Abstract
We introduce and examine three subclasses of the family of quantum nosignalling (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 classicaltoquantum correlations. We study various quantum versions of synchronous nosignalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum nonlocal games as a generalisation of nonlocal 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 classicaltoquantum strategies of the graph colouring game exist, as well as to noncommutative graph homomorphisms, where we identify quantum versions of noncommutative graph homomorphisms introduced by Stahlke.
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1 Introduction
Nonlocal games [19] 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 \(X\times Y\) of two finite sets, and the players produce as an output a pair (a, b) from a Cartesian product \(A\times B\). The combinations (x, y, a, b) that yield a win are determined by a predicate function \(\lambda : X\times Y \times A\times B \rightarrow \{0,1\}\). A probabilistic strategy is a family \(p = \{(p(a,bx,y))_{(a,b)\in A\times B}: (x,y)\in X\times Y\}\) of probability distributions, one for each input pair (x, y), where the value p(a, bx, 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 nosignalling condition, which ensures no communication between the players takes place during the course of the game, and are hence called nosignalling (NS) correlations.
In pseudotelepathy games [11], 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 \(\xi \) in the tensor product \(H_A\otimes H_B\) 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 \(p(a,bx,y) = \langle (E_{x,a}\otimes F_{y,b})\xi ,\xi \rangle \). Employing the commuting model of quantum mechanics leads, on the other hand, to the broader set of quantum commuting strategies, whose underlying nosignalling 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 nosignalling correlations:
The class \(\mathcal {C}_\textrm{qa}\) of approximately quantum correlations is the closure of the quantum class \(\mathcal {C}_\textrm{q}\)—known, due to the work of Slofstra [69] (see also [23]) to be strictly larger than \(\mathcal {C}_\textrm{q}\)—and \(\mathcal {C}_\textrm{ns}\) is the class of all nosignalling correlations, playing a fundamental role in generalised probabilistic theories [3, 4]. The longstanding question of whether \(\mathcal {C}_\textrm{qa}\) coincides with the class \(\mathcal {C}_\textrm{qc}\) of all quantum commuting correlations, known as Tsirelson’s problem, was recently settled in the negative in [38]. Due to the works [39] and [57], 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 twofold. Firstly, the resolution of the Connes embedding problem in [38] 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 nonlocal games with quantum inputs and quantum outputs. A number of versions of quantum games have already been examined. In [20], 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 [31], the focus is on multiple round quantum strategies that are available to players with quantum memory, while the quantumclassical and extended nonlocal games considered in [66] both have classical outputs (see also [15]). Here, we propose a framework for quantumtoquantum nonlocal games, which generalises directly (classical) nonlocal games. This allows us to define a quantum version of the graph homomorphism game (see [23, 51, 52, 62]), and leads to notions of quantum homomorphisms between (the widely studied at present [7, 8, 21, 22, 46, 70]) noncommutative graphs.
Our starting point is the definition of quantum nosignalling correlations given by Duan and Winter in [22]. Note that nosignalling (NS) correlations correspond precisely to (bipartite) classical information channels from \(X\times Y\) to \(A\times B\) with welldefined marginals. In [22], quantum nosignalling (QNS) correlations are thus defined as quantum channels \(M_{X\times Y}\rightarrow M_{A\times B}\) (here \(M_Z\) denotes the space of all \(Z\times Z\) complex matrices) whose marginal channels are welldefined. In Sect. 4, we define the quantum versions of the classes in (1.1), arriving at an analogous chain
The base for our definitions is a quantisation of positive operator valued measures, which we develop in Sect. 3. The stochastic operator matrices defined therein replace the families \((E_{x,a})_{x\in X,a\in A}\) of measurement operators that play a crucial role in the definitions of the classical classes (1.1). In Sect. 5, we define a universal operator system \(\mathcal {T}_{X,A}\), 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 \(\mathcal {V}_{X,A}\) of a given \(A\times X\)block operator isometry, which is a generalisation of the Brown algebra of a unitary matrix [12] (see also [30]). We describe \(\mathcal {T}_{X,A}\) 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 [27]. We show that any such quotient possesses the local lifting property [44]. This unifies a number of results in the literature, implying in particular [36, Theorem 4.9].
In Sect. 6, we provide operator theoretic descriptions of the classes \(\mathcal {Q}_\textrm{loc}\), \(\mathcal {Q}_\textrm{qa}\), \(\mathcal {Q}_\textrm{qc}\) and \(\mathcal {Q}_\textrm{ns}\), 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 \(\mathcal {Q}_\textrm{qc}\) arises from a state on the commuting tensor product \(\mathcal {T}_{X,A}\otimes _\textrm{c}\mathcal {T}_{Y,B}\), and that similar descriptions hold for the rest of the aforementioned classes. Along with the hierarchy (1.2), we introduce an intermediate chain
lying between (1.1) and (1.2), whose terms are classes of classicaltoquantum nosignalling (CQNS) correlations. We define their universal operator system, and provide analogous characterisations in terms of states on its tensor products; this is achieved in Sect. 7. In Sect. 8, we point out the canonical surjections \(\mathcal {Q}_\textrm{x}\rightarrow \mathcal{C}\mathcal{Q}_\textrm{x} \rightarrow \mathcal {C}_\textrm{x}\) (where \(\textrm{x}\) denotes any specific correlation class from the set \(\{\textrm{loc}, \textrm{q}, \textrm{qa}, \textrm{qc}, \textrm{ns}\}\)). 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 \(\mathcal {Q}_\textrm{loc}\) 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. [72, p. 358]). Thus, the channels from \(\mathcal {Q}_\textrm{q}\) 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 nonlocal 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 \(\mathcal {A}_{X,A}\)—the free product of X copies of the Adimensional abelian C*algebra. In Sect. 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 [32]. More precisely, if one restricts attention to QNS correlations that arise from the Brown algebra as opposed to the ternary ring of operators \(\mathcal {V}_{X,A}\), 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 [18]. 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 Sect. 10, we point out how QNS and CQNS correlations can be used as strategies for quantumtoquantum and classicaltoquantum 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. Employing the CQNS classes, we define new versions of quantum chromatic numbers of a classical graph G. The class \(\mathcal{C}\mathcal{Q}_\textrm{loc}\) yields the wellknown orthogonal rank \(\xi (G)\) of G [67]; thus, the chromatic numbers \(\xi _\textrm{q}(G)\) and \(\xi _\textrm{qc}(G)\), arising from \(\mathcal{C}\mathcal{Q}_\textrm{q}\) and \(\mathcal{C}\mathcal{Q}_\textrm{qc}\), respectively, can be thought of as entanglement assisted and commuting model versions of this classical graph parameter.
In Sect. 10.2, we define a noncommutative version of the graph homomorphism game [51]. We show that its perfect strategies from the class \(\mathcal {Q}_\textrm{loc}\) correspond precisely to noncommutative graph homomorphisms in the sense of Stahlke [70]. Thus, the perfect strategies from the larger classes in (1.2) can be thought of as quantum noncommutative graph homomorphisms. We note that special cases have been previously considered in [8] and [56], and that a different perspective, motivated by noncommutative topology, has been studied in [55]. The treatment in the latter papers was restricted to noncommutative graph isomorphisms, and the suggested approach was operatoralgebraic. We remedy this by suggesting, up to our knowledge, the first operational approach to noncommutative graph homomorphisms, thus aligning the noncommutative case with the case of quantum homomorphisms between classical graphs [51].
Finally, in Sect. 10.3, we introduce a quantum version of nonlocal 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 \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qa}, \textrm{qc}, \textrm{ns}\}\) are closed under channel composition, and prove that channel composition preserves traciality. Some of these results extend results previously proved in [58] in the case of classical nosignalling strategies.
2 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 \(\mathcal {B}(H)\) the space of all bounded linear operators on H and often write \(\mathcal {L}(H)\) if H is finite dimensional. If \(\xi ,\eta \in H\), we write \(\xi \eta ^*\) for the rank one operator given by \((\xi \eta ^*)(\zeta ) = \langle \zeta ,\eta \rangle \xi \). In addition to inner products, \(\langle \cdot ,\cdot \rangle \) will denote bilinear dualities between a vector space and its dual. We write \(\mathcal {B}(H)^+\) for the cone of positive operators in \(\mathcal {B}(H)\), denote by \(\mathcal {T}(H)\) its ideal of trace class operators, and by \(\textrm{Tr}\)—the trace functional on \(\mathcal {T}(H)\).
An operator system is a selfadjoint subspace \(\mathcal {S}\) of \(\mathcal {B}(H)\) for some Hilbert space H, containing the identity operator \(I_H\). The linear space \(M_n(\mathcal {S})\) of all n by n matrices with entries in \(\mathcal {S}\) can be canonically identified with a subspace of \(\mathcal {B}(H^n)\), where \(H^n\) is the direct sum of ncopies of H; we set \(M_n(\mathcal {S})^+ = M_n(\mathcal {S})\cap \mathcal {B}(H^n)^+\) and write \(\mathcal {S}_h\) for the real vector space of all hermitian elements of \(\mathcal {S}\). If K is a Hilbert space, \(\mathcal {T}\subseteq \mathcal {B}(K)\) is an operator system and \(\phi : \mathcal {S}\rightarrow \mathcal {T}\) is a linear map, we let \(\phi ^{(n)}: M_n(\mathcal {S})\rightarrow M_n(\mathcal {T})\) be the (linear) map given by \(\phi ^{(n)}((x_{i,j})_{i,j}) = (\phi (x_{i,j}))_{i,j}\). The map \(\phi \) is called positive (resp. unital) if \(\phi (\mathcal {S}^+)\subseteq \mathcal {T}^+\) (resp. \(\phi (I_H) = I_K\)), and completely positive if \(\phi ^{(n)}\) is positive for every \(n\in \mathbb {N}\). We call \(\phi \) a complete order embedding if it is injective and \(\phi ^{1}_{\phi (\mathcal {S})}: \phi (\mathcal {S}) \rightarrow \mathcal {S}\) is completely positive; we write \(\mathcal {S}\subseteq _\mathrm{c.o.i.} \mathcal {T}\). We note that \(\mathbb {C}\) is an operator system in a canonical way; a state of \(\mathcal {S}\) is a unital positive (linear) map \(\phi : \mathcal {S}\rightarrow \mathbb {C}\). We denote by \(S(\mathcal {S})\) the (convex) set of all states of \(\mathcal {S}\). We note that every operator system is an operator space in a canonical fashion, and denote by \(\mathcal {S}^\textrm{d}\) the dual Banach space of \(\mathcal {S}\), equipped with its canonical matrix order structure. Operator systems can be described abstractly via a set of axioms [59]; we refer the reader to [24, 59, 64] for details and for further background on operator space theory.
We denote by X the cardinality of a finite set X, let \(H^X = \oplus _{x\in X}H\) and denote by \(M_X\) the space of all complex matrices of size \(X\times X\); we identify \(M_X\) with \(\mathcal {L}(\mathbb {C}^X)\) and write \(I_X = I_{\mathbb {C}^{X}}\). For \(n\in \mathbb {N}\), we set \([n] = \{1,\dots ,n\}\) and \(M_n = M_{[n]}\). We write \((e_x)_{x\in X}\) for the canonical orthonormal basis of \(\mathbb {C}^{X}\), denote by \(\mathcal {D}_X\) the subalgebra of \(M_X\) of all diagonal, with respect to the basis \((e_x)_{x\in X}\), matrices, and let \(\Delta _X: M_X\rightarrow \mathcal {D}_X\) be the corresponding conditional expectation.
When \(\omega \) is a linear functional on \(M_X\), we often write \(\omega = \omega _X\). The canonical complete order isomorphism from \(M_X\) onto \(M_X^\textrm{d}\) maps an element \(\omega \in M_X\) to the linear functional \(f_{\omega }: M_X\rightarrow \mathbb {C}\) given by \(f_{\omega }(T) = \textrm{Tr}(T\omega ^\textrm{t})\) (here, and in the sequel, \(\omega ^\textrm{t}\) denotes the transpose of \(\omega \) in the canonical basis); see e.g. [63, Theorem 6.2]. We will thus consider \(M_X\) as selfdual space with pairing
On the other hand, note that the Banach space predual \(\mathcal {B}(H)_*\) can be canonically identified with \(\mathcal {T}(H)\); every normal functional \(\phi : \mathcal {B}(H) \rightarrow \mathbb {C}\) thus corresponds to a (unique) operator \(S_{\phi } \in \mathcal {T}(H)\) such that \(\phi (T) = \textrm{Tr}(TS_{\phi })\), \(T\in \mathcal {B}(H)\). 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 \(\omega ,\rho \in M_X\), \(S\in \mathcal {T}(H)\) and \(T\in \mathcal {B}(H)\), it will be convenient to continue writing
If X and Y are finite sets, we identify \(M_X\otimes M_Y\) with \(M_{X\times Y}\) and write \(M_{XY}\) in its place. Similarly, we set \(\mathcal {D}_{XY} = \mathcal {D}_X\otimes \mathcal {D}_Y\). Here, and in the sequel, we use the symbol \(\otimes \) to denote the algebraic tensor product of vector spaces. For an element \(\omega _X\in M_X\) and a Hilbert space H, we let \(L_{\omega _X}: M_X\otimes \mathcal {B}(H)\rightarrow \mathcal {B}(H)\) be the linear map given by \(L_{\omega _X}(S\otimes T) = \langle S,\omega _X\rangle T\). If \(H = \mathbb {C}^Y\) and \(\omega _Y\in M_Y\), we thus have linear maps \(L_{\omega _X}: M_{XY}\rightarrow M_Y\) and \(L_{\omega _Y}: M_{XY}\rightarrow M_X\); note that
and a similar formula holds for \(L_{\omega _Y}\). We let \(\textrm{Tr}_X: M_{XY}\rightarrow M_Y\) (resp. \(\textrm{Tr}_Y: M_{XY}\rightarrow M_X\)) be the partial trace, that is, \(\textrm{Tr}_X = L_{I_X}\) (resp. \(\textrm{Tr}_Y = L_{I_Y}\)).
Let X and A be finite sets. A classical information channel from X to A is a positive trace preserving linear map \(\mathcal {N}: \mathcal {D}_X\rightarrow \mathcal {D}_A\). It is clear that if \(\mathcal {N}: \mathcal {D}_X\rightarrow \mathcal {D}_A\) is a classical channel, then \(p(\cdot x):= \mathcal {N}(e_x e_x^*)\) is a probability distribution over A, and that \(\mathcal {N}\) is completely determined by the family \(\{(p(ax))_{a\in A}: x\in X\}\).
A quantum channel from \(M_X\) into \(M_A\) is a completely positive trace preserving map \(\Phi : M_X\rightarrow M_A\); such a \(\Phi \) will be called (X, A)classical if \(\Phi = \Delta _A\circ \Phi \circ \Delta _X\). A classical channel \(\mathcal {N}: \mathcal {D}_X\rightarrow \mathcal {D}_A\) gives rise to a (X, A)classical (quantum) channel \(\Phi _{\mathcal {N}}: M_X\rightarrow M_A\) by letting \(\Phi _{\mathcal {N}} = \mathcal {N} \circ \Delta _X\). Conversely, a quantum channel \(\Phi : M_X\rightarrow M_A\) induces a classical channel \(\mathcal {N}_{\Phi }: \mathcal {D}_X\rightarrow \mathcal {D}_A\) by letting \(\mathcal {N}_{\Phi } = \Delta _A\circ \Phi _{\mathcal {D}_X}\). Note that \(\mathcal {N}_{\Phi _{\mathcal {N}}} = \mathcal {N}\) for every classical channel \(\mathcal {N}\).
Let X, Y, A and B be finite sets. A quantum correlation over (X, Y, A, B) (or simply a quantum correlation if the sets are understood from the context) is a quantum channel \(\Gamma : M_{XY}\rightarrow M_{AB}\). Such a \(\Gamma \) is called a quantum nosignalling (QNS) correlation [22] if
and
We denote by \(\mathcal {Q}_\textrm{ns}\) the set of all QNS correlations; it is clear that \(\mathcal {Q}_\textrm{ns}\) is a closed convex subset of the cone \(\textrm{CP}(M_{XY},M_{AB})\) of all completely positive maps from \(M_{XY}\) into \(M_{AB}\).
Remark 2.1
A quantum channel \(\Gamma : M_{XY}\rightarrow M_{AB}\) is a QNS correlation if and only if
provided \(\rho '\), \(\rho ''\in M_{XY}\) are such that \(\textrm{Tr}_X\rho '=0\) and \(\textrm{Tr}_Y\rho ''=0\). Indeed, suppose that \(\Gamma \) is a QNS correlation and \(\rho '\in M_{XY}\), \(\textrm{Tr}_X\rho '=0\). Writing \(\rho '=\sum _{x,x',y,y'}\rho '_{x,x',y,y'}e_{x}e_{x'}^*\otimes e_ye_{y'}^*\), we have that \(\sum _{x\in X}\rho '_{x,x,y,y'}=0\) for all y, \(y'\in Y\).
Thus \(\textrm{Tr}\left( \sum _{x\in X}\rho '_{x,x,y,y'}e_xe_x^*\right) = 0\), and hence
Since \(\textrm{Tr}e_xe_{x'}^* = \delta _{x,x'}\), we also have \(\textrm{Tr}_A\Gamma (e_xe_{x'}^* \otimes e_ye_{y'}^*) = 0\) if \(x\ne x'\), for all \(y, y'\in Y\). It follows that \(\textrm{Tr}_A\Gamma (\rho ') = 0\). 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
where \((p(a,bx,y))_{(a,b)\in A\times B}\) is a probability distribution for each \((x,y)\in X\times Y\); classical correlations p thus correspond precisely to classical channels \(\mathcal {N}_p: \mathcal {D}_{XY}\rightarrow \mathcal {D}_{AB}\). A classical nosignalling correlation (or simply a nosignalling (NS) correlation) is a correlation \(p = ((p(a,bx,y))_{a,b})_{x,y}\) that satisfies the conditions
and
We denote by \(\mathcal {C}_\textrm{ns}\) the set of all NS correlations and identify its elements with classical channels from \(\mathcal {D}_{XY}\) to \(\mathcal {D}_{AB}\). Given a classical correlation p, we write \(\Gamma _p = \Phi _{\mathcal {N}_p}\); thus, \(\Gamma _p: M_{XY}\rightarrow M_{AB}\) is the \((X\times Y,A\times B)\)classical channel given by
Remark 2.2
If p is a classical correlation over (X, Y, A, B), then p is an NS correlation precisely when \(\Gamma _p\) is a QNS correlation. Indeed, if \(\textrm{Tr}\rho _X = 0\) and p satisfies (2.4) and (2.5), then
(2.3) is checked similarly. Conversely, assuming that \(\Gamma _p\) satisfies (2.2) and (2.3), the relations (2.4) and (2.5) are obtained by substituting in (2.6) \(\rho = e_{x}e_{x}^*\otimes e_{y}e_{y}^*  e_{x'}e_{x'}^*\otimes e_{y}e_{y}^*\) and \(\rho = e_{x}e_{x}^*\otimes e_{y}e_{y}^*  e_{x}e_{x}^*\otimes e_{y'}e_{y'}^*\). It follows that if \(\Gamma \) is a \((X\times Y, A\times B)\)classical QNS correlation, then \(\Gamma = \Gamma _p\) for some NS correlation p.
Let \(H_1,\dots ,H_n\) be Hilbert spaces, at most one of which is infinite dimensional, \(T\in \mathcal {B}(H_1\otimes \cdots \otimes H_n)\) and f be a bounded functional on \(\mathcal {B}(H_{i_1}\otimes \cdots \otimes H_{i_k})\), where \(k\le n\) and \(i_1,\dots ,i_k\) are distinct elements of [n] (not necessarily in increasing order). We will use the expression \(L_{f}(T)\), or \(\langle T,f\rangle \) (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 \(\mathcal {B}(H_{j_1}\otimes \cdots \otimes H_{j_l})\), where \(l\le n\) and the subset \(\{j_1,\dots ,j_l\}\) does not intersect \(\{i_1,\dots ,i_k\}\), then
Considering an element \(\omega \in M_X\) as a functional on \(M_X\) via (2.1), we have that if \(E = (E_{x,x'})_{x,x'}\in M_X\otimes \mathcal {B}(H)\), then
3 Stochastic Operator Matrices
Let X, Y, A and B be finite sets. A stochastic operator matrix over (X, A) is a positive operator \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) 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 \(E\in \mathcal {D}_X\otimes \mathcal {D}_A\otimes \mathcal {B}(\mathbb {C})\) coincide, after the natural identification of \(\mathcal {D}_X\otimes \mathcal {D}_A\) with the space of all \(X\times A\) matrices, with the rowstochastic scalarvalued matrices.
Let \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) be a stochastic operator matrix and \(E_{x,x',a,a'}\in \mathcal {B}(H)\), \(x,x'\in X\), \(a,a'\in A\), be the operators such that
we write \(E = (E_{x,x',a,a'})_{x,x',a,a'}\). Note that
Set
thus, \(E_{a,a'} = L_{e_{a} e_{a'}^*}(E)\), \(a,a'\in A\), and hence \(E_{a,a} \in \left( M_X\otimes \mathcal {B}(H)\right) ^+\), \(a\in A\). By Choi’s Theorem, stochastic operator matrices E are precisely the Choi matrices of unital completely positive maps \(\Phi _E: M_A\rightarrow M_X\otimes \mathcal {B}(H)\) defined by
Recall that a positive operatorvalued measure (POVM) on a Hilbert space H, indexed by A, is a family \((E_a)_{a\in A}\) of positive operators on H, such that \(\sum _{a\in A} E_a = I_H\). If \(E_a\) is a projection for each \(a\in A\), the family \((E_a)_{a\in A}\) is called a projection valued measure (PVM).
Theorem 3.1
Let H be a Hilbert space and \(E \in (M_X\otimes M_A\otimes \mathcal {B}(H))^+\). The following are equivalent:

(i)
E is a stochastic operator matrix;

(ii)
\((E_{a,a})_{a\in A}\) is a POVM in \(M_X\otimes \mathcal {B}(H)\);

(iii)
\(\textrm{Tr}_A L_{\omega _X}(E) = I_{H}\), for all states \(\omega _X \in M_X\);

(iv)
\(\textrm{Tr}_A L_{\omega _X}(E) = \textrm{Tr}(\omega _X) I_{H}\), for all \(\omega _X \in M_X\);

