ η) * (ζ 1 ), ζ d 2 = ζ 1 , θ(ξ ⊗ η)ζ d 2 = ζ 1 , ξ, ζ 2 η = ζ 2 , ξ ζ 1 , η = ξ d , ζ d 2 ζ 1 , η = ζ 1 , η ξ d , ζ d and hence In addition, We call a subspace of L(H d , H) satisfying these properties a twisted operator 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 −1 (S) of H ⊗ H is symmetric and skew. Given a graph G, let it is clear that U G is a symmetric skew subspace of C X ⊗ C X . We thus consider symmetric skew subspaces of C X ⊗ C X as a non-commutative version of graphs.
We write P U for the orthogonal projection from C X ⊗ C X onto U . Let U ⊥ ⊂ C X ⊗ C X d be the annihilator of U and write P U ⊥ ∈ L (C X ⊗ C X ) d 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 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 order isomorphism from M A onto M d A (see e.g. [63,Theorem 6.2]). On the other hand, the map ω d → ω t is a *-isomorphism from L (C A ) d onto M A . The composition of these maps, ω d → f w t , is thus a complete order isomorphism In the sequel, we identify these two spaces; note that, via this identification, (10.11) ρ, ω d = ρ, ω t = 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 : Given operator anti-systems S ⊆ M X and T ⊆ M A , Stahlke [71] defines a non-commutative graph homomorphism from S to T to be a quantum channel Φ : . . , m; if such Φ exists, he writes S → T . The appropriate version of this notion for twisted operator anti-systems -directly modelled 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 (C X ) d , C X and T ⊆ L (C A ) d , C A be twisted operator anti-systems. A homomorphism from S into T is a quantum channel such that M j SM 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 . Proof. Suppose that U loc → V and let Γ be a locally tracial QNS correlation for which (10.12) holds. By Theorem 9.9, there exist quantum channels Φ j : M X → M A , j = 1, . . . , k, such that Γ = k j=1 λ j Φ j ⊗ Φ ♯ j as a convex combination. We have since each of the terms in the sum on the left hand side is non-negative, selecting j with λ j > 0 and setting Φ = Φ j , we have It follows that Let ξ ∈ U and η ∈ V ⊥ be unit vectors; then ξξ * ≤ P U . In addition, By (10.14) and positivity, which, by (10.11), means that (M i ⊗ M j )ξ, η = 0, i, j = 1, . . . , m.
Thus, (M i ⊗ M j )ξ ∈ V for every ξ ∈ U and, by (10.7), Conversely, suppose that Φ : M X → M A is a quantum channel with a family of Kraus operators . . , m. The previous paragraphs show that for all unit vectors ξ ∈ U , η ∈ V ⊥ . It follows that for all unit vectors η ∈ V ⊥ . Taking infimum over all such η, we obtain for all unit vectors ξ ∈ U . Thus, by (10.11) and (10.10), is an orthonormal basis of U , we obtain (Φ ⊗ Φ ♯ )(P U ), P V ⊥ = 0. Proof. Write X and A for the vertex sets of G and H, respectively. Assume that G → H. By [71], S 0 for the set of Kraus operators such that M i S 0 G M * j ⊆ S 0 H , i, j = 1, . . . , m. Let J X : C X → C X be the map given by J X (η) =η. Then θ(e x ⊗ e y ) = J X • e y e * x • d −1 , x, y ∈ X. Therefore, implying M i S U G M d j ⊆ S U H ; by Proposition 10.10, U G loc → U H . The converse follows after reversing the arguments.
10.3. General quantum non-local games. We write P M for the projection 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 1 , P 2 ∈ P M , the projection P 1 ∨ P 1 (resp. P 1 ∧ P 2 ) has range the closed span (resp. the intersection) of the ranges of P 1 and P 2 . 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 ) for any family {P i } i∈I ⊆ P M . Note that if M is finite dimensional, then join continuity is equivalent to the preservation of finite joins.