(v)
there exists a Hilbert space K and operators \(V_{a,x}: H\rightarrow K\), \(x\in X\), \(a\in A\), such that \((V_{a,x})_{a,x} \in \mathcal {B}(H^{X},K^{A})\) is an isometry and
$$\begin{aligned} E_{x,x',a,a'} = V_{a,x}^* V_{a',x'}, \ \ \ x,x'\in X, a,a'\in A. \end{aligned}$$(3.3)
In particular, if E is a stochastic operator matrix, then \((E_{x,x,a,a})_{a\in A}\) is a POVM for every \(x\in X\).
Proof
(i)\(\Leftrightarrow \)(ii) and (iv)\(\Rightarrow \)(iii) are trivial, while (i)\(\Rightarrow \)(iii) is immediate from (2.7).
(iii)\(\Rightarrow \)(iv) By assumption, \(\textrm{Tr}_A L_{\omega }(E) = \textrm{Tr}(\omega ) I_{H}\) for every state \(\omega \in M_X.\) Write \(\omega = \sum _{i=1}^4 \lambda _i \omega _i\), where \(\omega _i\) is a state in \(M_X\) and \(\lambda _i\in \mathbb {C}\), \(i = 1,2,3,4\). Then,
(iii)\(\Rightarrow \)(i) By (2.7), for all \(\omega _X\in S(M_X)\) and all normal states \(\tau \) on \(\mathcal {B}(H)\), we have
By polarisation and linearity,
for all \(\sigma \in (M_X\otimes \mathcal {B}(H))_*\), and hence \(\textrm{Tr}_A (E) = I_X\otimes I_H\).
(i)\(\Rightarrow \)(v) Let \(\Phi = \Phi _E\) be the unital completely positive map given by (3.2). By Stinespring’s dilation theorem, there exist a Hilbert space \(\tilde{K}\), an isometry \(V: \mathbb {C}^X\otimes H\rightarrow \tilde{K}\) and a unital *homomorphism \(\pi : M_A\rightarrow \mathcal {B}(\tilde{K})\) such that \(\Phi (T) = V^* \pi (T) V\), \(T\in M_A\). Up to unitary equivalence, \(\tilde{K} = \mathbb {C}^A\otimes K\) for some Hilbert space K and \(\pi (T) = T\otimes I_K\), \(T\in M_A\). Write \(V_{a,x}: H\rightarrow K\), \(a\in A\), \(x\in X\), for the entries of V, when V is considered as a block operator matrix. For \(\xi ,\eta \in H\), \(x,x'\in X\) and \(a,a'\in A\), we have
(v)\(\Rightarrow \)(ii) Let \(\xi = \sum _{x\in X} \sum _{a\in A} e_x\otimes e_a \otimes \xi _{x,a}\), where \(\xi _{x,a}\in H\), \(x\in X\), \(a\in A\). Using (3.3), we have
and thus E is positive. Since V is an isometry, we have
Let \((E_{x,a})_{a\in A}\) be a POVM on a Hilbert space H for every \(x\in X\). A stochastic operator matrix of the form
will be called classical. A general stochastic operator matrix can thus be thought of as a coordinatefree version of a finite family of POVM’s.
Remarks. (i) In view of Theorem 3.1, stochastic operator matrices are precisely the positive completions E of partially defined diagonal block matrices \(D = (E_{a,a})_{a\in A}\) with entries in \(M_X\otimes \mathcal {B}(H)\) and \(\textrm{Tr}_A(D) = I\).
(ii) The following generalisation of Naimark’s dilation Theorem was proved in [60]: if \((E_{x,a})_{a\in A}\subseteq \mathcal {B}(H)\), \(x\in X\), are POVM’s, then there exist a Hilbert space \(\tilde{H}\), a PVM \((\tilde{E}_a)_{a\in A}\subseteq \mathcal {B}(\tilde{H})\) and isometries \(V_x: H\rightarrow \tilde{H}\), \(x\in X\), with orthogonal ranges such that
This can be seen as a corollary of Theorem 3.1: given POVM’s \((E_{x,a})_{a\in A}\subseteq \mathcal {B}(H)\), \(x\in X\), 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. Set \(\tilde{E}_a = e_a e_a^*\otimes I_H\), \(a\in A\), and let \(V_x\) be the column isometry \((V_{a,x})_{a\in A}: H\rightarrow K^A\), \(x\in X\). Then, \((\tilde{E}_a)_{a\in A}\) is a PVM fulfilling (3.5).
Let \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) be a stochastic operator matrix and \(\Phi = \Phi _E\) be given by (3.2). Recall that the predual \(\Phi _*: M_X\otimes \mathcal {T}(H) \rightarrow M_A\) of \(\Phi \) is the completely positive map satisfying \(\langle \Phi _*(\rho ),\omega \rangle = \langle \rho , \Phi (\omega )\rangle \), \(\rho \in M_X\otimes \mathcal {T}(H)\), \(\omega \in M_A\). For a state \(\sigma \in \mathcal {T}(H)\), set
then \(\Gamma _{E,\sigma }: M_X\rightarrow M_A\) is a quantum channel. We have
indeed, if \(a, a'\in A\), then
(3.6) now follows by linearity. By Choi’s Theorem, every quantum channel \(\Phi : M_X\rightarrow M_A\) has the form \(\Gamma _{E,1}\) for some stochastic operator matrix \(E\in M_X\otimes M_A\).
Remark 3.2
Let H be a Hilbert space and \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) be a stochastic operator matrix. The following are equivalent:

(i)
E is classical;

(ii)
for each state \(\sigma \in \mathcal {T}(H)\), the quantum channel \(\Gamma _{E,\sigma }: M_X\rightarrow M_A\) is (X, A)classical.
Proof
The channel \(\Gamma _{E,\sigma }\) is (X, A)classical if and only if \(\Gamma _{E,\sigma }(e_x e_{x'}^*) = 0\) whenever \(x\ne x'\) and
The latter equality holds for every \(\sigma \) if and only if \(E_{x,x',a,a'} = 0\) whenever \(x\ne x'\) and \(E_{x,x,a,a'} = 0\) whenever \(a\ne a'\), that is, if and only if E is classical.
4 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]).
4.1 Quantum commuting QNS correlations
Let H be a Hilbert space, and \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) and \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H)\) be stochastic operator matrices. The pair (E, F) will be called commuting if \(L_{\omega _X\otimes \omega _A}(E)\) and \(L_{\omega _Y\otimes \omega _B}(F)\) commute for all \(\omega _X\in M_X\), \(\omega _Y\in M_Y\), \(\omega _A\in M_A\) and \(\omega _B\in M_B\). Writing \(E = (E_{x,x',a,a'})_{x,x',a,a'}\) and \(F = (F_{y,y',b,b'})_{y,y',b,b'}\), we have that (E, F) is commuting if and only if
Proposition 4.1
Let H be a Hilbert space and \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\), \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H)\) form a commuting pair of stochastic operator matrices. There exists a unique operator \(E\cdot F\in M_{XY}\otimes M_{AB}\otimes \mathcal {B}(H)\) such that
for all \(\rho _X \in M_X\), \(\rho _Y\in M_Y\), \(\rho _A \in M_A\), \(\rho _B\in M_B\) and \(\sigma \in \mathcal {T}(H)\). Moreover,

(i)
\(E\cdot F\) is a stochastic operator matrix;

(ii)
\(\Vert E\cdot F\Vert \le \Vert E\Vert \Vert F\Vert \);

(iii)
If \(\sigma \in \mathcal {T}(H)\) is a state, then \(\Gamma _{E \cdot F,\sigma }\) is a QNS correlation.
Proof
Let
Denote by \(\mathcal {A}\) (resp. \(\mathcal {B}\)) the C*algebra, generated by \(E_{x,x',a,a'}\), \(x,x'\in X\), \(a,a'\in A\) (resp. \(F_{y,y',b,b'}\), \(y,y'\in Y\), \(b,b'\in B\)); by assumption, \(\mathcal {B}\subseteq \mathcal {A}'\). Let \(\pi _{\mathcal {A}}: M_{XA}(\mathcal {A}) \rightarrow M_{XYAB}(\mathcal {B}(H))\) (resp. \(\pi _{\mathcal {B}}: M_{YB}(\mathcal {B}) \rightarrow M_{XYAB}(\mathcal {B}(H))\)) be the *representation given by \(\pi _{\mathcal {A}}(S) = S\otimes I_{YB}\) (resp. \(\pi _{\mathcal {B}}(T) = T\otimes I_{XA}\)). Then, the ranges of \(\pi _{\mathcal {A}}\) and \(\pi _{\mathcal {B}}\) commute and hence the pair \((\pi _{\mathcal {A}},\pi _{\mathcal {B}})\) gives rise to a *representation \(\pi : M_{XA}(\mathcal {A})\otimes _{\max }M_{YB}(\mathcal {B}) \rightarrow M_{XYAB}(\mathcal {B}(H))\) with \(\pi (S\otimes T) = \pi _{\mathcal {A}}(S)\pi _{\mathcal {B}}(T)\), \(S\in M_{XA}(\mathcal {A})\), \(T\in M_{YB}(\mathcal {B})\). Thus, \(E \cdot F = \pi (E\otimes F) \in M_{XYAB}(\mathcal {B}(H))^+\). Inequality (ii) now follows from the contractivity of *representations. In addition,
that is, \(E\cdot F\) is a stochastic operator matrix. For \(x,x'\in X\), \(y,y'\in Y\), \(a,a'\in A\), \(b,b'\in B\) and \(\sigma \in \mathcal {T}(H)\), we have
and (4.1) follows by linearity.
To show (iii), let \(\sigma \in \mathcal {T}(H)\) be a state. Suppose that \(\rho _X\in M_X\) is traceless and \(\rho _Y\in M_Y\). For every \(\tau _B\in M_B\), by (4.1) and Theorem 3.1, we have
Thus, (2.2) is satisfied; by symmetry, so is (2.3). \(\square \)
If \(\xi \) is a unit vector in H, we set for brevity \(\Gamma _{E,F,\xi } = \Gamma _{E\cdot F, \xi \xi ^*}\).
Definition 4.2
A QNS correlation of the form \(\Gamma _{E,F,\xi }\), where (E, F) is a commuting pair of stochastic operator matrices acting on a Hilbert space H, and \(\xi \in H\) is a unit vector, will be called quantum commuting.
We denote by \(\mathcal {Q}_\textrm{qc}\) the set of all quantum commuting QNS correlations.
Proposition 4.3
In Definition 4.2 one can assume, without gain of generality, that \(\sigma \) is an arbitrary state.
Proof
Suppose that H is a Hilbert space and \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\), \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H)\) form a commuting pair of stochastic operator matrices. Let \(\sigma \) be a state in \(\mathcal {T}(H)\) and write \(\sigma = \sum _{i=1}^{\infty } \lambda _i \xi _i \xi _i^*\), where \((\xi _i)_{i=1}^{\infty }\) is sequence of unit vectors and \(\lambda _i \ge 0\), \(i\in \mathbb {N}\), are such that \(\sum _{i=1}^{\infty } \lambda _i = 1\). Set \(\tilde{H} = H\otimes \ell ^2\) and \(\xi = \sum _{i=1}^{\infty } \sqrt{\lambda _i} \xi _i \otimes e_i\); then, \(\xi \) is a unit vector in \(\tilde{H}\) and \(\langle \xi \xi ^*,T\otimes I_{\ell ^2}\rangle = \langle \sigma ,T\rangle \), \(T\in \mathcal {B}(H)\).
Let \(\tilde{E} = E\otimes I_{\ell ^2}\) and \(\tilde{F} = F\otimes I_{\ell ^2}\); thus, \(\tilde{E}\) and \(\tilde{F}\) are stochastic operator matrices acting on \(\tilde{H}\) that form a commuting pair. Moreover, if \(\rho _X\in M_X\), \(\rho _Y \in M_Y\), \(\sigma _A \in M_A\) and \(\sigma _B\in M_B\), then
\(\square \)
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 \((E_{x,a})_{a\in A}\), \(x\in X\), and \((F_{y,b})_{b\in B}\), \(y\in Y\), 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 \(\xi \in H\), such that
Suppose that the stochastic operator matrices \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) and \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H)\) are classical, and correspond to the families \((E_{x,a})_{a\in A}\), \(x\in X\), and \((F_{y,b})_{b\in B}\), \(y\in Y\), respectively, as in (3.4). It is clear that pair (E, F) is commuting and \(E\cdot F\) is classical. We have that \(\Gamma _{p} = \Gamma _{E,F,\xi }\). Indeed, by Remark 3.2, the QNS correlation \(\Gamma _{E,F,\xi }\) is classical; by Remark 2.2, \(\Gamma _{E,F,\xi } = \Gamma _{p'}\) for some NS correlation \(p'\). It is now straightforward that \(p'=p\).
4.2 Quantum QNS correlations
Let \(H_A\) and \(H_B\) be Hilbert spaces, and \(E\in M_X\otimes M_A\otimes \mathcal {B}(H_A)\) and \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H_B)\) be stochastic operator matrices; then
Reshuffling the terms of the tensor product, we consider \(E\otimes F\) as an element of \(M_{XY}\otimes M_{AB}\otimes \mathcal {B}(H_A\otimes H_B)\); to underline this distinction, the latter element will henceforth be denoted by \(E\odot F\). Note that if
and
(where the last containment is up to a suitable permutation of the tensor factors), then \((\tilde{E},\tilde{F})\) is a commuting pair of stochastic operator matrices, and \(E\odot F = \tilde{E}\cdot \tilde{F}\). By Proposition 4.1, \(E\odot F\) is a stochastic operator matrix on \(H_A\otimes H_B\) and, if \(\sigma \in \mathcal {T}(H_A\otimes H_B)\) is a state, then, by Proposition 4.1, \(\Gamma _{E\odot F,\sigma }\) is a QNS correlation.
Remark 4.5
It is straightforward to check that if \(\sigma = \sigma _A\otimes \sigma _B\), where \(\sigma _A\in \mathcal {T}(H_A)\) and \(\sigma _B\in \mathcal {T}(H_B)\) are states, then \(\Gamma _{E,\sigma _A}\otimes \Gamma _{F,\sigma _B} = \Gamma _{E\odot F,\sigma _A\otimes \sigma _B}.\)
Definition 4.6
(i) A QNS correlation \(\Gamma : M_{XY}\rightarrow M_{AB}\) is called quantum if there exist finite dimensional Hilbert spaces \(H_A\) and \(H_B\), stochastic operator matrices \(E\in M_X\otimes M_A\otimes \mathcal {L}(H_A)\) and \(F\in M_Y\otimes M_B\otimes \mathcal {L}(H_B)\) and a pure state \(\sigma \in \mathcal {L}(H_A\otimes H_B)\) such that \(\Gamma = \Gamma _{E\odot F,\sigma }\).
(ii) A QNS correlation will be called approximately quantum if it is the limit of a sequence of quantum QNS correlations.
We denote by \(\mathcal {Q}_\textrm{q}\) (resp. \(\mathcal {Q}_\textrm{qa}\)) the set of all quantum (resp. approximately quantum) QNS correlations. It is clear from the definitions that \(\mathcal {Q}_\textrm{q}\subseteq \mathcal {Q}_\textrm{qc}\). It will be shown later that \(\mathcal {Q}_\textrm{qc}\) is closed, and hence contains \(\mathcal {Q}_\textrm{qa}\).
Similarly to Proposition 4.3, it can be shown that quantum QNS correlations can equivalently be defined using arbitrary, as opposed to pure, states.
Remark 4.7
Recall that a classical NS correlation p over (X, Y, A, B) is called quantum if there exist finite dimensional Hilbert spaces \(H_A\) and \(H_B\), POVM’s \((E_{x,a})_{a\in A}\), on \(H_A\), \(x\in X\), \((F_{y,b})_{b\in A}\) on \(H_B\), \(y\in Y\), and a unit vector \(\xi \in H_A\otimes H_B\), such that
It is easy to verify that if the stochastic operator matrices \(E\in M_X\otimes M_A\otimes \mathcal {B}(H_A)\) and \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H_B)\) are classical, and determined by the families \((E_{x,a})_{a\in A}\), \(x\in X\), and \((F_{y,b})_{b\in B}\), \(y\in Y\), then \(E\odot F\) is classical and determined by the family \(\{(E_{x,a}\otimes F_{y,b})_{(a,b)\in A\times B}: (x,y)\in X\times Y\}\). As in Remark 4.4, it is easy to see that \(\Gamma _{p} = \Gamma _{E,F,\xi }\).
Proposition 4.8
The sets \(\mathcal {Q}_\textrm{q}\) and \(\mathcal {Q}_\textrm{qa}\) are convex.
Proof
Let \(E_i \in M_X\otimes M_A\otimes \mathcal {L}(K_{1,i})\) (resp. \(F_i \in M_Y\otimes M_B\otimes \mathcal {L}(K_{2,i})\)) be a stochastic operator matrix over (X, A) (resp. (Y, B)) and \(\sigma _i = \eta _i \eta _i^*\) be a pure state on \(K_{1,i}\otimes K_{2,i}\), \(i = 1,\dots ,n\). Fix \(\lambda _i \ge 0\), \(i = 1,\dots ,n\), with \(\sum _{i=1}^n \lambda _i = 1\). Let \(K_{k} = \oplus _{i=1}^n K_{k,i}\), \(k = 1,2\), \(E = \oplus _{i=1}^n E_i\), \(F = \oplus _{i=1}^n F_i\), and \(\eta = \oplus _{i=1}^n \sqrt{\lambda _i} \eta _i \in K_1\otimes K_2\), viewed as supported on the (i, i)terms of \(K_1\otimes K_2 \equiv \oplus _{i,j =1}^n K_{1,i}\otimes K_{2,j}\). Set \(\sigma = \eta \eta ^*\). Using distributivity, we consider E (resp. F) as a stochastic operator matrix in \(M_X\otimes M_A\otimes \mathcal {L}(K_{1})\) (resp. \(M_Y\otimes M_B\otimes \mathcal {L}(K_{2})\)). A direct verification shows that
thus, \(\mathcal {Q}_\textrm{q}\) is convex, and the convexity of \(\mathcal {Q}_\textrm{qa}\) follows from the fact that \(\mathcal {Q}_\textrm{qa} = \overline{\mathcal {Q}_\textrm{q}}\). \(\square \)
4.3 Local QNS correlations
It is clear that if \(\Phi : M_X\rightarrow M_A\) and \(\Psi : M_Y \rightarrow M_B\) are quantum channels, then the quantum channel \(\Gamma := \Phi \otimes \Psi \) is a QNS correlation.
Definition 4.9
A QNS correlation \(\Gamma : M_{XY}\rightarrow M_{AB}\) is called local if it is a convex combination of quantum channels of the form \(\Phi \otimes \Psi \), where \(\Phi : M_X\rightarrow M_A\) and \(\Psi : M_Y \rightarrow M_B\) are quantum channels.
We denote by \(\mathcal {Q}_\textrm{loc}\) the set of all local QNS correlations. The elements of \(\mathcal {Q}_\textrm{loc}\) are precisely the maps that arise via local operations and shared randomness (LOSR) (see e.g. [72, p. 358]).
Remark 4.10
We have that \(\mathcal {Q}_\textrm{loc}\) is a closed convex subset of \(\mathcal {Q}_\textrm{q}\).
Proof
Let \(\Phi : M_X\rightarrow M_A\) and \(\Psi : M_Y \rightarrow M_B\) be quantum channels and \(E\in M_X\otimes M_A\) and \(F\in M_Y\otimes M_B\) be the Choi matrices of \(\Phi \) and \(\Psi \), respectively. By Remark 4.5,
and hence \(\Phi \otimes \Psi \in \mathcal {Q}_\textrm{q}\).
Let \((\Gamma _k)_{k\in \mathbb {N}} \subseteq \mathcal {Q}_\textrm{loc}\) be a sequence with limit \(\Gamma \in \mathcal {Q}_\textrm{ns}\). Note that \(\Gamma _k\) all are elements of a real vector space of dimension \(2X^2 Y^2 A^2 B^2\). Let \(L = 2X^2 Y^2 A^2 B^2 + 1\). By Carathéodory’s theorem, \(\Gamma _k = \sum _{l=1}^L \lambda _l^{(k)} \Phi _l^{(k)} \otimes \Psi _l^{(k)}\) as a convex combination. By compactness, we may assume, by passing to subsequences as necessary, that \(\Phi _l^{(k)} \rightarrow _{k \rightarrow \infty } \Phi _l\), \(\Psi _l^{(k)} \rightarrow _{k\rightarrow \infty } \Psi _l\) and \(\lambda _l^{(k)} \rightarrow _{k\rightarrow \infty } \lambda _l\). Thus, \(\Gamma = \sum _{l=1}^L \lambda _l \Phi _l \otimes \Psi _l\) as a convex combination, that is, \(\Gamma \in \mathcal {Q}_\textrm{loc}\), showing that \(\mathcal {Q}_\textrm{loc}\) is closed. \(\square \)
Remark 4.11
Recall that a classical NS correlation p over (X, Y, A, B) is called local if there exist families of probability distributions \(\{(p_k^1(ax))_{a\in A}: x\in X\}\) and \(\{(p_k^2(by))_{b\in B}: y\in Y\}\) and positive scalars \(\lambda _k\), \(k = 1,\dots ,m\), such that \(\sum _{k=1}^m \lambda _k = 1\) and
It is clear that if \(\Phi _k\) (resp. \(\Psi _k\)) is the (X, A)classical (resp. (Y, B)classical) channel corresponding to \(p_1^k\) (resp. \(p_2^k\)), then \(\Gamma _p = \sum _{k=1}^m \lambda _k\Phi _k\otimes \Psi _k\) and hence \(\Gamma _p \in \mathcal {Q}_\textrm{loc}\).
If needed, we specify the dependence of \(\mathcal {Q}_\textrm{x}\) on the sets X, Y, A and B by using the notation \(\mathcal {Q}_\textrm{x}(X,Y,A,B)\), for \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qa}, \textrm{qc}, \textrm{ns}\}\).
5 The Operator System of a Stochastic Operator Matrix
Recall [37, 73] that a ternary ring is a complex vector space \(\mathcal {V}\), equipped with a ternary operation \([\cdot ,\cdot ,\cdot ]: \mathcal {V}\times \mathcal {V}\times \mathcal {V}\rightarrow \mathcal {V}\), linear on the outer variables and conjugate linear in the middle variable, such that
A ternary representation of \(\mathcal {V}\) is a linear map \(\theta : \mathcal {V} \rightarrow \mathcal {B}(H,K)\), for some Hilbert spaces H and K, such that
We call \(\theta \) nondegenerate if \(\text {span}\{\theta (u)^*\eta : u\in \mathcal {V}, \eta \in K\}\) is dense in H. A concrete ternary ring of operators (TRO) [73] is a subspace \(\mathcal {U}\subseteq \mathcal {B}(H,K)\) for some Hilbert spaces H and K such that \(S,T,R\in \mathcal {U}\) implies \(ST^*R\in \mathcal {U}\).
Let X and A be finite sets, and \(\mathcal {V}^0_{X,A}\) be the ternary ring, generated by elements \(v_{a,x}\), \(x\in X\), \(a\in A\), 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.
Let \(\theta : \mathcal {V}_{X,A}^0\rightarrow \mathcal {B}(H,K)\) be a nondegenerate ternary representation. Setting \(V_{a,x} = \theta (v_{a,x})\), \(x\in X\), \(a\in A\), (5.2) implies
conversely, a family \(\{V_{a,x}: x\in X, a\in A\}\subseteq \mathcal {B}(H,K)\) satisfying (5.3) clearly gives rise to a nondegenerate ternary representation \(\theta : \mathcal {V}^0_{X,A}\rightarrow \mathcal {B}(H,K)\). We therefore call such a family a representation of the relations (5.1). We note that the set of representations of (5.1) is nonempty. Indeed, consider isometries \(V_x\), \(x\in X\), with orthogonal ranges on some Hilbert space H, i.e. \(V_x^*V_{x'} = \delta _{x, x'}I_H\), \(x,x'\in X\). Fix \(a_0\in A\) and let \(V_{a,x} = \delta _{a,a_0}V_x\). Then, \(\sum _{a\in A} V_{a,x}^*V_{a,x'} = V_x^*V_{x'} = \delta _{x,x'}I_H\).
We note that (5.3) implies \(\Vert V_{a,x}\Vert \le 1\), \(x\in X\), \(a\in A\). We identify the family \(\{V_{a,x}: a\in A, x\in X\}\) with the isometry \(V = (V_{a,x})_{a,x}\) and write \(H_V = H\), \(K_V = K\) and \(\theta _V = \theta \). Two representations \(V = (V_{a,x})_{a,x}\) and \(W = (W_{a,x})_{a,x}\) are called equivalent if there exist unitary operators \(U_H: H_V\rightarrow H_W\) and \(U_K: K_V\rightarrow K_W\) such that \(W_{a,x}U_H = U_KV_{a,x}\), \(x\in X\), \(a\in A\).
Write \(\hat{\theta } = \oplus _V \theta _V\), where the direct sum is taken over all equivalence classes of representations of the relations (5.1), where the cardinality of the underlying Hilbert spaces are bounded by that of \(\mathcal {V}_{X,A}^0\). For \(u\in \mathcal {V}^0_{X,A}\), let \(\Vert u\Vert _0:= \Vert \hat{\theta }(u)\Vert \). As \(\Vert v_{a,x}\Vert \le 1\) and \(\mathcal {V}_{X,A}^0\) is generated by \(v_{a,x}\), \(a\in A\), \(x\in X\), we have that \(\Vert u\Vert _0<\infty \) for every \(u\in \mathcal {V}^0_{X,A}\). It is also clear that \(\Vert \cdot \Vert _0\) is a seminorm on \(\mathcal {V}_{X,A}^0\). Set \(N = \left\{ u\in \mathcal {V}^0_{X,A}: \Vert u\Vert _0 = 0\right\} \). We have that N is a ternary ideal of \(\mathcal {V}^0_{X,A}\), that is, \([u_1,u_2,u_3]\in N\) if \(u_i\in N\) for some \(i\in \{1,2,3\}\). The ternary product of \(\mathcal {V}^0_{X,A}\) thus induces a ternary product on \(\mathcal {V}^0_{X,A}/N\), and \(\hat{\theta }\) induces a ternary representation of \(\mathcal {V}^0_{X,A}/N\) that will be denoted in the same way. Letting \(\Vert u\Vert := \Vert \hat{\theta }(u)\Vert \), \(u\in \mathcal {V}^0_{X,A}/N\), we have that \(\Vert \cdot \Vert \) is a norm on \(\mathcal {V}^0_{X,A}/N\), and hence \(\mathcal {V}^0_{X,A}/N\) is a ternary preC*ring (see [73]). We let \(\mathcal {V}_{X,A}\) be the completion of \(\mathcal {V}^0_{X,A}/N\); thus, \(\mathcal {V}_{X,A}\) is a ternary C*ring [73]. Note that \(\hat{\theta }\) extends to a ternary representation of \(\mathcal {V}_{X,A}\) (denoted in the same way) onto a concrete TRO, and the equality \(\Vert u\Vert = \Vert \hat{\theta }(u)\Vert \) continues to hold for every \(u\in \mathcal {V}_{X,A}\). We thus have that \(\mathcal {V}_{X,A}\) is a TRO in a canonical fashion. It is clear that each \(\theta _V\) induces a ternary representation of \(\mathcal {V}_{X,A}\) onto a TRO, which will be denoted in the same way.
Let \(\mathcal {C}_{X,A}\) be the right C*algebra of \(\mathcal {V}_{X,A}\); if \(\mathcal {V}_{X,A}\) is represented faithfully as a concrete ternary ring of operators in \(\mathcal {B}(H,K)\) for some Hilbert spaces H and K (that is, \(\mathcal {V}_{X,A}\mathcal {V}_{X,A}^*\mathcal {V}_{X,A} \subseteq \mathcal {V}_{X,A}\)), the C*algebra \(\mathcal {C}_{X,A}\) may be defined by letting
Each representation \(V = (V_{a,x})_{a,x}\) of the relations (5.1) gives rise [35] to a unital *representation \(\pi _V\) of \(\mathcal {C}_{X,A}\) on \(H_V\) by letting
Lemma 5.1
The following hold true:

(i)
Every nondegenerate ternary representation of \(\mathcal {V}_{X,A}\) has the form \(\theta _V\), for some representation V of the relations (5.1);

(ii)
\(\hat{\theta }\) is a faithful ternary representation of \(\mathcal {V}_{X,A}\);

(iii)
Every unital *representation \(\pi \) of \(\mathcal {C}_{X,A}\) has the form \(\pi _V\), for some representation V of the relations (5.1).
Proof
(i) Suppose that \(\theta \) is a nondegenerate ternary representation of \(\mathcal {V}_{X,A}\). Letting \(q: \mathcal {V}_{X,A}^0\rightarrow \mathcal {V}_{X,A}\) be the quotient map, write \(\theta _0 = \theta \circ q\); thus, \(\theta _0\) is a nondegenerate ternary representation of \(\mathcal {V}_{X,A}^0\). Letting V be the representation of the relations (5.1) such that \(\theta _0 = \theta _V\), we have that \(\theta = \theta _V\).
(ii) follows from the fact that \(\Vert \hat{\theta }(u)\Vert =\Vert u\Vert \), \(u\in \mathcal {V}_{X,A}\).
(iii) Let \(\pi : \mathcal {C}_{X,A} \rightarrow \mathcal {B}(H)\) be a unital *representation. Then, there exists a ternary representation \(\theta : \mathcal {V}_{X,A}\rightarrow \mathcal {B}(H,K)\) such that \(\pi (S^*T) = \theta (S)^*\theta (T)\), \(S,T\in \mathcal {V}_{X,A}\) (see e.g. [5, Theorem 3.4] and [25, p. 1636]). Since \(\pi \) is unital, \(\theta \) is nondegenerate. By (i), there exists a representation V of the relations (5.1) such that \(\theta = \theta _V\), and hence \(\pi = \pi _V\). \(\square \)
Set \(e_{x,x',a,a'} = v_{a,x}^*v_{a',x'} \in \mathcal {C}_{X,A}\), \(x,x' \in X\), \(a,a'\in A\). We call the operator subsystem
of \(\mathcal {C}_{X,A}\) the BrownCuntz operator system. Note that relations (5.1) imply
Theorem 5.2
Let H be a Hilbert space and \(\phi : \mathcal {T}_{X,A}\rightarrow \mathcal {B}(H)\) be a linear map. The following are equivalent:

(i)
\(\phi \) is a unital completely positive map;

(ii)
\(\left( \phi (e_{x,x',a,a'})\right) _{x,x',a,a'}\) is a stochastic operator matrix;

(iii)
there exists a *representation \(\pi : \mathcal {C}_{X,A}\rightarrow \mathcal {B}(H)\) such that \(\phi = \pi _{\mathcal {T}_{X,A}}\).
Moreover, if \(\left( E_{x,x',a,a'}\right) _{x,x',a,a'}\) is a stochastic operator matrix acting on a Hilbert space H, then there exists a (unique) unital completely positive map \(\phi : \mathcal {T}_{X,A}\rightarrow \mathcal {B}(H)\) such that \(\phi (e_{x,x',a,a'}) = E_{x,x',a,a'}\) for all \(x,x',a,a'\).
Proof
(i)\(\Rightarrow \)(ii) By Arveson’s extension theorem and Stinespring’s theorem, there exist a Hilbert space K, a *representation \(\pi : \mathcal {C}_{X,A} \rightarrow \mathcal {B}(K)\) and an isometry \(W\in \mathcal {B}(H,K)\), such that \(\phi (u) = W^*\pi (u) W\), \(u\in \mathcal {T}_{X,A}\). By Lemma 5.1, \(\pi = \pi _V\) for some representation \(V = (V_{a,x})_{a,x}\) of the relations (5.1). By the proof of Theorem 3.1, \(E:=\left( \pi (e_{x,x',a,a'})\right) \in (M_X\otimes M_A\otimes \mathcal {B}(K))^+\), and hence
By (5.4) and Theorem 3.1, \(\left( \phi (e_{x,x',a,a'})\right) _{x,x',a,a'}\) is a stochastic operator matrix.
(ii)\(\Rightarrow \)(iii) By Theorem 3.1, there exist a Hilbert space K and an isometry \(V = (V_{a,x})_{a,x}\in \mathcal {B}(H^X,K^A)\) such that
The *representation \(\pi _V\) of \(\mathcal {C}_{X,A}\) is an extension of \(\phi \).
(iii)\(\Rightarrow \)(i) is trivial.
Suppose that \(E = \left( E_{x,x',a,a'}\right) _{x,x',a,a'}\) is a stochastic operator matrix acting on H. Letting V be the isometry, associated with E via Theorem 3.1, we have that \(\phi := \pi _V_{\mathcal {T}_{X,A}}\) satisfies the required conditions. \(\square \)
Let \(\mathcal {S}\) be an operator system. Recall that the pair \((C_u^*(\mathcal {S}),\iota )\) is called a universal C*cover of \(\mathcal {S}\), if \(C_u^*(\mathcal {S})\) is a unital C*algebra, \(\iota : \mathcal {S}\rightarrow C_u^*(\mathcal {S})\) is a unital complete order embedding, and whenever H is a Hilbert space and \(\phi : \mathcal {S}\rightarrow \mathcal {B}(H)\) is a unital completely positive map, there exists a *representation \(\pi _{\phi }: C^*_u(\mathcal {S}) \rightarrow \mathcal {B}(H)\) such that \(\pi _{\phi } \circ \iota = \phi \). It is clear that the universal C*cover is unique up to a *isomorphism. The following corollary is immediate from Theorem 5.2.
Corollary 5.3
The pair \((\mathcal {C}_{X,A}, \iota )\), where \(\iota \) is the inclusion map of \(\mathcal {T}_{X,A}\) into \(\mathcal {C}_{X,A}\), is the universal C*cover of \(\mathcal {T}_{X,A}\).
We will need the following slight extension of the equivalence (i)\(\Leftrightarrow \)(ii) of Theorem 5.2.
Proposition 5.4
Let H be a Hilbert space and \(\phi : \mathcal {T}_{X,A}\rightarrow \mathcal {B}(H)\) be a linear map. The following are equivalent:

(i)
\(\phi \) is a completely positive map;

(ii)
\(\left( \phi (e_{x,x',a,a'})\right) _{x,x',a,a'} \in \left( M_X\otimes M_A\otimes \mathcal {B}(H)\right) ^+\).
Proof
(i)\(\Rightarrow \)(ii) It follows from Theorem 3.1 and Lemma 5.1 (iii), with \(\pi _V\) a faithful *representation of \(\mathcal {C}_{X,A}\), that \((e_{x,x',a,a'})\in (M_X\otimes M_A\otimes \mathcal {C}_{X,A})^+\). Since \(\mathcal {T}_{X,A}\subseteq \mathcal {C}_{X,A}\) as an operator subsystem, we have
and (ii) follows.
(ii)\(\Rightarrow \)(i) Write \(E = \left( \phi (e_{x,x',a,a'})\right) _{x,x',a,a'}\) and let \(T = \phi (1)\). Since \(1 = \sum _{a\in A} e_{x,x,a,a}\) (where x is any element of X), we have that \(T\ge 0\). Note also that if \(x,x'\in X\) and \(x\ne x'\), then
Assume first that T is invertible. Let \(\psi : \mathcal {T}_{X,A} \rightarrow \mathcal {B}(H)\) be the map given by
Setting \(F = \left( \psi (e_{x,x',a,a'})\right) _{x,x',a,a'}\), we have that
Let \(\omega = (\omega _{x,x'}) \in M_X\) and \(\sigma \) be a state in \(\mathcal {T}(H)\). Using (5.6), we have
By Theorem 3.1, F is a stochastic operator matrix; by Theorem 5.2, \(\psi \) is completely positive. Since \(\phi (\cdot ) = T^{1/2} \psi (\cdot ) T^{1/2}\), so is \(\phi \).
Now relax the assumption that T be invertible. For every \(\epsilon > 0\), let \(\phi _{\epsilon }: \mathcal {T}_{X,A}\rightarrow \mathcal {B}(H)\) be the map, given by \(\phi _{\epsilon }(u) = \phi (u) + \epsilon I\). By the previous paragraph, \(\phi _{\epsilon }\) is completely positive. Since \(\phi = \lim _{\epsilon \rightarrow 0} \phi _{\epsilon }\) in the pointnorm topology, \(\phi \) is completely positive. \(\square \)
Let
we consider \(\mathcal {L}_{X,A}\) as an operator subsystem of \(M_{XA}\). For the next proposition, note that, by [17, Corollary 4.5], if \(\mathcal {T}\) is a finite dimensional operator system than its (matrix ordered) dual \(\mathcal {T}^\textrm{d}\) is an operator system, when equipped with any faithful state of \(\mathcal {T}\) as an Archimedean order unit. It is straightforward to verify that, in this case, \(\mathcal {T}^\textrm{dd}\cong _\mathrm{c.o.i.} \mathcal {T}\).
Proposition 5.5
The linear map \(\Lambda : \mathcal {T}_{X,A}^\textrm{d} \rightarrow \mathcal {L}_{X,A}\), given by
is a welldefined complete order isomorphism.
Proof
By Proposition 5.4, if \(\phi \in \mathcal {T}_{X,A}\rightarrow \mathbb {C}\) is a positive functional, then \(\left( \phi (e_{x,x',a,a'})\right) _{x,x',a,a'}\in \mathcal {L}_{X,A}^+\). Thus, the map \(\Lambda _+: \left( \mathcal {T}_{X,A}^\textrm{d}\right) ^+ \rightarrow \mathcal {L}_{X,A}^+\), given by
is welldefined. It is clear that \(\Lambda _+\) is additive and
Suppose that \(\phi \in \mathcal {T}_{X,A}^\textrm{d}\) is a hermitian functional. If \(\phi = \phi _1  \phi _2\), where \(\phi _1\) and \(\phi _2\) are positive functionals on \(\mathcal {T}_{X,A}\), set
The map \(\Lambda _h: \left( \mathcal {T}_{X,A}\right) _h^\textrm{d} \rightarrow \mathcal {L}_{X,A}\) is welldefined: if \(\phi _1  \phi _2 = \psi _1  \psi _2\), where \(\phi _1\), \(\phi _2\), \(\psi _1\) and \(\psi _2\) are positive functionals then, by the additivity of \(\Lambda _+\), we have that \(\Lambda _+(\phi _1) + \Lambda _+(\psi _2) = \Lambda _+(\psi _1) + \Lambda _+(\phi _2)\), that is, \(\Lambda _+(\phi _1)  \Lambda _+(\phi _2) = \Lambda _+(\psi _1)  \Lambda _+(\psi _2)\). It is straightforward that the map \(\Lambda _h\) is \(\mathbb {R}\)linear, and thus it extends to a (\(\mathbb {C}\))linear map \(\Lambda : \mathcal {T}_{X,A}^\textrm{d}\rightarrow \mathcal {L}_{X,A}\).
Suppose that \((\phi _{i,j})_{i,j=1}^m\in M_m\left( \mathcal {T}_{X,A}^\textrm{d}\right) ^+\); thus, the map \(\Phi : \mathcal {T}_{X,A}\rightarrow M_m\), given by \(\Phi (u) = (\phi _{i,j}(u))_{i,j=1}^m\), is completely positive. By Proposition 5.4, \((\phi _{i,j}(e_{x,x',a,a'}))_{i,j} \in \left( M_{XA}\otimes M_m\right) ^+\). This shows that \(\Lambda \) is completely positive.
If \(\Lambda (\phi ) = 0\), then \(\phi (e_{x,x',a,a'}) = 0\) for all \(x,x'\in X\) and all \(a,a'\in A\), implying \(\phi = 0\); thus, \(\Lambda \) is injective. Since \(\mathcal {L}_{X,A}\) 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 \(\mathcal {L}_{X,A}\) is in the range of \(\Lambda \); it follows that \(\Lambda \) is surjective.
Finally, suppose that \(\phi _{i,j}\in \mathcal {T}_{X,A}^\textrm{d}\), \(i,j=1,\dots ,m\), are such that the matrix \(\left( \Lambda (\phi _{i,j})\right) _{i,j=1}^m\) is a positive element of \(M_m\left( \mathcal {L}_{X,A}\right) \). Let \(\Phi : \mathcal {T}_{X,A}\rightarrow M_m\) be given by \(\Phi (u) = (\phi _{i,j}(u))_{i,j=1}^m\). Then, \(\left( \Phi (e_{x,x',a,a'})\right) \in M_m\left( \mathcal {L}_{X,A}\right) ^+\). By Proposition 5.4, \(\Phi \) is completely positive, that is, \(\left( \phi _{i,j}\right) _{i,j=1}^m\in M_m\left( \mathcal {T}_{X,A}^\textrm{d}\right) ^+\). Thus, \(\Lambda ^{1}\) is completely positive, and the proof is complete. \(\square \)
Let \(\mathcal {S}\) be an operator system. A kernel in \(\mathcal {S}\) [44] is a linear subspace \(J\subseteq \mathcal {S}\), for which there exists an operator system \(\mathcal {T}\) and a unital completely positive map \(\psi : \mathcal {S}\rightarrow \mathcal {T}\) such that \(J = \ker (\psi )\). If J is a kernel in \(\mathcal {S}\), the quotient space \(\mathcal {S}/J\) can be equipped with a unique operator system structure with the property that whenever \(\mathcal {T}\) is an operator system and \(\phi : \mathcal {S}\rightarrow \mathcal {T}\) is a completely positive map annihilating J, the induced map \(\tilde{\phi }: \mathcal {S}/J\rightarrow \mathcal {T}\) is completely positive. If \(\mathcal {T}\) is an operator system, a surjective map \(\phi : \mathcal {S}\rightarrow \mathcal {T}\) is called complete quotient, if the map \(\tilde{\phi }\) is a complete order isomorphism. We refer the reader to [44] for further details.
Let
Corollary 5.6
The space \(J_{X,A}\) is a kernel in \(M_{XA}\) and the operator system \(\mathcal {T}_{X,A}\) is completely order isomorphic to the quotient \(M_{XA}/J_{X,A}\).
Proof
By Proposition 5.5, the map \(\Lambda : \mathcal {T}_{X,A}^\textrm{d} \rightarrow M_{XA}\) is a complete order embedding. By [29, Proposition 1.8], the dual \(\Lambda ^*: M_{XA}^\textrm{d} \rightarrow \mathcal {T}_{X,A}\) of \(\Lambda \) is a complete quotient map. Identifying \(M_{XA}^\textrm{d}\) with \(M_{XA}\) canonically, for an element \(f\in M_{XA}^\textrm{d}\), we have
Thus, \(\ker (\Lambda ^*) = J_{X,A}\). \(\square \)
6 Descriptions via Tensor Products
In this section, we provide a description of the classes of QNS correlations, introduced in Sect. 4, analogous to the description of the classes of NS correlations given in [49] (see also [27] and [62]). We will use the tensor theory of operator systems developed in [43]. If \(\mathcal {S}\) and \(\mathcal {T}\) are operator systems, \(\mathcal {S}\otimes _{\min }\mathcal {T}\) denotes the minimal tensor product of \(\mathcal {S}\) and \(\mathcal {T}\): if \(\mathcal {A}\) and \(\mathcal {B}\) are unital C*algebras, \(\mathcal {A}\otimes _{\min } \mathcal {B}\) is the spatial tensor product of \(\mathcal {A}\) and \(\mathcal {B}\), and \(\mathcal {S}\subseteq _\mathrm{c.o.i.} \mathcal {A}\) and \(\mathcal {T}\subseteq _\mathrm{c.o.i.} \mathcal {B}\), then \(\mathcal {S}\otimes _{\min }\mathcal {T}\subseteq _\mathrm{c.o.i.} \mathcal {A}\otimes _{\min }\mathcal {B}\). The commuting tensor product \(\mathcal {S}\otimes _\textrm{c} \mathcal {T}\) sits completely order isomorphically in the maximal tensor product \(C^*_u(\mathcal {S}) \otimes _{\max } C^*_u(\mathcal {T})\) of the universal C*covers of \(\mathcal {S}\) and \(\mathcal {T}\), while the maximal tensor product \(\mathcal {S}\otimes _{\max } \mathcal {T}\) is characterised by the property that it linearises jointly completely positive maps \(\theta : \mathcal {S}\times \mathcal {T}\rightarrow \mathcal {B}(H)\). We refer the reader to [43] for more details and further background.
Let X, Y, A and B be finite sets. As in Sect. 5, we write \(e_{x,x',a,a'}\), \(x,x'\in X\), \(a,a'\in A\), for the canonical generators of \(\mathcal {T}_{X,A}\). Similarly, we write \(f_{y,y',b,b'}\), \(y,y'\in Y\), \(b,b'\in B\), for the canonical generators of \(\mathcal {T}_{Y,B}\). Given a linear functional \(s: \mathcal {T}_{X,A} \otimes \mathcal {T}_{Y,B} \rightarrow \mathbb {C}\) (or a linear functional \(s: \mathcal {C}_{X,A} \otimes \mathcal {C}_{Y,B} \rightarrow \mathbb {C}\)), we let \(\Gamma _s: M_{XY}\rightarrow M_{AB}\) be the linear map given by
Remark 6.1
The correspondence \(s\rightarrow \Gamma _s\) is a linear map from the dual \((\mathcal {T}_{X,A} \otimes \mathcal {T}_{Y,B})^\textrm{d}\) of \(\mathcal {T}_{X,A} \otimes \mathcal {T}_{Y,B}\) into the space \(\mathcal {L}(M_{XY},M_{AB})\) of all linear transformations from \(M_{XY}\) into \(M_{AB}\).
Theorem 6.2
Let X, Y, A, B be finite sets and \(\Gamma : M_{XY}\rightarrow M_{AB}\) be a linear map. The following are equivalent:

(i)
\(\Gamma \) is a QNS correlation;