Let H be a Hilbert space and P be an orthogonal projection on H with range U . As in Subsection 10.2, we denote by U ⊥ the annihilator of U in the space 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-toquantum non-local game, a classical non-local game) if ϕ is join continuous 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 strategy for the quantum (resp. classical-to-quantum, classical) non-local game ϕ if (10.15) Λ(P ), ϕ (P ) ⊥ = 0, P ∈ P M XY (resp. P ∈ P D XY ). 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), (0, Q) ∈ B for all P ∈ P B(H) , Q ∈ P B(K) , and (P 1 , Q 1 ), (P 2 , Q 2 ) ∈ B ⇒ (P 1 ∨ P 2 , Q 1 ∧ Q 2 ) ∈ B and (P 1 ∧ P 2 , Q 1 ∨ Q 2 ) ∈ B. Thus, quantum non-local games (resp. classicalto-quantum non-local games, classical non-local games) can be alternatively defined as bilattices; we have chosen to use maps instead because they are more convenient to work with when compositions are considered (see Definition 10.15). Conditions (10.15) are reminiscent of J. A. Erdos' characterisation [25] of reflexive spaces of operators, introduced by L. N. Loginov and V. S. Shulman in [47]. As shown in [25], a subspace S ⊆ B(H, K) (H and K being Hilbert spaces) is reflexive in the sense of [47] if and only if there exists a join continuous zero-preserving map ϕ : P B(H) → P B(K) such that S coincides with the the space Op( ϕ ) = T ∈ B(H, K) : ϕ (P ) ⊥ T P = 0, for all P ∈ P B(H) .
(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 be referred to as the empty game. It is clear that the set of perfect strategies for 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. The quantum graph colouring game considered in Subsection 10.1 is the classicalto-quantum non-local game ϕ : P D XX → P M AA , given by ϕ (e x e * x ⊗ e y e * y ) =

1
|A| Ω ⊥ A if x ∼ y I otherwise.
Similarly, letting U ⊆ C X ⊗ C X and V ⊆ C A ⊗ C A be symmetric skew spaces, we define the homomorphism game U → V to be the quantum nonlocal game ψ, given by For x ∈ {loc, q, qc}, we have that U x → V if and only if the game U → V has a perfect strategy of class Q x .
Proposition 10.14. An NS correlation p is a perfect strategy for the nonlocal game (with rule function) λ if and only if N p is a perfect strategy for ϕ λ .
Proof. Note that, if (x, y) ∈ X × Y then P βx,y(λ) ⊥ 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 N p P {(x,y)} , P βx,y(λ) ⊥ = 0, (x, y) ∈ X × Y.
Thus, N p (P {(x,y)} ), ϕ λ (P ) ⊥ = 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 N p (P ), ϕ λ (P ) ⊥ = 0. The converse is direct from the first paragraph.
Definition 10.15. Let X, Y , A, B, Z and W be finite sets and ϕ 1 (resp. ϕ 2 ) be a game from XY to AB (resp. from AB to ZW ). The composition of ϕ 1 and ϕ 2 is the game ϕ 2 • ϕ 1 from XY to ZW .
It is clear that ϕ 2 • ϕ 1 is well-defined in all cases except when ϕ 1 is a quantum game, while ϕ 2 is a classical-to-quantum game.
Lemma 10. 16. Let X, A and Z be finite sets, H and K be Hilbert spaces and E ∈ M X ⊗ M A ⊗ B(H) and F ∈ M A ⊗ M Z ⊗ B(K) be stochastic operator matrices. Set G x,x ′ ,z,z ′ = 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 ) toH A (resp.K Z ) for some Hilbert spaceH (resp.K), such that 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 = a∈A W z,a ⊗ V a,x , x ∈ X, z ∈ Z.
For x, x ′ ∈ X, we have 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 By Theorem 3.1, G is a stochastic operator matrcx acting on K ⊗ H.
We call the stochastic operator matrix G from Lemma 10.16 the composition of F and E and denote it by F • E.
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 results contained in [57,Theorem 3.7].
Corollary 10.18. Let λ 1 (resp. λ 2 ) be the rule functions of non-local games from 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, 2, then p 2 • p 1 is a perfect strategy for λ 2 • λ 1 from the class C x . 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}.
Acknowledgement. It is our pleasure to thank Michael Brannan, Li Gao, Marius Junge, Dan Stahlke and Andreas Winter for fruitful discussions on the 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 correlations and study a version of the homomorphism game from a non-commutative to a classical graph. Although there are similarities between our approaches, there is no duplication of results in the current paper with those in [9].