(ii)
there exists a state \(s: \mathcal {T}_{X,A} \otimes _{\max } \mathcal {T}_{Y,B} \rightarrow \mathbb {C}\) such that \(\Gamma = \Gamma _s\).
Proof
(i)\(\Rightarrow \)(ii) Let \(\Gamma : M_{XY}\rightarrow M_{AB}\) be a QNS correlation and write
for some \(C^{x',x,y',y}_{a,a',b,b'}\in \mathbb {C}\), \(x,x'\in X\), \(y,y'\in Y\), \(a,a'\in A\), \(b,b'\in B\). It follows from (2.2) and (2.3) that the Choi matrix \(C:= \left( C^{x,x',y,y'}_{a,a',b,b'}\right) \) of \(\Gamma \) satisfies the following conditions (see also [22]):

(a)
\(C\in M_{XYAB}^+\);

(b)
there exists \(c_{b,b'}^{y,y'} \in \mathbb {C}\) such that \(\sum _{a\in A} C^{x,x',y,y'}_{a,a,b,b'} = \delta _{x,x'} c_{b,b'}^{y,y'}\), \(y,y'\in Y\), \(b,b'\in B\);

(c)
there exists \(d_{a,a'}^{x,x'}\in \mathbb {C}\) such that \(\sum _{b\in B} C^{x,x',y,y'}_{a,a',b,b} = \delta _{y,y'} d_{a,a'}^{x,x'}\), \(x,x'\in X\), \(a,a'\in A\).
By condition (b), \(L_{\omega _{YB}}(C)\in \mathcal {L}_{X,A}\) for every \(\omega _{YB}\in M_{YB}\), while by condition (c), \(L_{\omega _{XA}}(C)\in \mathcal {L}_{Y,B}\) for every \(\omega _{XA}\in M_{XA}\). Thus,
by the injectivity of the minimal operator system tensor product, \(C\in \left( \mathcal {L}_{X,A}\otimes _{\min }\mathcal {L}_{Y,B}\right) ^+\).
By [29, Proposition 1.9] and Proposition 5.5,
via the identification \(\Lambda \) given by (5.8). The state s of \(\mathcal {T}_{X,A}\otimes _{\max }\mathcal {T}_{Y,B}\) corresponding to C via (6.2) satisfies
Thus, \(\Gamma = \Gamma _s\).
(ii)\(\Rightarrow \)(i) Let s be a state of \(\mathcal {T}_{X,A}\otimes _{\max }\mathcal {T}_{Y,B}\), and define \(C^{x,x',y,y'}_{a,a',b,b'}\) via (6.3); thus, C is the Choi matrix of \(\Gamma _s\). By (5.5) and the definition of the maximal tensor product,
and hence the matrix \(C:= \left( C^{x,x',y,y'}_{a,a',b,b'}\right) \) is positive; by Choi’s Theorem, \(\Gamma _s\) is completely positive. Relations (5.4) imply that \(\Gamma _s\) is trace preserving and that conditions (b) and (c) hold. Suppose that \(\rho _X = (\rho _{x,x'})_{x,x'} \in M_X\) has zero trace and \(\rho _Y = (\rho ^{y,y'})_{y,y'} \in M_Y\). We have
that is, (2.2) holds; similarly, (c) implies (2.3). \(\square \)
Theorem 6.3
Let X, Y, A, B be finite sets and \(\Gamma : M_{XY}\rightarrow M_{AB}\) be a linear map. The following are equivalent:

(i)
\(\Gamma \) is a quantum commuting QNS correlation;

(ii)
there exists a state \(s: \mathcal {T}_{X,A} \otimes _\textrm{c} \mathcal {T}_{Y,B} \rightarrow \mathbb {C}\) such that \(\Gamma = \Gamma _s\);

(iii)
there exists a state \(s: \mathcal {C}_{X,A} \otimes _{\max } \mathcal {C}_{Y,B} \rightarrow \mathbb {C}\) such that \(\Gamma = \Gamma _s\).
Proof
(i)\(\Rightarrow \)(iii) Let H be a Hilbert space, \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\), \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H)\) form a commuting pair of stochastic operator matrices, and \(\tau \in \mathcal {T}(H)^+\) be such that \(\Gamma = \Gamma _{E\cdot F,\tau }\). By Theorem 5.2, there exist representations \(\pi _X\) and \(\pi _Y\) of \(\mathcal {C}_{X,A}\) and \(\mathcal {C}_{Y,B}\), respectively, such that \(E_{x,x',a,a'} = \pi _X(e_{x,x',a,a'})\) and \(F_{y,y',b,b'} = \pi _Y(e_{y,y',b,b'})\) for all \(x,x'\in X\), \(y,y'\in Y\), \(a,a'\in A\), \(b,b'\in B\). Since \(\mathcal {C}_{X,A}\) (resp. \(\mathcal {C}_{Y,B}\)) is generated by the elements \(e_{x,x',a,a'}\), \(x,x'\in X\), \(a,a'\in A\) (resp. \(f_{y,y',b,b'}\), \(y,y'\in Y\), \(b,b'\in B\)), \(\pi _X\) and \(\pi _Y\) have commuting ranges. Let \(\pi _X\times \pi _Y\) be the (unique) *representation \(\mathcal {C}_{X,A}\otimes _{\max }\mathcal {C}_{Y,B}\rightarrow \mathcal {B}(H)\) such that \((\pi _X\times \pi _Y)(u\otimes v) = \pi _X(u)\pi _Y(v)\), \(u\in \mathcal {C}_{X,A}\), \(v\in \mathcal {C}_{Y,B}\). By (4.2),
Letting \(s(w) = \langle (\pi _X\times \pi _Y)(w),\tau \rangle \), \(w\in \mathcal {C}_{X,A}\otimes _{\max }\mathcal {C}_{Y,B}\), we have \(\Gamma = \Gamma _s\).
(iii)\(\Rightarrow \)(i) Let s be a state on \(\mathcal {C}_{X,A}\otimes _{\max }\mathcal {C}_{Y,B}\) and write \(\pi _s\) and \(\xi _s\) for the corresponding GNS representation of \(\mathcal {C}_{X,A}\otimes _{\max }\mathcal {C}_{Y,B}\) and for its cyclic vector, respectively. Then, \(E:=(\pi _s(e_{x,x',a,a'}\otimes 1))_{x,x',a,a'}\) and \(F:=(\pi _s(1\otimes f_{y,y',b,b'})_{y,y',b,b'}\) form a commuting pair of stochastic operator matrices; moreover, for \(x,x'\in X\), \(y,y'\in Y\), \(a,a'\in A\) and \(b,b'\in B\), we have
(ii)\(\Leftrightarrow \)(iii) By Corollary 5.3 and [43, Theorem 6.4], \(\mathcal {T}_{X,A}\otimes _\textrm{c} \mathcal {T}_{Y,B}\) sits completely order isomorphically in \(\mathcal {C}_{X,A}\otimes _{\max } \mathcal {C}_{Y,B}\); thus the states of \(\mathcal {T}_{X,A}\otimes _\textrm{c} \mathcal {T}_{Y,B}\) are precisely the restrictions of the states of \(\mathcal {C}_{X,A}\otimes _{\max } \mathcal {C}_{Y,B}\). \(\square \)
Corollary 6.4
The set \(\mathcal {Q}_\textrm{qc}\) is closed and convex.
Proof
By Theorem 6.3 and Remark 6.1, the map \(s\rightarrow \Gamma _s\) is an affine bijection from the state space of \(\mathcal {T}_{X,A} \otimes _\textrm{c} \mathcal {T}_{Y,B}\) onto \(\mathcal {Q}_\textrm{qc}\). It is straightforward that it is also a homeomorphism, when its domain is equipped with the weak* topology. Since the state space of \(\mathcal {T}_{X,A} \otimes _\textrm{c} \mathcal {T}_{Y,B}\) is weak* compact, its range is (convex and) closed. \(\square \)
Theorem 6.5
Let X, Y, A, B be finite sets and \(\Gamma : M_{XY}\rightarrow M_{AB}\) be a linear map. The following are equivalent:

(i)
\(\Gamma \) is an approximately quantum QNS correlation;

(ii)
there exists a state \(s: \mathcal {T}_{X,A} \otimes _{\min } \mathcal {T}_{Y,B} \rightarrow \mathbb {C}\) such that \(\Gamma = \Gamma _s\);

(iii)
there exists a state \(s: \mathcal {C}_{X,A} \otimes _{\min } \mathcal {C}_{Y,B} \rightarrow \mathbb {C}\) such that \(\Gamma = \Gamma _s\).
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.
(iii)\(\Rightarrow \)(i) Let \(\pi _X: \mathcal {C}_{X,A}\rightarrow \mathcal {B}(H_X)\) and \(\pi _Y: \mathcal {C}_{Y,B}\rightarrow \mathcal {B}(H_Y)\) be faithful *representations. Then, \(\pi _X\otimes \pi _Y: \mathcal {C}_{X,A} \otimes _{\min } \mathcal {C}_{Y,B}\rightarrow \mathcal {B}(H_X\otimes H_Y)\) is a faithful *representation of \(\mathcal {C}_{X,A} \otimes _{\min } \mathcal {C}_{Y,B}\). Let s be a state satisfying (iii). By [40, Corollary 4.3.10], s can be approximated in the weak* topology by elements of the convex hull of vector states on \((\pi _X\otimes \pi _Y)(\mathcal {C}_{X,A} \otimes _{\min } \mathcal {C}_{Y,B})\); thus, given \(\varepsilon > 0\), there exist unit vectors \(\xi _1,\ldots ,\xi _n\in H_X\otimes H_Y\) and positive scalars \(\lambda _1,\ldots ,\lambda _n\) with \(\sum _{i=1}^n\lambda _i=1\) such that
for all \(x,x'\in X\), \(y,y'\in Y\), \(a,a'\in A\) and \(b,b'\in B\). Let \(\xi = \oplus _{i=1}^n\sqrt{\lambda _i}\xi _i\in \mathbb C^{n}\otimes (H_X\otimes H_Y)\); then, \(\Vert \xi \Vert = 1\). Set \(E_{x,x',a,a'} = I_n \otimes \pi _X(e_{x,x',a,a'})\) and \(F_{y,y',b,b'} = \pi _Y(f_{y,y',b,b'})\). Then, \((E_{x,x'a,a'})_{x,x',a,a'}\) (resp. \((F_{y,y',b,b'})_{y,y',b,b'}\)) is a stochastic operator matrix on \(\mathbb {C}^n\otimes H_X\) (resp. \(H_Y\)), and
It follows that \(\Gamma _s\) is in the closure of the set of correlations of the form \(\Gamma _{E\odot F,\xi }\), where E and F act on, possibly infinite dimensional, Hilbert spaces H and K. Given such a correlation \(\Gamma _{E\odot F,\xi }\), let \((P_{\alpha })_{\alpha }\) (resp. \((Q_{\beta })_{\beta }\)) be a net of finite rank projections on H (resp. K) such that \(P_{\alpha } \rightarrow _{\alpha } I_H\) (resp. \(Q_{\beta } \rightarrow _{\beta } I_K\)) in the strong operator topology. Set \(H_{\alpha } = P_{\alpha }H\) (resp. \(K_{\beta } = Q_{\beta }K\)), \(E_{\alpha } = (I_X\otimes I_A\otimes P_{\alpha })E(I_X\otimes I_A\otimes P_{\alpha })\) (resp. \(F_{\beta } = (I_Y\otimes I_B\otimes Q_{\beta })F(I_Y\otimes I_B\otimes Q_{\beta })\)), and \(\xi _{\alpha ,\beta } = \frac{1}{\Vert (P_{\alpha }\otimes Q_{\beta })\xi \Vert }(P_{\alpha }\otimes Q_{\beta })\xi \) (note that \(\xi _{\alpha ,\beta }\) is eventually welldefined). Then, \(E_{\alpha }\) (resp. \(F_{\beta }\)) is a stochastic operator matrix acting on \(H_{\alpha }\) (resp. \(K_{\beta }\)), and \(\Gamma _{E_{\alpha }\odot F_{\beta },\xi _{\alpha ,\beta }} \rightarrow _{(\alpha ,\beta )} \Gamma _{E\odot F,\xi }\) along the product net. It follows that \(\Gamma _s \in \mathcal {Q}_\textrm{qa}\).
(i)\(\Rightarrow \)(iii) Given \(\varepsilon > 0\), let E and F be stochastic operator matrices acting on finite dimensional Hilbert spaces \(H_X\) and \(H_Y\), respectively, and \(\xi \in H_X\otimes H_Y\) be a unit vector, such that
for all \(x,x'\in X\), \(y,y'\in Y\), \(a,a'\in A\) and \(b,b'\in B\). By Lemma 5.1, there exists a *representation \(\pi _X\) (resp. \(\pi _Y\)) of \(\mathcal {C}_{X,A}\) (resp. \(\mathcal {C}_{Y,B}\)) on \(H_X\) (resp. \(H_Y\)) such that \(E_{x,x',a,a'} = \pi _X(e_{x,x',a,a'})\) (resp. \(F_{y,y',b,b'} = \pi _Y(f_{y,y',b,b'})\)), \(x,x'\in X\), \(a,a'\in A\) (resp. \(y,y'\in Y\), \(b,b'\in B\)). Let \(s_\varepsilon \) be the state on \(\mathcal {C}_{X,A}\otimes _{\min }\mathcal {C}_{Y,B}\) given by
and s be a cluster point of the sequence \(\{s_{1/n}\}_n\) in the weak* topology. Then,
giving \(\Gamma = \Gamma _s\).
(ii)\(\Leftrightarrow \)(iii) follows from the fact that \(\mathcal {T}_{X,A} \otimes _{\min } \mathcal {T}_{Y,B} \subseteq _\mathrm{c.o.i} \mathcal {C}_{X,A} \otimes _{\min } \mathcal {C}_{Y,B}.\) \(\square \)
Recall [63] that, given any Archimedean ordered unit (AOU) space V, there exists a (unique) operator system \(\textrm{OMIN}(V)\) (resp. \(\textrm{OMAX}(V)\)) 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 \(\mathcal {T}\) into V (resp. from V into an operator system \(\mathcal {T}\)) is automatically completely positive as a map from \(\mathcal {T}\) into \(\textrm{OMIN}(V)\) (resp. from \(\textrm{OMAX}(V)\) into \(\mathcal {T}\)). If V is in addition an operator system, we denote by \(\textrm{OMIN}(V)\) (resp. \(\textrm{OMAX}(V)\)) 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. An element \(u\in \textrm{OMAX}(V)\otimes _{\max } \textrm{OMAX}(W)\) is positive if and only if \(u = \sum _{i=1}^k v_i \otimes w_i\), for some \(v_i\in V^+\), \(w_i\in W^+\), \(i = 1,\dots ,k\).
Proof
Let D be the set of all sums of elementary tensors \(v\otimes w\) with \(v \in V^+\) and \(w\in W^+\). We claim that if, for every \(\epsilon > 0\), there exists \(u_{\epsilon }\in D\) such that \(\Vert u_{\epsilon }\Vert \rightarrow _{\epsilon \rightarrow 0} 0\) and \(u + u_{\epsilon } \in D\) for every \(\epsilon > 0\), then \(u\in D\). Assume, without loss of generality, that \(\Vert u_{\epsilon }\Vert \le 1\) for all \(\epsilon > 0\). Set \(L = 2\dim (V)\dim (W) + 1\) and, using Carathéodory’s theorem, write
where \(v_j^{(\epsilon )} \in V^+\), \(w_j^{(\epsilon )} \in W^+\) and \(\Vert v_j^{(\epsilon )}\Vert = \Vert w_j^{(\epsilon )}\Vert \) for all \(j = 1,\dots ,L\) and all \(\epsilon > 0\). Since \(v_j^{(\epsilon )} \otimes w_j ^{(\epsilon )} \le u + u_{\epsilon }\) and \(\Vert u + u_{\epsilon }\Vert \le \Vert u\Vert + 1\) for all \(\epsilon > 0\), we have \(\Vert v_j^{(\epsilon )}\Vert \le \sqrt{\Vert u\Vert + 1}\) and \(\Vert w_j^{(\epsilon )}\Vert \le \sqrt{\Vert u\Vert + 1}\), \(j = 1,\dots ,L\). By compactness, we may assume that \(v_j^{(\epsilon )} \rightarrow _{\epsilon \rightarrow 0} v_j\) and \(w_j^{(\epsilon )} \rightarrow _{\epsilon \rightarrow 0} w_j\) for all \(j = 1,\dots ,L\). It follows that \(u = \sum _{j=1}^L v_j \otimes w_j \in D\).
Let
for some \(a_p\in M_n\), \(v_p\in V^+\), \(p = 1,\dots , l\), and \(b_q\in M_m^+\), \(w_q\in W^+\), \(q = 1,\dots , r\). If \(\alpha \in M_{1,nm}\), then
Suppose that \(S\in M_n(\textrm{OMAX}(V))^+\) and \(\alpha \in M_{1,nm}\). By the definition of the maximal tensor product [43], if \(\epsilon > 0\), then \(S + \epsilon 1_n\) has the form of \(S_0\) in (6.4). Hence
Since \(\alpha \left( 1_n \otimes T_0\right) \alpha ^* \in D\), the previous paragraph shows that
Now let \(T\in M_m(\textrm{OMAX}(W))^+\), and write \(T + \epsilon 1_m\) in the form of \(T_0\) in (6.4). Then,
By the previous paragraph, \(\alpha \left( S \otimes 1_m\right) \alpha ^*\in D\); by the first paragraph, \(\alpha \left( S\otimes T\right) \alpha ^* \in D\).
Let \(u\in \left( \textrm{OMAX}(V)\otimes _{\max } \textrm{OMAX}(W)\right) ^+\). By the definition of the maximal tensor product [43], for every \(\epsilon > 0\), there exist \(n,m\in \mathbb {N}\), \(S\in M_n(\textrm{OMAX}(V))^+\), \(T\in M_m(\textrm{OMAX}(W))^+\) and \(\alpha \in M_{1,nm}\), such that \(u + \epsilon 1 = \alpha \left( S\otimes T\right) \alpha ^*\). By the previous and the first paragraph, \(u\in D\). \(\square \)
Theorem 6.7
Let X, Y, A, B be finite sets and \(\Gamma : M_{XY}\rightarrow M_{AB}\) be a linear map. The following are equivalent:

(i)
\(\Gamma \) is a local QNS correlation;

(ii)
there exists a state \(s: \textrm{OMIN}(\mathcal {T}_{X,A}) \otimes _{\min } \textrm{OMIN}(\mathcal {T}_{Y,B}) \rightarrow \mathbb {C}\) such that \(\Gamma = \Gamma _s\).
Proof
(ii)\(\Rightarrow \)(i) Let \(s: \textrm{OMIN}(\mathcal {T}_{X,A}) \otimes _{\min } \textrm{OMIN}(\mathcal {T}_{Y,B}) \rightarrow \mathbb {C}\) be a state. Using [41, Theorem 9.9] and [29, Proposition 1.9], we can identify s with an element of \(\big (\textrm{OMAX}(\mathcal {T}_{X,A}^\textrm{d}) \otimes _{\max } \textrm{OMAX}(\mathcal {T}_{Y,B}^\textrm{d})\big )^+\). By Lemma 6.6, there exist states \(\phi _i\in \left( \mathcal {T}_{X,A}^\textrm{d}\right) ^+\) and \(\psi _i\in \left( \mathcal {T}_{Y,B}^\textrm{d}\right) ^+\), and nonnegative scalars \(\lambda _i\), \(i = 1,\dots ,m\), such that \(s\equiv \sum _{i=1}^m \lambda _i \phi _i\otimes \psi _i\). Set \(E_i = (\phi _i(e_{x,x',a,a'}))_{x,x',a,a'}\) (resp. \(F_i = (\psi _i(f_{y,y',b,b'}))_{y,y',b,b'}\)), and let \(\Phi _i: M_X\rightarrow M_A\) (resp. \(\Psi _i: M_Y\rightarrow M_B\)) be the quantum channel with Choi matrix \(E_i\) (resp. \(F_i\)), \(i = 1,\dots ,m\). Then, \(\Gamma _s = \sum _{i=1}^m \lambda _i \Phi _i\otimes \Psi _i\).
(i)\(\Rightarrow \)(ii) Write \(\Gamma = \sum _{i=1}^m \lambda _i \Phi _i\otimes \Psi _i\) as a convex combination of quantum channels \(\Phi _i: M_X\rightarrow M_A\) and \(\Psi _i: M_Y\rightarrow M_B\), \(i = 1,\dots ,m\), and let s be a functional on \(\mathcal {T}_{X,A}\otimes \mathcal {T}_{Y,B}\) such that \(\Gamma = \Gamma _s\). Let \(E_i \in (M_X\otimes M_A)^+\) (resp. \(F_i \in (M_Y\otimes M_B)^+\)) be the Choi matrix of \(\Phi _i\) (resp. \(\Psi _i\)); thus, \(E_i\) (resp. \(F_i\)) is a stochastic operator matrix acting on \(\mathbb C\). By Theorem 5.2, there exist positive functionals \(\phi _i: \mathcal {T}_{X,A}\rightarrow \mathbb {C}\) and \(\psi _i: \mathcal {T}_{Y,B}\rightarrow \mathbb {C}\) such that \((\phi _i(e_{x,x',a',a}))_{x,x',a,a'} = E_i\) and \((\psi _i(f_{y,y',b',b}))_{y,y',b,b'} = F_i\), \(i = 1,\dots ,m\). It is now straightforward to see that s is the functional corresponding to \(\sum _{i=1}^m \lambda _i\phi _i\otimes \psi _i\) and is hence, by Lemma 6.6, a state on \( \textrm{OMIN}(\mathcal {T}_{X,A}) \otimes _{\min } \textrm{OMIN}(\mathcal {T}_{Y,B})\). \(\square \)
7 ClassicaltoQuantum Nosignalling Correlations
In this section, we consider the set of classicaltoquantum nosignalling correlations, and provide descriptions of its various subclasses in terms of canonical operator systems.
7.1 Definition and subclasses
Let X, Y, A and B be finite sets and H be a Hilbert space.
Definition 7.1
A family \(\Theta = (\sigma _{x,y})_{x\in X, y\in Y}\) of states in \(M_{AB}\) is called a classicaltoquantum nosignalling (CQNS) correlation if
for all \(x,x',x''\in X\) and all \(y,y',y''\in Y\).
A stochastic operator matrix \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) will be called semiclassical if \(L_{e_x e_{x'}^*}(E) = 0\) whenever \(x\ne x'\), that is, if
for some \(E_x\in \left( M_A\otimes \mathcal {B}(H)\right) ^+\) with \(\textrm{Tr}_A E_x = I_H\), \(x\in X\). We write \(E = (E_x)_{x\in X}\); note that, in its own right, \(E_x\) is a stochastic operator matrix in \(\mathcal {L}(\mathbb {C})\otimes M_A\otimes \mathcal {B}(H)\), \(x\in X\).
Suppose that \(E = (E_x)_{x\in X}\) and \(F = (F_y)_{y\in Y}\) form a commuting pair of semiclassical stochastic operator matrices, acting on a Hilbert space H and \(\sigma \) is a vector state on \(\mathcal {B}(H)\). The family formed by the states
is a CQNS correlation; indeed, by Proposition 4.1, \(\textrm{Tr}_A \sigma _{x,y} = L_{\sigma }(F_y)\) and \(\textrm{Tr}_B \sigma _{x,y} = L_{\sigma }(E_x)\) for all x, y. We call the CQNS correlations of this form quantum commuting. Similarly, if \((E_x)_{x\in X}\) (resp. \((F_y)_{y\in Y}\)) is a semiclassical stochastic operator matrix on \(H_A\) (resp. \(H_B\)) and \(\sigma \) is a vector state on \(\mathcal {L}(H_A\otimes H_B)\), the family formed by
will be called a quantum CQNS correlation. A CQNS correlation \(\Theta = (\sigma _{x,y})_{x\in X, y\in Y}\) will be called approximately quantum if there exist quantum CQNS correlations \(\Theta _n = (\sigma _{x,y}^{(n)})_{x\in X, y\in Y}\), \(n\in \mathbb {N}\), such that
Finally, \(\Theta \) will be called local if there exist states \(\sigma _{i,x}^{A}\) (resp. \(\sigma _{i,y}^{B}\)) in \(M_A\) (resp. \(M_B\)) and scalars \(\lambda _i > 0\), \(i = 1,\dots ,m\), such that
If \(\mathcal {E}: \mathcal {D}_{XY}\rightarrow M_{AB}\) is a (classicaltoquantum) channel, we set \(\Gamma _{\mathcal {E}} = \mathcal {E}\circ \Delta _{XY}\); thus, \(\Gamma _{\mathcal {E}}\) is a (quantum) channel from \(M_{XY}\) to \(M_{AB}\). Given a CQNS correlation \(\Theta = \left( \sigma _{x,y}\right) _{x\in X, y\in Y}\), we let \(\mathcal {E}_{\Theta }: \mathcal {D}_{XY}\rightarrow M_{AB}\) be the channel given by
and \(\Gamma _{\Theta } = \Gamma _{\mathcal {E}_{\Theta }}\). In the sequel, we will often identify \(\Theta \) with the channel \(\mathcal {E}_{\Theta }\). For \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qa}, \textrm{qc}, \textrm{ns}\}\), we write \(\mathcal{C}\mathcal{Q}_\textrm{x}\) for the set of all CQNS correlations of class \(\textrm{x}\); thus, the elements of \(\mathcal{C}\mathcal{Q}_\textrm{x}\) will often be considered as channels from \(\mathcal {D}_{XY}\) 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 brevity
and
Lemma 7.2
Let H be a Hilbert space, \(E \in M_X\otimes M_A\otimes \mathcal {B}(H)\) be a stochastic operator matrix and \(\sigma \in \mathcal {T}(H)\) be a state. Set \(E' = \tilde{\Delta }_X(E)\) and \(E'' = \tilde{\Delta }_{X,A}(E)\). Then, \(E'\) (resp. \(E''\)) is a semiclassical (resp. classical) stochastic operator matrix,
Moreover, if \(F \in M_Y\otimes M_B\otimes \mathcal {B}(H)\) is a stochastic operator matrix that forms a commuting pair with E, then
Proof
Note that if \(E = \sum _{x,x'\in X} \sum _{a,a'\in A} e_x e_{x'}^* \otimes e_a e_{a'}^* \otimes E_{x,x',a,a'}\), then
We now have
for all \(x,x'\in X\) and all \(a,a'\in A\). The second identity in (7.3) is equally straightforward. Finally, for (7.4), notice that if \(E = \left( E_{x,x',a,a'}\right) \) and \(F = \left( F_{y,y',b,b'}\right) \), then both sides of the identity are equal to
\(\square \)
Theorem 7.3
Fix \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qa}, \textrm{qc}, \textrm{ns}\}\). If \(\Gamma \in \mathcal {Q}_\textrm{x}\), then \(\Gamma _{\mathcal {D}_{XY}}\in \mathcal{C}\mathcal{Q}_\textrm{x}\); conversely, if \(\mathcal {E}\in \mathcal{C}\mathcal{Q}_\textrm{x}\), then \(\Gamma _{\mathcal {E}} \in \mathcal {Q}_\textrm{x}\). Moreover, for a channel \(\mathcal {E}: \mathcal {D}_{XY}\rightarrow M_{AB}\), we have that

(i)
\(\mathcal {E} \in \mathcal{C}\mathcal{Q}_\textrm{qc}\) if and only if \(\Gamma _{\mathcal {E}} = \Gamma _{E\cdot F,\sigma }\), where (E, F) is a commuting pair of semiclassical stochastic operator matrices, acting on a Hilbert spaces H, and \(\sigma \) is a normal state on \(\mathcal {B}(H)\);

(ii)
\(\mathcal {E} \in \mathcal{C}\mathcal{Q}_\textrm{q}\) if and only if \(\Gamma _{\mathcal {E}} = \Gamma _{E\odot F,\sigma }\), where E and F are semiclassical stochastic operator matrices, acting on finite dimensional Hilbert spaces \(H_A\) and \(H_B\), respectively, and \(\sigma \) is a state on \(\mathcal {L}(H_A\otimes H_B)\).
Proof
It is trivial that if \(\Gamma \in \mathcal {Q}_\textrm{ns}\), then \(\Gamma _{\mathcal {D}_{XY}}\in \mathcal{C}\mathcal{Q}_\textrm{ns}\). Conversely, suppose that \(\mathcal {E}\in \mathcal{C}\mathcal{Q}_\textrm{ns}\), and let \(\rho _X\in M_X\) and \(\rho _Y\in M_Y\) be states, with \(\textrm{Tr}(\rho _X) = 0\). By (7.1),
and, by symmetry, \(\Gamma _{\mathcal {E}} \in \mathcal {Q}_\textrm{ns}\).
Let \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) and \(F\in M_Y\otimes M_B\otimes \mathcal {B}(H)\) form a commuting pair of stochastic operator matrices and \(\sigma \in \mathcal {T}(H)\) be a state. It follows from Lemma 7.2 that
Conversely, suppose that \(\mathcal {E}_{\Theta }\in \mathcal{C}\mathcal{Q}_\textrm{qc}\), where \(\Theta = (\sigma _{x,y})_{x\in X, y\in Y}\) is a CQNS correlation. Let H, \(\sigma \), E and F be such that (7.2) holds; then, \(\Gamma _{\Theta } = \Gamma _{E\cdot F,\sigma }\). A similar argument applies in the case \(\textrm{x} = \textrm{q}\), and the case \(\textrm{x} = \textrm{qa}\) follows from the fact that the map \(\mathcal {E} \rightarrow \mathcal {E}\circ \Delta _{XY}\), from \(\mathcal {L}(\mathcal {D}_{XY},M_{AB})\) into \(\mathcal {L}(M_{XY},M_{AB})\), is continuous. Finally, if \(\sigma _{x,y} = \sigma _x\otimes \sigma ^y\), where \(\sigma _x\in M_A\) (resp. \(\sigma ^y\in M_B\)) is a state, \(x\in X\) (resp. \(y\in Y\)), and \(\Phi : M_X\rightarrow M_A\) (resp. \(\Psi : M_Y\rightarrow M_B\)) is the channel given by \(\Phi (e_x e_{x'}^*) = \delta _{x,x'} \sigma _x\) (resp. \(\Psi (e_y e_{y'}^*) = \delta _{y,y'} \sigma ^y\)), then \(\Gamma _{\mathcal {E}} = \Phi \otimes \Psi \), and the case \(\textrm{x} = \textrm{loc}\) follows. \(\square \)
7.2 Description in terms of states
We next introduce an operator system, universal for classicaltoquantum nosignalling correlations in a similar manner that \(\mathcal {T}_{X,A}\) 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 \(x\in X\), write \(\{e_{x,a,a'}: a,a'\in A\}\) for the canonical matrix unit system of the xth copy of \(M_A\), and let
considered as an operator subsystem of \(\mathcal {B}_{X,A}\).
Given operator systems \(\mathcal {S}_1,\dots ,\mathcal {S}_n\), their coproduct \(\mathcal {S} = \mathcal {S}_1\oplus _1\cdots \oplus _1\mathcal {S}_n\) is an operator system, equipped with complete order embeddings \(\iota _i: \mathcal {S}_i \rightarrow \mathcal {S}\), characterised by the universal property that whenever \(\mathcal {R}\) is an operator system and \(\phi _i: \mathcal {S}_i \rightarrow \mathcal {R}\) is a unital completely positive map, \(i = 1,\dots ,n\), there exists a unique unital completely positive map \(\phi : \mathcal {S}\rightarrow \mathcal {R}\) such that \(\phi \circ \iota _i = \phi _i\), \(i = 1,2,\dots ,n\). We refer the reader to [42, Section 8] for a detailed account of the coproduct of operator systems.
Remark 7.4
Let \(\mathcal {A}_i\), \(i = 1,\dots ,n\), be unital C*algebras and \(\mathcal {S} = \textrm{span}\{a_i: a_i\in \mathcal {A}_i, i = 1,\dots ,n\}\), considered as an operator subsystem of the free product \(\mathcal {A}_1*_1\cdots *_1 \mathcal {A}_n\), amalgamated over the unit. It was shown in [28, Theorem 5.2] that \(\mathcal {S} \cong _\mathrm{c.o.i.}\mathcal {A}_1 \oplus _1 \cdots \oplus _1 \mathcal {A}_n\). In particular, we have
An application of [62, Lemma 2.8] now shows that
Theorem 7.5
Let H be a Hilbert space and \(\phi : \mathcal {R}_{X,A}\rightarrow \mathcal {B}(H)\) be a linear map. The following are equivalent:

(i)
\(\phi \) is a unital completely positive map;

(ii)
\(\left( \left( \phi (e_{x,a,a'})\right) _{a,a'\in A}\right) _{x\in X}\) is a semiclassical stochastic operator matrix.
Proof
(i)\(\Rightarrow \)(ii) The restriction \(\phi _x\) of \(\phi \) to the xth copy of \(M_A\) is a unital completely positive map. By Choi’s Theorem, \(\left( \phi _x(e_{x,a,a'})\right) _{a,a'}\) is a stochastic operator matrix in \(M_A\otimes \mathcal {B}(H)\) for every \(x\in X\); thus, \(\left( \left( \phi (e_{x,a,a'})\right) _{a,a'\in A}\right) _{x\in X}\) is a semiclassical stochastic operator matrix.
(ii)\(\Rightarrow \)(i) For each \(x\in X\), let \(\phi _x: M_A\rightarrow \mathcal {B}(H)\) be the linear map defined by letting \(\phi _x(e_{a}e_{a'}^*) = \phi (e_{x,a,a'})\). By Choi’s Theorem, \(\phi _x\) is a (unital) completely positive map. By the universal property of the coproduct, there exists a (unique) unital completely positive map \(\psi : \mathcal {R}_{X,A}\rightarrow \mathcal {B}(H)\) whose restriction to the xth copy of \(M_A\) coincides with \(\phi _x\). It follows that \(\psi = \phi \), and hence \(\phi \) is completely positive. \(\square \)
Remark 7.6
By [28, Theorem 5.1], \(\mathcal {R}_{X,A}\) is an operator system quotient of \(M_{XA}\). Now [29, Proposition 1.8] shows that if
then the linear map \(\Lambda _\textrm{cq}: \mathcal {R}_{X,A}^\textrm{d} \rightarrow \mathcal {Q}_{X,A}\), given by
is a welldefined unital complete order isomorphism.
We denote the canonical generators of \(\mathcal {R}_{Y,B}\) by \(f_{y,b,b'}\), \(y\in Y\), \(b,b'\in B\). Given a functional \(t: \mathcal {R}_{X,A} \otimes \mathcal {R}_{Y,B} \rightarrow \mathbb {C}\), we let \(\mathcal {E}_t: \mathcal {D}_{XY}\rightarrow M_{AB}\) be the linear map defined by
We note that \(t\rightarrow \mathcal {E}_t\) is a linear map from \((\mathcal {R}_{X,A} \otimes \mathcal {R}_{Y,B})^*\) into \(\mathcal {L}(\mathcal {D}_{XY},M_{AB})\).
Theorem 7.5 and the universal property of the coproduct imply the existence of a unital completely positive map \(\beta _{X,A}: \mathcal {R}_{X,A}\rightarrow \mathcal {T}_{X,A}\) such that
Similarly, the matrix \((\delta _{x,x'} e_{x,a,a'})_{x,x',a,a'}\) is stochastic, and Theorem 5.2 implies the existence of a unital completely positive map \(\beta _{X,A}': \mathcal {T}_{X,A}\rightarrow \mathcal {R}_{X,A}\) such that
It is clear that
Theorem 7.7
The map \(t\rightarrow \mathcal {E}_t\) is an affine isomorphism

(i)
from the state space of \(\mathcal {R}_{X,A} \otimes _{\max } \mathcal {R}_{Y,B}\) onto \(\mathcal{C}\mathcal{Q}_\textrm{ns}\);

(ii)
from the state space of \(\mathcal {R}_{X,A} \otimes _\textrm{c} \mathcal {R}_{Y,B}\) onto \(\mathcal{C}\mathcal{Q}_\textrm{qc}\);

(iii)
from the state space of \(\mathcal {R}_{X,A} \otimes _{\min } \mathcal {R}_{Y,B}\) onto \(\mathcal{C}\mathcal{Q}_\textrm{qa}\);

(iv)
from the state space of \(\textrm{OMIN}\left( \mathcal {R}_{X,A}\right) \otimes _{\min } \textrm{OMIN}\left( \mathcal {R}_{Y,B}\right) \) onto \(\mathcal{C}\mathcal{Q}_\textrm{loc}\).
Proof
It is clear that the map \(t\rightarrow \mathcal {E}_t\) is bijective. It is also straightforward to see that, for a linear functional \(s: \mathcal {T}_{X,A}\otimes \mathcal {T}_{Y,B}\rightarrow \mathbb {C}\), we have \(\Gamma _s_{\mathcal {D}_{XY}} = \mathcal {E}_t\), where \(t = s\circ \left( \beta _{X,A}\otimes \beta _{Y,B}\right) \). The claims now follow from Theorems 6.2, 6.3, 6.5, 6.7, 7.3 and the functoriality of the involved tensor products. \(\square \)
As a consequence of Theorem 7.7, we see that the sets \(\mathcal{C}\mathcal{Q}_\textrm{qc}\) and \(\mathcal{C}\mathcal{Q}_\textrm{loc}\) are closed (as are \(\mathcal{C}\mathcal{Q}_\textrm{ns}\) and \(\mathcal{C}\mathcal{Q}_\textrm{qa}\)).
Remark. As in Theorems 6.3 and 6.5, the classes \(\mathcal{C}\mathcal{Q}_\textrm{qc}\) and \(\mathcal{C}\mathcal{Q}_\textrm{qa}\) can be equivalently described via states on the C*algebraic tensor products \(\mathcal {B}_{X,A} \otimes _{\max } \mathcal {B}_{Y,B}\) and \(\mathcal {B}_{X,A} \otimes _{\min } \mathcal {B}_{Y,B}\), respectively. For the class \(\mathcal{C}\mathcal{Q}_\textrm{qa}\), this is a direct consequence of the injectivity of the minimal tensor product in the operator system category, while for the class \(\mathcal{C}\mathcal{Q}_\textrm{qc}\), this is a consequence of Remark 7.4.
8 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 \(\ell ^{\infty }_A\). Note that, by [28, Theorem 5.2] (see Remark 7.4), \(\mathcal {S}_{X,A}\) is an operator subsystem of \(\mathcal {A}_{X,A}\). We let \((e_{x,a})_{a\in A}\) be the canonical basis of the xth copy of \(\ell ^{\infty }_A\) inside \(\mathcal {S}_{X,A}\); thus, \(\mathcal {S}_{X,A}\) is generated, as a vector space, by \(\{e_{x,a}: x\in X, a\in A\}\), and the relations
are satisfied. Note that, by the universal property of the operator system coproduct, \(\mathcal {S}_{X,A}\) is characterised by the following property: whenever H is a Hilbert space and \(\{E_{x,a}: x\in X, a\in A\}\) is a family of positive operators on H such that \((E_{x,a})_{a\in A}\) is a POVM for every \(x\in X\), there exists a (unique) unital completely positive map \(\phi : \mathcal {S}_{X,A}\rightarrow \mathcal {B}(H)\) such that \(\phi (e_{x,a}) = E_{x,a}\), \(x\in X\), \(a\in A\).
We denote by \(\mathfrak {E}\) the map sending a quantum channel \(\Gamma : M_{XY}\rightarrow M_{AB}\) to \(\Gamma _{\mathcal {D}_{XY}}\) (and recall that \(\mathfrak {N}\) stands for the map sending \(\Gamma \) to \(\mathcal {N}_{\Gamma } = \Delta _{AB}\circ \Gamma _{\mathcal {D}_{XY}}\)); Remark 8.1 below justifies calling \(\mathfrak {E}\) and \(\mathfrak {N}\) 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 \(\mathcal {C}_\textrm{ns}\) with the set \(\{\mathcal {N}_p: p \text{ an } \text{ NS } \text{ correlation }\}\).
Remark 8.1
Let X, Y, A and B be finite sets, \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qa}, \textrm{qc}, \textrm{ns}\}\), \(p\in \mathcal {C}_\textrm{x}\) and \(\mathcal {E}\in \mathcal{C}\mathcal{Q}_\textrm{x}\). The following hold:

(i)
\(p \in \mathcal {C}_\textrm{x}\) \(\Leftrightarrow \) \(\mathcal {E}_p \in \mathcal{C}\mathcal{Q}_\textrm{x}\) \(\Leftrightarrow \) \(\Gamma _p \in \mathcal {Q}_\textrm{x}\);

(ii)
\(\mathcal {E} \in \mathcal{C}\mathcal{Q}_\textrm{x}\) \(\Leftrightarrow \) \(\Gamma _{\mathcal {E}} \in \mathcal {Q}_\textrm{x}\).
Moreover, the maps \(\mathfrak {E}: \mathcal {Q}_\textrm{x} \rightarrow \mathcal{C}\mathcal{Q}_\textrm{x}\) and \(\mathfrak {N}: \mathcal{C}\mathcal{Q}_\textrm{x} \rightarrow \mathcal {C}_\textrm{x}\) are welldefined and surjective.
We identify an element \(\mathcal {N}\) of \(\mathcal {C}_\textrm{x}\) with the corresponding classicaltoquantum channel from \(\mathcal {D}_{XY}\) into \(M_{AB}\), and an element \(\mathcal {E}\) of \(\mathcal{C}\mathcal{Q}_\textrm{x}\) 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, \(\mathcal {C}_\textrm{loc} \ne \mathcal {C}_\textrm{q}\) for all subsets X, Y, A, B of cardinality at least 2. By Remark 8.1, we have that \(\mathcal{C}\mathcal{Q}_\textrm{loc} \ne \mathcal{C}\mathcal{Q}_\textrm{q}\) and \(\mathcal {Q}_\textrm{loc} \ne \mathcal {Q}_\textrm{q}\). By [69], \(\mathcal {C}_\textrm{q} \ne \mathcal {C}_\textrm{qa}\) for some finite sets X, Y, A and B (see also [23]) and hence \(\mathcal{C}\mathcal{Q}_\textrm{q} \ne \mathcal{C}\mathcal{Q}_\textrm{qa}\) and \(\mathcal {Q}_\textrm{q} \ne \mathcal {Q}_\textrm{qa}\) for a suitable choice of sets. The inequality \(\mathcal {C}_\textrm{qc} \ne \mathcal {C}_\textrm{ns}\) is wellknown (it follows e.g. from [27, Theorem 7.11]), implying that \(\mathcal{C}\mathcal{Q}_\textrm{qc} \ne \mathcal{C}\mathcal{Q}_\textrm{ns}\) and \(\mathcal {Q}_\textrm{qc} \ne \mathcal {Q}_\textrm{ns}\).
It was recently shown [38] that the inequality \(\mathcal {C}_\textrm{qa} \ne \mathcal {C}_\textrm{qc}\) also holds true for suitable sets X, Y, A and B, thus resolving the longstanding Tsirelson problem and, by [39] and [57], the Connes embedding problem, in the negative. It thus follows from Remark 8.1 that, for this choice of sets, \(\mathcal{C}\mathcal{Q}_\textrm{qa} \ne \mathcal{C}\mathcal{Q}_\textrm{qc}\) and \(\mathcal {Q}_\textrm{qa} \ne \mathcal {Q}_\textrm{qc}\). We next strengthen these inequalities.
Lemma 8.2
Let \(X_i\) and \(A_i\) be finite sets, \(i = 1,2\), with \(X_1\subseteq X_2\) and \(A_1\subseteq A_2\). There exist unital completely positive maps \(\iota _1: \mathcal {S}_{X_1,A_1}\rightarrow \mathcal {S}_{X_2,A_2}\) and \(\iota _2: \mathcal {S}_{X_2,A_2}\rightarrow \mathcal {S}_{X_1,A_1}\) such that \(\iota _2\circ \iota _1 = \textrm{id}\).
Proof
Denote the canonical generators of \(\mathcal {S}_{X_1,A_1}\) by \(e_{x,a}\), and of \(\mathcal {S}_{X_2,A_2}\)—by \(f_{x,a}\). By induction, it suffices to prove the claim in two cases.
Case 1. \(X_1 = X_2\) and \(A_2 = A_1 \cup \{a_2\}\), where \(a_2\not \in A_1\).
Let \(a_1\in A_1\). Define the maps \(\iota _1\) and \(\iota _2\) by setting
and
Case 2. \(A_2 = A_1\) and \(X_2 = X_1 \cup \{x_2\}\), where \(x_2\not \in X_1\).
Let \(x_1\in X_1\). Define \(\iota _1(e_{x,a}) = f_{x,a}\), \(x\in X_1\), \(a\in A_1\), and
By the universal property of the operator systems \(\mathcal {S}_{X,A}\), \(\iota _1\) and \(\iota _2\) are unital completely positive maps, and the condition \(\iota _2\circ \iota _1 = \textrm{id}\) is readily verified. \(\square \)
Theorem 8.3
For all finite sets X, Y, A and B of sufficiently large cardinality, the following hold true:

(i)
\(\mathcal {Q}_\textrm{qa}(X,Y,A,B) \ne \mathcal {Q}_\textrm{qc}(X,Y,A,B)\);

(ii)
\(\mathcal{C}\mathcal{Q}_\textrm{qa}(X,Y,A,B) \ne \mathcal{C}\mathcal{Q}_\textrm{qc}(X,Y,A,B)\);

(iii)
\(\mathcal {T}_{X,A}\otimes _{\min } \mathcal {T}_{Y,B} \ne \mathcal {T}_{X,A}\otimes _\textrm{c} \mathcal {T}_{Y,B}\);

(iv)
\(\mathcal {R}_{X,A}\otimes _{\min } \mathcal {R}_{Y,B} \ne \mathcal {R}_{X,A}\otimes _\textrm{c} \mathcal {R}_{Y,B}\);

(v)
\(\mathcal {B}_{X,A}\otimes _{\min } \mathcal {B}_{Y,B} \ne \mathcal {B}_{X,A}\otimes _{\max } \mathcal {B}_{Y,B}\);

(vi)
\(\mathcal {C}_{X,A}\otimes _{\min } \mathcal {C}_{Y,B} \ne \mathcal {C}_{X,A}\otimes _{\max } \mathcal {C}_{Y,B}\).
Proof
By [38], there exist (finite) sets \(X_0\), \(Y_0\), \(A_0\) and \(B_0\) and an NS correlation
Using [49, Corollary 3.2], let s be a state on \(\mathcal {S}_{X_0,A_0}\otimes _\textrm{c} \mathcal {S}_{Y_0,B_0}\) such that
for all \(x\in X_0\), \(y\in Y_0\), \(a\in A_0\) and \(b\in B_0\). Assume that \(X_0\subseteq X\), \(Y_0\subseteq Y\), \(A_0\subseteq A\) and \(B_0\subseteq B\). Write \(\iota _i^A\) (resp. \(\iota _i^B\)), \(i = 1,2\), for the maps arising from Lemma 8.2 for the operator systems \(\mathcal {S}_{X_0,A_0}\) and \(\mathcal {S}_{X,A}\) (resp. \(\mathcal {S}_{Y_0,B_0}\) and \(\mathcal {S}_{Y,B}\)). By the functoriality of the commuting tensor product, the map \(t:= s\circ (\iota _2^A\otimes \iota _2^B)\) is a state on \(\mathcal {S}_{X,A}\otimes _\textrm{c} \mathcal {S}_{Y,B}\). The NS correlation \(q \in \mathcal {C}_\textrm{qc}(X,Y,A,B)\) arising from t as in (8.1) does not belong to the class \(\mathcal {C}_\textrm{qa}\). Indeed, if \(q\in \mathcal {C}_\textrm{qa}\), then, by [49, Corollary 3.3], t is a state on \(\mathcal {S}_{X,A}\otimes _{\min } \mathcal {S}_{Y,B}\), and hence \(s = t\circ (\iota _1^A\otimes \iota _1^B)\) (see Lemma 8.2) is a state on \(\mathcal {S}_{X_0,A_0}\otimes _{\min } \mathcal {S}_{Y_0,B_0}\) which, in view of [49, Corollary 3.3], contradicts the fact that p is not approximately quantum.
It follows that \(\mathcal {C}_\textrm{qa}(X,Y,A,B) \ne \mathcal {C}_\textrm{qc}(X,Y,A,B)\) 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 7.7. Finally, (v) follows from (iv) and Remark 7.4, and (vi) follows from (iii), Corollary 5.3 and [43, Theorem 6.4]. \(\square \)
Recall that an operator system \(\mathcal {S}\) is said to possess the operator system local lifting property (OSLLP) [44] if, whenever \(\mathcal {A}\) is a unital C*algebra, \(\mathcal {I}\subseteq \mathcal {A}\) is a twosided ideal, \(\mathcal {T}\subseteq \mathcal {S}\) is a finite dimensional operator subsystem and \(\varphi : \mathcal {T}\rightarrow \mathcal {A}/\mathcal {I}\) is a unital completely positive map, there exists a unital completely positive map \(\psi : \mathcal {T}\rightarrow \mathcal {A}\) such that \(\varphi = q\circ \psi \) (here \(q: \mathcal {A}\rightarrow \mathcal {A}/\mathcal {I}\) denotes the quotient map). We conclude this section with showing that the operator systems we introduced possess OSLPP.
Proposition 8.4
Let \(\mathcal {S}\) be an operator system quotient of \(M_k\), for some \(k\in \mathbb {N}\), and H be a Hilbert space. Then, \(\mathcal {S} \otimes _{\min }\mathcal {B}(H) \cong _\mathrm{c.o.i.} \mathcal {S} \otimes _{\max }\mathcal {B}(H)\), and hence \(\mathcal {S}\) possesses OSLLP.
Proof
Let \(J\subseteq M_k\) be a kernel such that \(\mathcal {S} = M_k/J\); write \(q: M_k\rightarrow \mathcal {S}\) for the quotient map. By [29, Proposition 1.8], the dual \(q^*: \mathcal {S}^\textrm{d}\rightarrow M_k^\textrm{d}\) is a complete order embedding.
Fix \(u\in M_n\left( \mathcal {S}\otimes _{\min }\mathcal {B}(H)\right) ^+\); after a canonical identification, we consider u as an element of \(\left( \mathcal {S}\otimes _{\min } M_n\left( \mathcal {B}(H)\right) \right) ^+\). Let \(\{S_1,\dots ,S_m\}\) be a basis of \(\mathcal {S}\), and write \(u = \sum _{i=1}^m S_i \otimes T_i\), for some \(T_i\in M_n\left( \mathcal {B}(H)\right) \), \(i = 1,\dots ,m\). By [42, Proposition 6.1], the map \(\phi _u: \mathcal {S}^\textrm{d} \rightarrow M_n\left( \mathcal {B}(H)\right) \), given by \(\phi _u(f) = \sum _{i=1}^m f(S_i) T_i\), is completely positive. By Arveson’s extension theorem, there exists a completely positive map \(\psi : M_k^\textrm{d} \rightarrow M_n\left( \mathcal {B}(H)\right) \) with \(\psi \circ q^* = \phi _u\). Let \(S'_i\in M_k\) be such that \(q(S'_i) = S_i\), \(i = 1,\dots ,m\), and let \(\{S_i': i = m+1,\dots ,k^2\}\) be a basis of J. Then, \(\{S'_1,\dots ,S'_m,S'_{m+1},\dots ,S'_{k^2}\}\) is a basis of \(M_k\). Let
be an element such that
by [42, Proposition 6.1], \(v\in \left( M_k\otimes _{\min } M_n\left( \mathcal {B}(H)\right) \right) ^+\). Since \(M_k\) is nuclear, v belongs to \(\left( M_k \otimes _{\max } M_n\left( \mathcal {B}(H)\right) \right) ^+\). Let \(w = (q\otimes \textrm{id})(v)\); by the functoriality of the maximal tensor product, \(w\in (\mathcal {S} \otimes _{\max } M_n(\mathcal {B}(H)))^+\). We have
For all \(f\in \mathcal {S}^\textrm{d}\), we have
It follows that \(T_i = T_i'\), \(i = 1,\dots ,m\), and hence \(u = w\). Thus, \(u\in M_n\left( \mathcal {S}\otimes _{\max } \mathcal {B}(H)\right) ^+\), and it follows from [44, Theorem 8.6] that \(\mathcal {S}\) possesses OSLLP. \(\square \)
Proposition 8.4, combined with Corollary 5.6 and Remark 7.6, yields the following corollary.
Corollary 8.5
Let X and A be finite sets. Then, \(\mathcal {T}_{X,A}\) and \(\mathcal {R}_{X,A}\) possess OSLPP.
Remark. It is worth noting the different nature of the C*algebras \(\mathcal {A}_{X,A}\) and \(\mathcal {B}_{X,A}\) on one hand, and \(\mathcal {C}_{X,A}\) on the other. This is best seen in the special case where \(X = 1\), when \(\mathcal {A}_{X,A} \cong \mathcal {D}_A\), \(\mathcal {B}_{X,A} \cong M_A\) and \(\mathcal {C}_{X,A} \cong C^*_u(M_A)\).
9 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 \(p = \left\{ \left( p(a,bx,y)\right) _{a,b\in A}: x,y\in X\right\} \) is called synchronous [61] if
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].
Theorem 9.1
Let p be an NS correlation. Then,

(i)
p is synchronous and quantum commuting if and only if there exists a trace \(\tau : \mathcal {A}_{X,A}\rightarrow \mathbb {C}\) such that
$$\begin{aligned} p(a,bx,y) = \tau \left( e_{x,a}e_{y,b}\right) , \ \ \ x,y\in X, a,b\in A; \end{aligned}$$(9.1) 
(ii)
p is synchronous and quantum if and only if there exist a finite dimensional C*algebra \(\mathcal {A}\), a trace \(\tau _{\mathcal {A}}\) on \(\mathcal {A}\) and a *homomorphism \(\pi : \mathcal {A}_{X,A} \rightarrow \mathcal {A}\) such that (9.1) holds for the trace \(\tau = \tau _{\mathcal {A}}\circ \pi \);

(iii)
p is synchronous and local if and only if there exist an abelian C*algebra \(\mathcal {A}\), a trace \(\tau _{\mathcal {A}}\) on \(\mathcal {A}\) and a *homomorphism \(\pi : \mathcal {A}_{X,A} \rightarrow \mathcal {A}\) such that (9.1) holds for the trace \(\tau = \tau _{\mathcal {A}}\circ \pi \).
9.1 Fair correlations
If \(\mathcal {A}\) is a unital C*algebra, we write \(\mathcal {A}^\textrm{op}\) for the opposite C*algebra of \(\mathcal {A}\); recall that \(\mathcal {A}^\textrm{op}\) has the same underlying set (whose elements will be denoted by \(u^\textrm{op}\), for \(u\in \mathcal {A}\)), the same involution, linear structure and norm, and multiplication given by \(u^\textrm{op} v^\textrm{op} = (vu)^\textrm{op}\), \(u,v\in \mathcal {A}\). For a subset \(\mathcal {S}\subseteq \mathcal {A}\), we let \(\mathcal {S}^\textrm{op} = \left\{ u^\textrm{op}: u\in \mathcal {S}\right\} \).
For a Hilbert space H, we denote by \(H^\textrm{d}\) its Banach space dual; if K is a(nother) Hilbert space and \(T \in \mathcal {B}(H,K)\), we denote by \(T^\textrm{d}\) its adjoint, acting from \(K^\textrm{d}\) into \(H^\textrm{d}\). We note the relation
It is straightforward to see that if \(\mathcal {A}\) is a C*algebra and \(\pi : \mathcal {A}\rightarrow \mathcal {B}(H)\) is a (faithful) *representation, then the map \(\pi ^\textrm{op}: \mathcal {A}^\textrm{op}\rightarrow \mathcal {B}(H^\textrm{d})\), given by \(\pi ^\textrm{op}(u^\textrm{op}) = \pi (u)^\textrm{d}\), is a (faithful) *representation. Note that the transposition map \(u\rightarrow (u^t)^\textrm{op}\) is a *isomorphism between \(M_X\) and \(M_X^\textrm{op}\). It was shown in [45] that there exists a *isomorphism \(\partial _{\mathcal {A}}: \mathcal {A}_{X,A}\rightarrow \mathcal {A}_{X,A}^\textrm{op}\) such that \(\partial _{\mathcal {A}}(e_{x,a}) = e_{x,a}^\textrm{op}\), \(x\in X\), \(a\in A\). The following analogous statements for \(\mathcal {C}_{X,A}\) and \(\mathcal {B}_{X,A}\) will be needed later.
Lemma 9.2
Let X and A be finite sets.

(i)
There exists a *isomorphism \(\partial : \mathcal {C}_{X,A}\rightarrow \mathcal {C}_{X,A}^\textrm{op}\) such that
$$\begin{aligned} \partial (e_{x,x',a,a'}) = e_{x',x,a',a}^\textrm{op}, \ \ \ x,x'\in X, a,a'\in A. \end{aligned}$$ 
(ii)
There exists a *isomorphism \(\partial _{\mathcal {B}}: \mathcal {B}_{X,A}\rightarrow \mathcal {B}_{X,A}^\textrm{op}\) such that
$$\begin{aligned} \partial _{\mathcal {B}}(e_{x,a,a'}) = e_{x,a',a}^\textrm{op}, \ \ \ x\in X, a,a'\in A. \end{aligned}$$
Proof
(i) Let \(\pi : \mathcal {C}_{X,A}\rightarrow \mathcal {B}(H)\) be a faithful *representation. Write \(E_{x,x',a,a'} = \pi (e_{x,x',a,a'})\), \(x,x'\in X\), \(a,a'\in A\). Using Theorem 3.1, let K be a Hilbert space and \((V_{a,x})_{a,x}: H^X\rightarrow K^A\) be an isometry such that \(E_{x,x',a,a'} = V_{a,x}^* V_{a',x'}\), \(x,x'\in X\), \(a,a'\in A\). Let \(W_{a,x} = \left( V_{a,x}^\textrm{d}\right) ^*\); thus, \(W_{a,x}\in \mathcal {B}(H^\textrm{d},K^\textrm{d})\), \(x\in X\), \(a\in A\). Using (9.2), we have
thus, \(\left( W_{a,x}\right) _{a,x}\) is an isometry. By Theorem 3.1, if \(F_{x,x',a,a'} = W_{a,x}^*W_{a',x'}\), \(x,x'\in X\), \(a,a'\in A\), then \(\left( F_{x,x',a,a'}\right) _{x,x',a,a'}\) is a stochastic operator matrix. Note that
By the universal property of \(\mathcal {C}_{X,A}\), there exists a *homomorphism \(\pi ': \pi \left( \mathcal {C}_{X,A}\right) \rightarrow \mathcal {B}\left( H^\textrm{d}\right) \) such that
By the paragraph before Lemma 9.2, \(\pi '\circ \pi \) can be regarded as a *homomorphism from \(\mathcal {C}_{X,A}\) into \(\mathcal {C}_{X,A}^\textrm{op}\), which maps \(e_{x,x',a,a'}\) to \(e_{x',x,a',a}^\textrm{op}\). The claim follows by symmetry.
(ii) The words of the form \(e_{x_1,a_1,a_1'}\ldots e_{x_k,a_k,a_k'}\) span a dense \(*\)subalgebra of \(\mathcal {B}_{X,A}\). As \(u\mapsto (u^t)^\textrm{op}\) is a *isomorphism from \(M_A\) to \(M_A^\textrm{op}\) that maps the matrix unit \(e_{a} e_{a'}^*\) to \(\left( e_{a'} e_{a}^*\right) ^\textrm{op}\), the universal property of the free product implies that the map \(\partial _{\mathcal {B}}\) given by
extends to the desired *isomorphism. \(\square \)
If \(\mathcal {U}\) is a subspace of a \(C^*\)algebra \(\mathcal {A}\), we call a linear functional \(s: \mathcal {U}\otimes \mathcal {U}^\textrm{op} \rightarrow \mathbb {C}\) fair if
It will be convenient to write \(t_Y\) for the transpose map on \(M_Y\). A state \(\rho \in M_{XY}\) will be called fair if \(\textrm{Tr}_X\left( (\textrm{id}\otimes t_Y)(\rho )\right) = \textrm{Tr}_Y\left( (\textrm{id}\otimes t_Y)(\rho )\right) \). We write \(\Sigma _{X} = \{\rho \in M_{XY}^+: \rho \text{ a } \text{ fair } \text{ state }\}\), and observe that an element \(\rho = (\rho _{x,x',y,y'})\in M_{XY}^+\) belongs to \(\Sigma _{X}\) precisely when
that is, when
We let \(\Sigma _{X}^\textrm{cl} = \Sigma _{X}\cap \mathcal {D}_{XY}\); thus, a state \(\rho = (\rho _{x,y})_{x,y}\in \mathcal {D}_{XY}^+\) is in \(\Sigma _{X}^\textrm{cl}\) precisely when
It follows from (9.4) and (9.5) that
Definition 9.3
A QNS correlation \(\Gamma : M_{XY}\rightarrow M_{AB}\) (resp. a CQNS correlation \(\mathcal {E}: \mathcal {D}_{XY}\rightarrow M_{AB}\), an NS correlation \(\mathcal {N}: \mathcal {D}_{XY}\rightarrow \mathcal {D}_{AB}\)) is called fair if \(\Gamma \left( \Sigma _{X}\right) \subseteq \Sigma _{A}\) (resp. \(\mathcal {E}\left( \Sigma _{X}^\textrm{cl}\right) \subseteq \Sigma _{A}\), \(\mathcal {N}\left( \Sigma _{X}^\textrm{cl}\right) \subseteq \Sigma _{A}^\textrm{cl}\)).
Theorem 9.4
Let \(\Gamma \) be a QNS correlation.

(i)
\(\Gamma \) is fair if and only if there exists a state \(s: \mathcal {T}_{X,A}\otimes _{\max } \mathcal {T}_{X,A}\rightarrow \mathbb C\) such that \(\Gamma = \Gamma _s\) and the state \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair;

(ii)
\(\Gamma \) is fair and belongs to \(\mathcal {Q}_\textrm{qc}\) if and only if there exists a state \(s: \mathcal {T}_{X,A}\otimes _\textrm{c} \mathcal {T}_{X,A}\rightarrow \mathbb C\) such that \(\Gamma = \Gamma _s\) and the state \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair;

(iii)
\(\Gamma \) is fair and belongs to \(\mathcal {Q}_\textrm{qa}\) if and only if there exists a state \(s: \mathcal {T}_{X,A}\otimes _{\min } \mathcal {T}_{X,A}\rightarrow \mathbb C\) such that \(\Gamma = \Gamma _s\) and the state \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair;

(iv)
\(\Gamma \) is fair and belongs to \(\mathcal {Q}_\textrm{loc}\) if and only if there exists a state \(s: \textrm{OMIN}(\mathcal {T}_{X,A}) \otimes _{\min } \textrm{OMIN}(\mathcal {T}_{Y,B}) \rightarrow \mathbb C\) such that \(\Gamma = \Gamma _s\) and the state \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair.
Proof
We only show (i); the proofs of (ii)(iv) are similar. Let \(\Gamma \) be a QNS correlation. By Theorem 6.2, there exists a state \(s\in \mathcal {T}_{X,A}\otimes _{\max } \mathcal {T}_{X,A}\rightarrow \mathbb C\) such that \(\Gamma = \Gamma _s\). The condition
is equivalent to
\(b,b'\in B\). Note that
and hence
Thus, letting \(\mu _{y,y'}^{(1)} = \sum _{x\in X} \rho _{x,x,y,y'}\), we have that the left hand side of (9.7) coincides with
Similarly, letting \(\mu _{x,x'}^{(2)} = \sum _{y\in X}\rho _{x,x',y,y}\), we have that the right hand side of (9.7) coincides with
that is, with
Let now \(\rho \in \Sigma _{X}\). By (9.4), \(\mu _{y,y'}^{(1)} = \mu _{y',y}^{(2)}\). Hence, if \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair, then \(\Gamma (\rho )\in \Sigma _{A}\), that is, \(\Gamma \) is fair.
Conversely, assuming that \(\Gamma \) is fair, the previous paragraph shows that
for any u of the form \(u = \sum _{y,y'\in X}(\sum _x\rho _{x,x,y,y'})e_{y,y',b,b'}\) with \(\rho \in \Sigma _{X}\). Letting \(\rho = e_xe_x^*\otimes e_xe_x^*\in \Sigma _{X}\) we conclude that (9.8) holds for \(u = e_{x,x,b,b'}\), \(x\in X\), \(b,b'\in A\). Letting \(\rho = 1\otimes \omega ^t + \omega \otimes 1\), where \(\omega = \alpha (e_ze_{z}^*+e_{z'}e_{z'}^*)+\beta e_ze_{z'}^*+\bar{\beta }e_{z'}e_{z}^*\), \(z\ne z'\), with \(\alpha \ge \beta \), we obtain that (9.8) holds for \(u = \alpha (2\sum _{y\in X}e_{y,y,b,b'}+Xe_{z,z,b,b'}+Xe_{z',z',b,b'})+\beta X e_{z',z,b,b'}+\bar{\beta }Xe_{z,z',b,b'}\). From this we deduce that (9.8) holds for any \(u = e_{y,y',b,b'}\), \(y,y'\in X\), \(b,b'\in A\). \(\square \)
Let \(\mathcal {S}\subseteq \mathcal {B}(K)\) be an operator system. We let \(\mathcal {S}^\textrm{op} = \{u^\textrm{d}: u\in \mathcal {S}\}\), considered as an operator subsystem of \(\mathcal {B}(K^\textrm{d})\). Note that \(\mathcal {S}^\textrm{op}\) is welldefined: if \(\phi : \mathcal {S}\rightarrow \mathcal {B}(\tilde{K})\) is a unital complete isometry, then the map \(\tilde{\phi }: \mathcal {S}^\textrm{op}\rightarrow \mathcal {B}(\tilde{K}^\textrm{d})\), given by \(\tilde{\phi }(u^\textrm{d}) = \phi (u)^\textrm{d}\), is also unital and completely isometric. We thus write \(u^\textrm{op} = u^\textrm{d}\) in the (abstract) operator system \(\mathcal {S}^\textrm{op}\).
For a linear map \(\Phi : M_X\rightarrow M_A\), let \(\Phi ^{\sharp }: M_X\rightarrow M_A\) be the linear map given by \(\Phi ^{\sharp }(\omega ) = \Phi (\omega ^\textrm{t})^\textrm{t}\).
Lemma 9.5
Let \(\mathcal {S}\) be an operator system.

(i)
If \(\phi : \mathcal {S}\rightarrow \mathcal {B}(H)\) be a unital completely positive map, then the map \(\phi ^\textrm{op}: \mathcal {S}^\textrm{op} \rightarrow \mathcal {B}(H^\textrm{d})\), given by \(\phi ^\textrm{op}(u^\textrm{op}) = \phi (u)^\textrm{d}\), is unital and completely positive.

(ii)
Up to a canonical *isomorphism, \(C^*_u(\mathcal {S}^\textrm{op}) = C^*_u(\mathcal {S})^\textrm{op}\).

(iii)
If \(\Phi : M_X\rightarrow M_A\) is a completely positive map, then so is \(\Phi ^{\sharp }\).
Proof
(i) Represent \(\mathcal {S}\subseteq \mathcal {B}(K)\) as a concrete operator system. Then, \(\mathcal {S}^\textrm{op}\subseteq \mathcal {B}(K^\textrm{d})\). Suppose that \(u_{i,j}\in \mathcal {S}\), \(i,j = 1,\dots ,n\), are such that \((u_{i,j}^\textrm{d})_{i,j}\in M_n(\mathcal {B}(K^\textrm{d}))^+\). Then, \((u_{j,i})_{i,j} = (u_{i,j}^\textrm{d})_{i,j}^\textrm{d}\in M_n(\mathcal {B}(K))^+\) and hence \((\phi (u_{j,i}))_{i,j}\in M_n(\mathcal {B}(H))^+\). Thus,
(ii) Suppose that \(\psi : \mathcal {S}^\textrm{op}\rightarrow \mathcal {B}(H)\) is a unital completely positive map. By (i), \(\psi ^\textrm{op}: \mathcal {S}\rightarrow \mathcal {B}(H^\textrm{d})\) is (unital and) completely positive. By the universal property of the maximal C*cover, there exists a *homomorphism \(\pi : C^*_u(\mathcal {S}) \rightarrow \mathcal {B}(H^\textrm{d})\) extending \(\psi ^\textrm{op}\). It follows that \(\pi ^\textrm{op}: C^*_u(\mathcal {S})^\textrm{op} \rightarrow \mathcal {B}(H)\) is a *homomorphism that extends \(\psi \). Thus, \(C^*_u(\mathcal {S})^\textrm{op}\) satisfies the universal property of the C*cover of \(\mathcal {S}^\textrm{op}\).
(iii) The transposition is a (unital) complete order isomorphism from \(M_X\) onto \(M_X^\textrm{op}\). The statement follows after observing that, under the latter identification, \(\Phi ^{\sharp }\) coincides with \(\Phi ^\textrm{op}\). \(\square \)
Corollary 9.6
A local QNS correlation \(\Gamma \) is fair if and only if \(\Gamma = \sum _{i=1}^m\) \(\lambda _i \Phi _i\otimes \Psi _i\) for some quantum channels \(\Phi _i, \Psi _i: M_X\rightarrow M_A\) and scalars \(\lambda _i\ge 0\), \(i = 1,\dots ,m\), \(\sum _{i=1}^m \lambda _i = 1\), such that
Proof
Suppose that \(\Gamma \) is fair and, using Theorem 9.4, write \(\Gamma = \Gamma _s\), where s is a state on \(\textrm{OMIN}(\mathcal {T}_{X,A}) \otimes _{\min } \textrm{OMIN}(\mathcal {T}_{Y,B})\) such that \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair. As in the proof of Theorem 6.7, identify s with a convex combination \(\sum _{i=1}^m \lambda _i \phi _i\otimes \psi _i\), where \(\phi _i\) and \(\psi _i\) are states on \(\mathcal {T}_{X,A}\), \(i = 1,\dots ,m\); then, the fairness condition is equivalent to
Let \(\Phi _i\) and \(\Psi _i\) be the quantum channels from \(M_X\) to \(M_A\), corresponding to \(\phi _i\) and \(\psi _i\), respectively; then, \(\Gamma = \sum _{i=1}^m\lambda _i \Phi _i\otimes \Psi _i\). Let \(\tilde{\psi }_i: u \mapsto \psi _i((\partial ^{1}(u^\textrm{op}))\), \(u\in \mathcal {T}_{X,A}\). By Lemma 9.2, \(\tilde{\psi }_i\) is a state. Moreover,
that is, the quantum channel \(\Psi _i^\sharp \) corresponds to \(\tilde{\psi }_i\). Identity (9.9) now follows from (9.10). The converse implication follows by reversing the previous steps.
Corollary 9.7

(i)
A CQNS correlation \(\mathcal {E}\) is fair if and only if there is a state \(t: \mathcal {R}_{X,A}\otimes _{\max } \mathcal {R}_{X,A}\rightarrow \mathbb C\) such that \(t\circ (\textrm{id}\otimes \partial _{\mathcal {B}})^{1}\) is fair and \(\mathcal {E} = \mathcal {E}_t\). Similar descriptions hold for fair correlations in the classes \(\mathcal{C}\mathcal{Q}_\textrm{qc}\), \(\mathcal{C}\mathcal{Q}_\textrm{qa}\) and \(\mathcal{C}\mathcal{Q}_\textrm{loc}\).

(ii)
An NS correlation p is fair if and only if there is a state \(t: \mathcal {S}_{X,A}\otimes _{\max } \mathcal {S}_{X,A}\rightarrow \mathbb C\) such that \(t(u\otimes 1) = t(1\otimes u)\), \(u\in \mathcal {S}_{X,A}\), and
$$\begin{aligned} p(a,bx,y) = t(e_{x,a}\otimes e_{y,b}), \ \ \ x,y\in X, a,b\in A. \end{aligned}$$Similar descriptions hold for fair correlations in the classes \(\mathcal {C}_\textrm{qc}\), \(\mathcal {C}_\textrm{qa}\) and \(\mathcal {C}_\textrm{loc}\).
Proof
We only give details for (i). Let \(\mathcal {E}: \mathcal {D}_{XY}\rightarrow M_{AB}\) be a fair CQNS correlation. By (9.6), \(\mathcal {E} \circ \Delta _{XY}: M_{XY}\rightarrow M_{AB}\) is a fair QNS correlation. By Theorem 9.4 (i), \(\mathcal {E} \circ \Delta _{XY} = \Gamma _s\), for some state s on \(\mathcal {T}_{X,A}\otimes _{\max } \mathcal {T}_{X,A}\) such that \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair. It follows that \(\mathcal {E} = \mathcal {E}_t\), where \(t:= s\circ (\beta _{X,A}\otimes \beta _{X,A})\) is a state on \(\mathcal {R}_{X,A}\otimes _{\max } \mathcal {R}_{X,A}\) and \(t\circ (\textrm{id}\otimes \partial _{\mathcal {B}})^{1}\) is fair. Conversely, if \(\mathcal {E} = \mathcal {E}_t\) for some state t on \(\mathcal {R}_{X,A}\otimes _{\max } \mathcal {R}_{X,A}\) such that \(t\circ (\textrm{id}\otimes \partial _{\mathcal {B}})^{1}\) is fair, then \(\Gamma _{\mathcal {E}} = \Gamma _s\), where \(s:= t\circ (\beta '_{X,A}\otimes \beta '_{X,A})\) and \(s\circ (\textrm{id}\otimes \partial )^{1}\) is fair. By Theorem 9.4 (i), \(\Gamma _{\mathcal {E}}\) is fair, and hence so is \(\mathcal {E}\). The statements regarding \(\mathcal{C}\mathcal{Q}_\textrm{qc}\), \(\mathcal{C}\mathcal{Q}_\textrm{qa}\) and \(\mathcal{C}\mathcal{Q}_\textrm{loc}\) follow after a straightforward modification of the argument.
Remark. It follows from Theorems 9.1, 9.4 and Corollary 9.7 that fair correlations can be viewed as a noncommutative, and less restrictive, version of synchronous correlations.
9.2 Tracial QNS correlations
Let \(\mathcal {A}\) be a unital C*algebra, \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) be a state and \(\mathcal {A}^\textrm{op}\) be the opposite C*algebra of \(\mathcal {A}\). By the paragraph before Theorem 6.2.7 in [13], the linear functional \(s_{\tau }: \mathcal {A}\otimes _{\max }\mathcal {A}^\textrm{op}\rightarrow \mathbb {C}\), given by \(s_{\tau }(u\otimes v^\textrm{op}) = \tau (uv)\), is a state.
A positive element \(E\in M_X\otimes M_A\otimes \mathcal {A}\) will be called a stochastic \(\mathcal {A}\)matrix if \((\textrm{id}\otimes \textrm{id}\otimes \pi )(E)\) is a stochastic operator matrix for some faithful *representation of \(\mathcal {A}\). Such an E will be called semiclassical if it belongs to \(\mathcal {D}_X\otimes M_A\otimes \mathcal {A}\).
Let \(E = (g_{x,x',a,a'})_{x,x',a,a'}\) be a stochastic \(\mathcal {A}\)matrix, and set
Lemma 9.2 shows that \(E^\textrm{op}\) is a stochastic \(\mathcal {A}^\textrm{op}\)matrix. Thus, after a permutation of the tensor factors, we can consider \(E\otimes E^\textrm{op}\) as an element of \(\left( M_{XA}\otimes M_{XA}\otimes (\mathcal {A}\otimes _{\max } \mathcal {A}^\textrm{op})\right) ^+\). By Theorem 5.2, there exists a *homomorphism \(\pi _E: \mathcal {C}_{X,A}\rightarrow \mathcal {A}\), such that \(\pi _E(e_{x,x',a,a'}) = g_{x,x',a,a'}\) for all \(x,x',a,a'\). By Corollary 5.3 and Lemma 9.5, \(C^*_u(\mathcal {T}_{X,A}^\textrm{op}) \equiv \mathcal {C}_{X,A}^\textrm{op}\); thus,
Write
we have that \(f_{E,\tau }\) is a state on \(\mathcal {T}_{X,A}\otimes _\textrm{c}\mathcal {T}_{X,A}\), and
In the sequel, we write \(\Gamma _{E,\tau } = \Gamma _{f_{E,\tau }}\); by Theorem 6.3, \(\Gamma _{E,\tau } \in \mathcal {Q}_\textrm{qc}\). By Theorem 5.2, we may assume, without loss of generality, that \(\mathcal {A} = \mathcal {C}_{X,A}\) and \(E = (e_{x,x',a,a'})_{x,x',a,a'}\). In this case, we will abbreviate \(\Gamma _{E,\tau }\) to \(\Gamma _{\tau }\).
Definition 9.8
A QNS correlation \(\Gamma \) is called

(i)
tracial if \(\Gamma = \Gamma _{\tau }\), where \(\tau : \mathcal {C}_{X,A}\rightarrow \mathbb {C}\) is a trace;

(ii)
quantum tracial if there exists a finite dimensional C*algebra \(\mathcal {A}\), a trace \(\tau _{\mathcal {A}}\) on \(\mathcal {A}\) and a *homomorphism \(\pi : \mathcal {C}_{X,A}\rightarrow \mathcal {A}\) such that \(\Gamma = \Gamma _{\tau _{\mathcal {A}}\circ \pi }\);

(iii)
locally tracial if there exists an abelian C*algebra \(\mathcal {A}\), a state \(\tau _{\mathcal {A}}\) on \(\mathcal {A}\) and a *homomorphism \(\pi : \mathcal {C}_{X,A}\rightarrow \mathcal {A}\) such that \(\Gamma = \Gamma _{\tau _{\mathcal {A}}\circ \pi }\).
Theorem 9.9
Let X and A be finite sets.

(i)
If \(\Gamma \) is a quantum tracial QNS correlation, then \(\Gamma \in \mathcal {Q}_\textrm{q}\);

(ii)
A QNS correlation \(\Gamma : M_{XX}\rightarrow M_{AA}\) is locally tracial if and only if there exists quantum channels \(\Phi _j: M_X\rightarrow M_A\), \(j = 1,\dots ,k\), such that
$$\begin{aligned} \Gamma = \sum _{j=1}^k \lambda _j \Phi _j \otimes \Phi _j^{\sharp } \end{aligned}$$(9.12)as a convex combination. In particular, if \(\Gamma \) is a locally tracial QNS correlation, then \(\Gamma \in \mathcal {Q}_\textrm{loc}\).
Proof
(i) Suppose that H is a finite dimensional Hilbert space on which \(\mathcal {A}\) acts faithfully and let \(\pi :\mathcal {C}_{X,A}\rightarrow \mathcal {A}\) be as in Definition 9.8 (ii). Let \(E_{x,x',a,a'} = \pi (e_{x,x',a,a'})\) and \(E = \left( E_{x,x',a,a'}\right) _{x,x',a,a'}\). By the proof of Lemma 9.2, \(E^\textrm{op}:= \left( E^\textrm{d}_{x',x,a',a}\right) _{x,x',a,a'}\) is a stochastic operator matrix. Let \(\sigma \) be any positive functional on \(\mathcal {L}(H\otimes H^\textrm{d})\) that extends the state \(s_{\tau _{\mathcal {A}}}\) which, by nuclearity, may be considered as a state on \(\mathcal {A}\otimes _{\min }\mathcal {A}^\textrm{op}\). Then, \(\Gamma _{\tau } = \Gamma _{E\odot E^\textrm{op},\sigma }\) and, by the paragraph before Remark 4.7, \(\Gamma _{\tau }\in \mathcal {Q}_\textrm{q}\).
(ii) Suppose that \(\Phi _j: M_X\rightarrow M_A\), \(j = 1,\dots ,k\), are quantum channels and \(\Gamma \) is the convex combination (9.12). Letting \(\left( \lambda _{x,x',a,a'}^{(j)}\right) _{a,a'} = \Phi _j\left( e_x e_{x'}^*\right) \), \(x,x'\in X\), we have that the matrix \(C_j = \left( \lambda _{x,x',a,a'}^{(j)}\right) _{x,x',a,a'}\) is a stochastic \(\mathbb {C}\)matrix. By Theorem 5.2, there exists a (unique) *representation \(\pi _j: \mathcal {C}_{X,A}\rightarrow \mathbb {C}\) such that \(\pi _j(e_{x,x',a,a'}) = \lambda _{x,x',a,a'}^{(j)}\), \(x,x'\in X\), \(a,a'\in A\). Let \(\pi : \mathcal {C}_{X,A} \rightarrow \mathcal {D}_k\) be the *representation given by
and let \(\tau _k: \mathcal {D}_k\rightarrow \mathbb {C}\) be the state defined by \(\tau _k\left( (\mu _j)_{j=1}^k\right) = \sum _{j=1}^k \lambda _j \mu _j\). We have
Conversely, let \(\mathcal {A}\) be a unital abelian C*algebra, \(\tau _{\mathcal {A}}: \mathcal {A}\rightarrow \mathbb {C}\) a state, and \(\pi : \mathcal {C}_{X,A}\rightarrow \mathcal {A}\) a *homomorphism such that \(\Gamma = \Gamma _{\tau _{\mathcal {A}}\circ \pi }\). Without loss of generality, assume that \(\mathcal {A} = C(\Omega )\), where \(\Omega \) is a compact Hausdorff topological space, and \(\mu \) is a Borel probability measure on \(\Omega \) such that \(\tau _{\mathcal {A}}(f) = \int _{\Omega } fd\mu \), \(f\in \mathcal {A}\). Set \(h_{x,x',a,a'} = \pi (e_{x,x',a,a'})\), \(x,x'\in X\), \(a,a'\in A\). For each \(s\in \Omega \), let \(\Phi (s): M_X\rightarrow M_A\) be the quantum channel given by \(\Phi (s)\left( e_{x}e_{x'}^*\right) = \left( h_{x,x',a,a'}(s)\right) _{a,a'}\). We have
It follows that \(\Gamma \) 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 \(\Gamma \) has the form (9.12).
Remark 9.10
(i) Every tracial QNS correlation \(\Gamma = \Gamma _{E,\tau }\) is fair. Indeed, writing \(E = (g_{x,x',a,a'})\), we have
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 \(\mathcal {A}\) (resp. \(\mathcal {B}\)) is a unital C*algebra, \(\tau _{\mathcal {A}}\) (resp. \(\tau _{\mathcal {B}}\)) a trace on \(\mathcal {A}\) and E (resp. F) a stochastic \(\mathcal {A}\)matrix (resp. a stochastic \(\mathcal {B}\)matrix). Let \(\lambda \in (0,1)\), \(\mathcal {C} = \mathcal {A}\oplus \mathcal {B}\), \(\tau : \mathcal {C}\rightarrow \mathbb {C}\) be given by \(\tau (u\oplus v) = \lambda \tau _{\mathcal {A}}(u) + (1\lambda ) \tau _{\mathcal {B}}(v)\), and \(G = E\oplus F\), considered as an element of \(M_X\otimes M_A\otimes \mathcal {C}\). Then, G is a stochastic \(\mathcal {C}\)matrix and
(iii) It is straightforward from Theorem 9.1 that if \(p\in \mathcal {C}_\textrm{qc}\) (resp. \(p\in \mathcal {C}_\textrm{q}\), \(p\in \mathcal {C}_\textrm{loc}\)) is synchronous, then \(\Gamma _p\) is a tracial (resp. quantum tracial, locally tracial) QNS correlation. By [23, Theorem 4.2], the set \(\mathcal {C}_\textrm{q}^\textrm{s}\) of synchronous quantum NS correlations is not closed if \(X = 5\) and \(A = 2\). Let \(p\in \overline{\mathcal {C}_\textrm{q}^\textrm{s}}{\setminus } \mathcal {C}_\textrm{q}^\textrm{s}\). Then, p is a synchronous NS correlation and does not lie in \(\mathcal {C}_\textrm{q}\). Assume that \(\Gamma _p\) is quantum tracial. By Theorem 9.9, \(\Gamma _p\in \mathcal {Q}_\textrm{q}\) and hence, by Remark 8.1, \(p\in \mathcal {C}_\textrm{q}\), 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 \((\Gamma _n)_{n\in \mathbb {N}}\) is a sequence of tracial QNS correlations converging to the QNS correlation \(\Gamma \), let \(\mathcal {A}_n\) be a unital C*algebra with a trace \(\tau _n\), and \(E_n = \left( g^{(n)}_{x,x',a,a'}\right) \) be a stochastic \(\mathcal {A}_n\)matrix such that \(\Gamma _n = \Gamma _{E_n,\tau _n}\). Let \(\mathcal {A}\) be the tracial ultraproduct of the family \(\{(\mathcal {A}_n,\tau _n)\}_{n\in \mathbb {N}}\) with respect to a nontrivial ultrafilter \(\mathfrak {u}\) [34, Section 4]. Write \(\tau \) for the trace on \(\mathcal {A}\) and \(E = (g_{x,x',a,a'})\) for the class of \(\oplus _{n\in \mathbb {N}} E_n\) in \(\mathcal {A}\). Then,
We next show that the class of all tracial QNS correlations, as well as each of the subclasses of quantum tracial and locally tracial QNS correlations, have natural classes of invariant states. Given a unital C*algebra \(\mathcal {A}\), a trace \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) and a stochastic \(\mathcal {A}\)matrix \(E = (g_{z,z'})_{z,z'} \in \mathcal {L}(\mathbb {C})\otimes M_Z\otimes \mathcal {A}\), let \(\omega ^{E,\tau } = \left( \omega ^{E,\tau }_{z,z',u,u'}\right) \in M_{ZZ}\) be defined by
Equivalently, let \(E^\textrm{op}\) be the stochastic \(\mathcal {A}^\textrm{op}\)matrix \(\left( g_{u',u}^\textrm{op}\right) \), and recall that \(s_{\tau }: \mathcal {A}\otimes _{\max }\mathcal {A}^\textrm{op} \rightarrow \mathbb {C}\) is the state given by \(s_{\tau }(u\otimes v^\textrm{op}) = \tau (uv)\). Then, \(\omega ^{E,\tau } = L_{s_{\tau }}\left( E\otimes E^\textrm{op}\right) ,\) where
is the corresponding slice. It follows that \(\omega ^{E,\tau }\) is a state.
Definition 9.11
Let Z be a finite set. A state \(\omega \in M_{ZZ}\) is called

(i)
C*reciprocal if there exists a unital C*algebra \(\mathcal {A}\), a trace \(\tau \) on \(\mathcal {A}\) and a stochastic \(\mathcal {A}\)matrix \(E\in M_Z\otimes \mathcal {A}\) such that \(\omega = \omega ^{E,\tau }\);

(ii)
quantum reciprocal if it is C*reciprocal, and the C*algebra \(\mathcal {A}\) from (i) can be chosen to be finite dimensional;

(iii)
locally reciprocal if it is C*reciprocal, and the C*algebra \(\mathcal {A}\) from (i) can be chosen to be abelian.
We will denote by \(\Upsilon (Z)\) (resp. \(\Upsilon _\textrm{q}(Z)\), \(\Upsilon _\textrm{loc}(Z)\)) the set of all C*reciprocal (resp. quantum reciprocal, locally reciprocal) states in \(M_{ZZ}\).
Theorem 9.12
Let \(\Gamma \) be a QNS correlation.

(i)
If \(\Gamma \) is tracial, then \(\Gamma \left( \Upsilon (X)\right) \subseteq \Upsilon (A)\);

(ii)
If \(\Gamma \) is quantum tracial, then \(\Gamma \left( \Upsilon _\textrm{q}(X)\right) \subseteq \Upsilon _\textrm{q}(A)\);

(iii)
If \(\Gamma \) is locally tracial, then \(\Gamma \left( \Upsilon _\textrm{loc}(X)\right) \subseteq \Upsilon _\textrm{loc}(A)\).
Proof
(i) Let \(\tau \) be a trace on \(\mathcal {C}_{X,A}\), \(\mathcal {A}\) be a C*algebra, \(\tau _{\mathcal {A}}\) be a trace on \(\mathcal {A}\), and \(E = (g_{x,x'})_{x,x'}\in M_X\otimes \mathcal {A}\) be a stochastic \(\mathcal {A}\)matrix. Set \(\omega = \Gamma _{\tau }\left( \omega ^{E,\tau _{\mathcal {A}}}\right) \) and write \(\omega = \left( \omega _{a,a',b,b'}\right) _{a,a',b,b'}\). Let \(\mathcal {B} = \mathcal {A}\otimes _{\max } \mathcal {C}_{X,A}\) and \(\tau _{\mathcal {B}} = \tau _{\mathcal {A}}\otimes \tau \) be the product trace on \(\mathcal {B}\) [13, Proposition 3.4.7]. Set
thus, \(F:= (h_{a,a'})_{a,a'}\in M_A\otimes \mathcal {B}\). Moreover,
To see that F is positive, we assume that \(\mathcal {A}\) and \(\mathcal {C}_{X,A}\) are faithfully represented and let \(V_x\) and \(V_{a,x}\) be operators such that \((V_x)_x\) is a row isometry, \((V_{a,x})_{a,x}\) is an isometry, \(g_{x,x'} = V_x^*V_{x'}\) and \(e_{x,x',a,a'} = V_{a,x}^*V_{a',x'}\), \(x,x' \in X\), \(a,a'\in A\). Letting \(W = \left( \sum _{x\in X} V_x\otimes V_{a,x}\right) _{a\in A}\), considered as row operator, we have that \(F = W^*W\). Hence, F is a stochastic \(\mathcal {B}\)matrix. In addition, for \(a,a',b,b'\in A\) we have
implying that \(\omega = \omega ^{F,\tau _{\mathcal {B}}}\).
(ii) and (iii) follow from the fact that if the C*algebra \(\mathcal {A}\) is finite dimensional (resp. abelian) and \(\tau \) factors through a finitedimensional (resp. abelian) C*algebra, then so does \(\tau _{\mathcal {B}}\).
Remark 9.13
(i) The state \(\frac{1}{X^2}I_{XX}\) is locally reciprocal, and hence it follows from Theorem 9.12 that
Similar descriptions hold for \(\Upsilon _\textrm{q}(A)\) and \(\Upsilon (A)\). Remark 9.10 thus implies that the sets \(\Upsilon (A)\), \(\Upsilon _\textrm{q}(A)\) and \(\Upsilon _\textrm{loc}(A)\) are convex and the sets \(\Upsilon (A)\) and \(\Upsilon _\textrm{loc}(A)\) are closed.
(ii) Recall that a state \(\rho \in M_{XX}\) is called de Finetti [18] if there exist states \(\omega _i\in M_X\), \(i = 1,\dots , k\), such that \(\rho = \sum _{j=1}^k \lambda _j \omega _j \otimes \omega _j\) as a convex combination. By (9.13) and Theorem 9.9,
Thus, the locally reciprocal states can be viewed as twisted de Finetti states. The presence of the transposition in our case is required in view of 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 [33, 53], to which we refer the reader for the definition used here). Indeed, factorisable channels have Choi matrices of the form \(\tau (g_{x,x'}h_{y',y})_{x,x',y,y'}\), where \(\tau \) is a faithful normal trace on a von Neumann algebra \(\mathcal {A}\), and \((g_{x,x'})_{x,x'}\) and \((h_{y,y'})_{y,y'}\) are matrix unit systems—a special type of stochastic \(\mathcal {A}\)matrices (see [53, Proposition 3.1]). Equivalently, the Choi matrices of factorisable channels \(\Phi : M_X\rightarrow M_X\) can be described [33, Definition 3.1] as the matrices of the form \(\left( \tau (v_{a,x}^*v_{a',x'})\right) _{x,x',a,a'}\), where \(V = (v_{a,x})_{a,x}\in M_X(\mathcal {A})\) is a unitary matrix. Note that if E is the stochastic operator matrix corresponding to V, then the QNS correlation \(\Gamma = \Gamma _{E,\tau }\) has marginal channels \(\Gamma _A(\cdot ) = \Gamma (\cdot \otimes I)\) and \(\Gamma _B(\cdot ) = \Gamma (I\otimes \cdot )\) that coincide with \(\Phi \). We can thus view tracial QNS correlations as generalised couplings of factorisable channels. Here, by a coupling of the pair \((\Phi ,\Psi )\) of channels, we mean a channel \(\Gamma \) with \(\Gamma _A = \Phi \) and \(\Gamma _B = \Psi \)—a generalisation of classical coupling of probability distributions in the sense of optimal transport [71].
9.3 Tracial CQNS correlations
In this subsection, we define a tracial version of CQNS correlations. Let \(\mathcal {A}\) be a unital C*algebra, \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) be a trace and \(E\in \mathcal {D}_X\otimes M_A\otimes \mathcal {A}\) be a semiclassical stochastic \(\mathcal {A}\)matrix. Write \(E = (g_{x,a,a'})_{x,a,a'}\); thus, \((g_{x,a,a'})_{a,a'}\in \left( M_A\otimes \mathcal {A}\right) ^+\) and \(\sum _{a\in A} g_{x,a,a} = 1\), for each \(x\in X\). Set \(E^\textrm{op} = (g_{x,a',a}^\textrm{op})_{x,a,a'}\); thus, \(E^\textrm{op} \in \mathcal {D}_X\otimes M_A\otimes \mathcal {A}^\textrm{op}\) and Lemma 9.2 shows that \(E^\textrm{op}\) is a semiclassical stochastic \(\mathcal {A}^\textrm{op}\)matrix. Let \(\phi _{E,x}: M_A\rightarrow \mathcal {A}\) be the unital completely positive map given by \(\phi _{E,x}(e_ae_{a'}^*) = g_{x,a,a'}\). By Boca’s Theorem [6], there exists a unital completely positive map \(\phi _E: \mathcal {B}_{X,A}\rightarrow \mathcal {A}\) such that \(\phi _E(e_{x,a,a'}) = g_{x,a,a'}\), \(x\in X\), \(a,a'\in A\). Let \(\phi _E^\textrm{op}: \mathcal {B}_{X,A}^\textrm{op}\rightarrow \mathcal {A}^\textrm{op}\) be the map given by \(\phi _E^\textrm{op}(u^\textrm{op}) = \phi _{E}(u)^\textrm{op}\), which is completely positive by Lemma 9.5. Write
thus, by (7.6) \(f_{E,\tau }\) is a state on \(\mathcal {R}_{X,A}\otimes _\textrm{c}\mathcal {R}_{X,A}\). Note that
In the sequel, we write \(\mathcal {E}_{E,\tau } = \mathcal {E}_{f_{E,\tau }}\); by Theorem 7.7, \(\mathcal {E}_{E,\tau } \in \mathcal{C}\mathcal{Q}_\textrm{qc}\).
Definition 9.14
A CQNS correlation \(\mathcal {E}\) is called

(i)
tracial if \(\mathcal {E} = \mathcal {E}_{E,\tau }\), where \(E\in \mathcal {D}_X\otimes M_A\otimes \mathcal {A}\) is a semiclassical stochastic \(\mathcal {A}\)matrix for some unital C*algebra \(\mathcal {A}\) and \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) 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.
Proposition 9.15
Let \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{AA}\) be a CQNS correlation.

(i)
If \(\mathcal {E}\) is quantum tracial, then \(\mathcal {E}\in \mathcal{C}\mathcal{Q}_\textrm{q}\);

(ii)
\(\mathcal {E}\) is locally tracial if and only if there exist channels \(\mathcal {E}_j: \mathcal {D}_X\rightarrow M_A\), \(j = 1,\dots ,k\), such that
$$\begin{aligned} \mathcal {E} = \sum _{j=1}^k \lambda _j \mathcal {E}_j\otimes \mathcal {E}_j^{\sharp }. \end{aligned}$$(9.14)In particular, if \(\mathcal {E}\) is locally tracial, then \(\mathcal {E}\in \mathcal{C}\mathcal{Q}_\textrm{loc}\).
Proof
(i) Suppose that \(\mathcal {E}\) is quantum tracial and write \(\mathcal {E} = \mathcal {E}_{E,\tau }\), where \(E = (g_{x,a,a'})_{x,a,a'} \in \mathcal {D}_X\otimes M_A\otimes \mathcal {A}\) is a semiclassical stochastic \(\mathcal {A}\)matrix for some finite dimensional C*algebra \(\mathcal {A}\) and a trace \(\tau : \mathcal {A}\rightarrow \mathbb {C}\). The matrix \(\tilde{E} = \left( \delta _{x,x'}g_{x,a,a'}\right) _{x,x',a,a'}\) is a stochastic matrix in \(M_X\otimes M_A\otimes \mathcal {A}\) and hence gives rise, via Theorem 5.2, to a canonical *homomorphism \(\pi _{\tilde{E}}: \mathcal {C}_{X,A}\rightarrow \mathcal {A}\). Letting \(\tilde{\tau } = \tau \circ \pi _{\tilde{E}}\), we have that \(\tilde{\tau }\) is a trace on \(\mathcal {C}_{X,A}\) and \(\Gamma _{\mathcal {E}} = \Gamma _{\tilde{\tau }}\). Thus, \(\Gamma _{\mathcal {E}} \in \mathcal {Q}_\textrm{q}\). By Remark 8.1, \(\mathcal {E}\in \mathcal{C}\mathcal{Q}_\textrm{q}\).
(ii) We fix \(\mathcal {A}\), \(\tau \) and E as in (i), with \(\mathcal {A}\) abelian. The trace \(\tilde{\tau }\), defined in the proof of (i), now factors through an abelian C*algebra, and hence \(\Gamma _{\mathcal {E}}\) is locally tracial. By Theorem 9.9, there exists quantum channels \(\Phi _j: M_X\rightarrow M_A\), \(j = 1,\dots ,k\), such that \(\Gamma _{\mathcal {E}} = \sum _{j=1}^k \Phi _j\otimes \Phi _j^{\sharp }\) as a convex combination. Letting \(\mathcal {E}_j = \Phi _j_{\mathcal {D}_X}\), \(j = 1,\dots ,k\), we see that \(\mathcal {E}\) has the form (9.14).
Conversely, suppose that \(\mathcal {E}\) has the form (9.14). By Theorem 9.9, there exists an abelian C*algebra \(\mathcal {A}\), a *representation \(\pi : \mathcal {C}_{X,A}\rightarrow \mathcal {A}\) and a trace \(\tau \) on \(\mathcal {A}\) such that \(\Gamma _{\mathcal {E}} = \Gamma _{\tau \circ \pi }\). The stochastic operator matrix \(E = \left( \pi (e_{x,x,a,a'})\right) _{x,a,a'}\) is semiclassical and \(\mathcal {E} = \mathcal {E}_{E,\tau }\).
We now specialise Definition 9.11 to states in \(\mathcal {D}_{XX}\), that is, bipartite probability distributions. A probability distribution \(q = (q(x,y))_{x,y\in X}\) on \(X\times X\) will be called C*reciprocal if there exists a C*algebra \(\mathcal {A}\), a POVM \((g_x)_{x\in X}\) in \(\mathcal {A}\) and a trace \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) such that \(q(x,y) = \tau (g_x g_y)\), \(x,y\in X\). If \(\mathcal {A}\) can be chosen to be finite dimensional (resp. abelian), we call q quantum reciprocal (resp. locally reciprocal). We denote by \(\Upsilon ^\textrm{cl}(X)\) (resp. \(\Upsilon ^\textrm{cl}_\textrm{q}(X)\), \(\Upsilon ^\textrm{cl}_\textrm{loc}(X)\)) the (convex) set of all C*reciprocal (resp. quantum reciprocal, locally reciprocal) probability distributions on \(X\times X\).
It can be seen as in Remark 9.13 that the class of locally reciprocal probability distributions coincides with the wellknown class of exchangeable probability distributions, that is, the convex combinations of the form
where \(q_i\) is a probability distribution on X, \(i = 1,\dots ,n\). Thus, C*reciprocal and quantum reciprocal probability distributions can be viewed as quantum versions of exchangeable distributions.
It is straightforward to see that, writing \(\Delta = \Delta _{XX}\), we have
These relations, combined with Theorem 9.12, easily yield the following proposition, whose proof is omitted.
Proposition 9.16
Let \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{AA}\) be a CQNS correlation.

(i)
If \(\mathcal {E}\) is tracial, then \(\mathcal {E}\left( \Upsilon ^\textrm{cl}(X)\right) \subseteq \Upsilon (A)\);

(ii)
If \(\mathcal {E}\) is quantum tracial, then \(\mathcal {E}\left( \Upsilon _\textrm{q}^\textrm{cl}(X)\right) \subseteq \Upsilon _\textrm{q}(A)\);

(iii)
If \(\mathcal {E}\) is locally tracial, then \(\mathcal {E}\left( \Upsilon _\textrm{loc}^\textrm{cl}(X)\right) \subseteq \Upsilon _\textrm{loc}(A)\).
9.4 Tracial NS correlations
The correlation classes introduced in Sects. 9.2 and 9.3 have a natural NS counterpart. For a C*algebra \(\mathcal {A}\), equipped with a trace \(\tau \), and a classical stochastic \(\mathcal {A}\)matrix \(E\in \mathcal {D}_X\otimes \mathcal {D}_A\otimes \mathcal {A}\), say, \(E = (g_{x,a})_{x,a}\) (so that \(g_{x,a}\in \mathcal {A}^+\) for all \(x\in X\) and all \(a\in A\) and \(\sum _{a\in A} g_{x,a} = 1\), \(x\in X\)), write
Similar arguments to the ones in Sects. 9.2 and 9.3 show that \(p_{E,\tau }\in \mathcal {C}_\textrm{qc}\).
Definition 9.17
An NS correlation p is called

(i)
tracial if it is of the form \(p_{E,\tau }\), where E is a classical stochastic \(\mathcal {A}\)matrix for some unital C*algebra \(\mathcal {A}\) and \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) is a trace;

(ii)
quantum tracial if it is tracial and the C*algebra \(\mathcal {A}\) in (i) can be chosen to be finite dimensional;

(iii)
locally tracial if it it is tracial and the C*algebra \(\mathcal {A}\) 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 \(p \in \mathcal {C}_\textrm{q}\);

(ii)
p is locally tracial if and only if \(p = \sum _{j=1}^k \lambda _j q_j\otimes q_j\), where \(q_j = \{q_j(\cdot x): x\in X\}\), is a family of probability distributions, \(j = 1,\dots ,k\). In particular, if p is locally tracial, then \(p\in \mathcal {C}_\textrm{loc}\).
Proposition 9.19
Let \(\mathcal {N}: \mathcal {D}_{XX}\rightarrow \mathcal {D}_{AA}\) be an NS correlation.

(i)
If \(\mathcal {N}\) is tracial, then \(\mathcal {N} \left( \Upsilon ^\textrm{cl}(X)\right) \subseteq \Upsilon ^\textrm{cl}(A)\).

(ii)
If \(\mathcal {N}\) is quantum tracial, then \(\mathcal {N}\left( \Upsilon _\textrm{q}^\textrm{cl}(X)\right) \subseteq \Upsilon _\textrm{q}^\textrm{cl}(A)\).

(iii)
If \(\mathcal {N}\) is locally tracial, then \(\mathcal {N}\left( \Upsilon _\textrm{loc}^\textrm{cl}(X)\right) \subseteq \Upsilon _\textrm{loc}^\textrm{cl}(A)\).
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 \(\mathcal {E}\) be a CQNS correlation. The following hold:

(i)
p is tracial (resp. quantum tracial, locally tracial, fair) if and only if \(\mathcal {E}_p\) is tracial (resp. quantum tracial, locally tracial, fair), if and only if \(\Gamma _p\) is tracial (resp. quantum tracial, locally tracial, fair);

(ii)
\(\mathcal {E}\) is tracial (resp. quantum tracial, locally tracial, fair) if and only if \(\Gamma _{\mathcal {E}}\) is tracial (resp. quantum tracial, locally tracial, fair).
Moreover,

(iii)
the map \(\mathfrak {N}\) 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 \(\mathfrak {C}\) 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 \(p(a,bx,y) = \tau (g_{x,a}g_{y,b})\), \(x,y\in X\), \(a,b\in A\), for some trace \(\tau \) on a unital C*algebra \(\mathcal {A}\) and matrix \(F = (g_{x,a})_{x,a} \in \left( \mathcal {D}_X\otimes \mathcal {D}_A\otimes \mathcal {A}\right) ^+\) with \(\sum _{a\in A} g_{x,a} = 1\), \(x\in X\). The matrix \(F' = (\delta _{a,a'}g_{x,a})_{x,a,a'} \in \mathcal {D}_X\otimes M_A\otimes \mathcal {A}\) is a semiclassical stochastic \(\mathcal {A}\)matrix and, trivially, \(\mathcal {E}_p = \mathcal {E}_{F',\tau }\). Similarly, the family \(F'' = (\delta _{a,a'}\delta _{x,x'}g_{x,a})_{x,x',a,a'} \in M_X\otimes M_A\otimes \mathcal {A}\) is a stochastic \(\mathcal {A}\)matrix and \(\Gamma _p = \Gamma _{F'',\tau }\).
Conversely, suppose that \(\Gamma _p = \Gamma _{E,\tau }\), where \(E = (g_{x,x',a,a'})_{x,x',a,a'}\) is a stochastic \(\mathcal {A}\)matrix and \(\tau \) is a trace on the unital C*algebra \(\mathcal {A}\). Then, \(E':= (g_{x,x,a,a'})_{x,a,a'}\) (resp. \(E'':= (g_{x,x,a,a})_{x,a}\)) is a semiclassical (resp. classical) stochastic \(\mathcal {A}\)matrix such that \(\mathcal {E}_p = \mathcal {E}_{E',\tau }\) (resp. \(p = p_{E'',\tau }\)).
(ii) is similar to (i).
(iii) follows from the fact that if E is a stochastic \(\mathcal {A}\)matrix and \(\tau \) is a trace on \(\mathcal {A}\) such that \(\Gamma = \Gamma _{E,\tau }\), then \(\mathfrak {C}(\Gamma ) = \mathcal {E}_{E',\tau }\), 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
For (i), observe that \(\mathcal {E}_p=\Delta _{A,B}\circ \mathcal {E}_p\) and hence
showing that p is fair if and only if so is \(\mathcal {E}_p\). As \(\Gamma _p = \Gamma _{\mathcal {E}_p}\), the equivalence with fairness of \(\Gamma _p\) 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, \(\mathcal {N}_p\) is a tracial NS correlation. In fact, the synchronous quantum commuting NS correlations arise precisely from classical stochastic \(\mathcal {A}\)matrices \((g_{x,a})_{x,a}\), where each \((g_{x,a})_{a\in A}\) 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 [23]. It can be shown, using results on the completely positive semidefinite cone of matrices [14, 46] that \(\Upsilon ^\textrm{cl}_\textrm{loc} \ne \Upsilon ^\textrm{cl}_\textrm{q}\) [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 \(p = \{p(\cdot  x): x\in X\}\) and \(q = \{q(\cdot  x): x\in X\}\) be families of distributions so that, for some \(x\in X\), we have that \(\textrm{supp}p(\cdot x)\cap \textrm{supp}q(\cdot x) = \emptyset \). Then, \(\tilde{p} = 1/2(p\otimes q + q\otimes p)\) is a fair NS correlation. However, \(\tilde{p}\) is not tracial; indeed, assuming the contrary, we have that \(\tilde{p} = \sum _{j=1}^m \lambda _j p_j\otimes p_j\) as a convex combination, where \(\{p_j\}_{j=1}^m\) consists of families of probability distributions indexed by X. Since
we have \(\sum _{j=1}^m \lambda _j p_j(ax)^2 = 0\), and hence \(p_j(ax)=0\), for all \(a\in A\) 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 \(q\otimes q\), where \(q = \{q(\cdot x): x\in X\}\) is a family of probability distributions with at least one x having \(\textrm{supp}q(\cdot x) > 1\), is not synchronous, but is locally tracial; thus, the first inclusion in the first, second and third rows are strict.
10 Correlations as Strategies for Nonlocal Games
In this section, we discuss how QNS correlations can be viewed as perfect strategies for quantum nonlocal games, extending the analogous viewpoint on NS correlations to the quantum case. Let X, Y, A and B be finite sets. A nonlocal 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) \(\lambda : X\times Y\times A \times B \rightarrow \{0,1\}\). 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 \((x,y)\in X\times Y\) and the players produce a pair \((a,b)\in A\times B\); they win the round if and only if \(\lambda (x,y,a,b) = 1\). An NS correlation p on \(X\times Y \times A\times B\) is called a perfect strategy for the game \(\lambda \) if
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 \(p(\cdot ,\cdot x,y)\), 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 \(x,y\in X\), we write \(x\sim y\) if \(\{x,y\}\) is an edge of G. By assumption, \(x\sim y\) implies \(x\ne y\); we write \(x\simeq y\) if \(x\sim y\) or \(x = y\). A classical colouring of G is a map \(f: X\rightarrow A\), where A is a finite set, such that
The chromatic number \(\chi (G)\) of G is the minimal cardinality A of a set A for which a classical colouring \(f: X\rightarrow A\) of G exists.
The graph colouring game for G (called henceforth the Gcolouring game) [16] is the nonlocal game with \(Y = X\), \(B = A\), and rules

(i)
\(x = y \ \ \Longrightarrow \ \ a = b\);

(ii)
\(x \sim y \ \ \Longrightarrow \ \ a \ne b\).
Thus, an NS correlation \(p = \left\{ (p(a,bx,y))_{a,b\in A}: x,y\in X\right\} \) is a perfect strategy of the Gcolouring game if

(S)
p is synchronous;

(P)
\(x \sim y \ \Rightarrow \ p(a,ax,y) = 0\) for all a.
It is easy to see that if p is a perfect strategy of the Gcolouring game from the class \(\mathcal {C}_\textrm{loc}\), then G possesses a classical colouring from the set A. Thus, the perfect strategies for the Gcolouring game from \(\mathcal {C}_\textrm{x}\), where \(\textrm{x} \in \{\textrm{loc}, \textrm{q}, \textrm{qc}\}\) can be thought of as classical \(\textrm{x}\)colourings of G. The \(\textrm{x}\)chromatic number of G is the parameter
in particular, \(\chi _\textrm{loc}(G = \chi (G)\) (see [16, 50, 62] and the references therein).
We call p a Gproper correlation if condition (P) is satisfied. For a finite set A, we let \(\Omega _A\) be the nonnormalised 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 Gproper if and only
Proof
The claim is immediate from the fact that
Remark 10.1 allows to generalise the classical \(\textrm{x}\)colourings of a graph G to the quantum setting as follows.
Definition 10.2
Let G be a graph with vertex set X. A CQNS correlation \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{AA}\) is called Gproper if
A Gproper CQNS correlation \(\mathcal {E}\) is called

(i)
a quantum \(\textrm{loc}\)colouring of G by A if \(\mathcal {E}\) is locally tracial;

(ii)
a quantum \(\textrm{q}\)colouring of G by A if \(\mathcal {E}\) is quantum tracial;

(iii)
a quantum \(\textrm{qc}\)colouring of G by A if \(\mathcal {E}\) is tracial.
For \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qc}\}\), let
be the quantum \(\textrm{x}\)chromatic number of G.
Remark. The complete quantum graph on n vertices was introduced in [70] as the subspace \(\mathcal {Q}_n = \{\Omega _n\}^{\perp }\), where \(\Omega _n\) is the maximally entangled state in \(\mathbb {C}^n\otimes \mathbb {C}^n\). In view of Definition 10.8 below, we see that the existence of a Gproper correlation \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{n^2}\) of a specified type \(\textrm{t}\) is equivalent to the existence of a homomorphism of type \(\textrm{t}\) from the quantisation (10.10) of G into \(\mathcal {Q}_n\).
Recall [67] that an orthogonal representation of a graph G with vertex set X is a family \((\xi _x)_{x\in X}\) of unit vectors in \(\mathbb {C}^k\) such that
The orthogonal rank \(\xi (G)\) of G is given by
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 \(\mathbb {C}^k\);

(ii)
there exists a quantum \(\textrm{loc}\)colouring of G by a set A with \(A = k\).
Proof
(i)\(\Rightarrow \)(ii) Suppose that \((\xi _x)_{x\in X}\subseteq \mathbb {C}^k\) is an orthogonal representation of G. Let \(\mathcal {E}_0: \mathcal {D}_{X}\rightarrow M_{A}\) be the quantum channel given by
and set \(\mathcal {E} = \mathcal {E}_0\otimes \mathcal {E}_0^{\sharp }\); by Proposition 9.15, \(\mathcal {E}\) is locally tracial. If \(x\sim y\), then
thus, \(\mathcal {E}\) is a quantum \(\textrm{loc}\)colouring of G.
(ii)\(\Rightarrow \)(i) Suppose that \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{AA}\) is a quantum \(\textrm{loc}\)colouring of G, and write \(\mathcal {E} = \sum _{j=1}^k \lambda _j \mathcal {E}_j\otimes \mathcal {E}_j^{\sharp }\) as a convex combination with positive coefficients, where \(\mathcal {E}_j: \mathcal {D}_X\rightarrow M_A\) is a quantum channel, \(j = 1,\dots ,k\). Suppose that \(x\sim y\). Then,
and hence, by the nonnegativity of each of the terms of the sum,
Let \(\xi _x\) be a unit eigenvector of \(\mathcal {E}_1(e_x e_x^*)\), corresponding to a positive eigenvalue, \(x\in X\). Condition (10.1) implies that \(\left\langle \xi _x \xi _x^* \otimes \left( \xi _y \xi _y^*\right) ^\textrm{t},\Omega _A\right\rangle = 0\), which in turn means, by the arguments in the previous paragraph, that \(\langle \xi _x,\xi _y\rangle = 0\).
By Proposition 10.3, \(\xi _\textrm{loc}(G) = \xi (G)\). Thus, the parameters \(\xi _\textrm{q}\) and \(\xi _\textrm{qc}\) can be viewed as quantum versions of the orthogonal rank.
Proposition 10.4
Let G be a graph. Then,

(i)
\(\xi _\textrm{qc}(G) \le \xi _\textrm{q}(G) \le \xi _\textrm{loc}(G)\), and

(ii)
\(\xi _\textrm{x}(G)\le \chi _\textrm{x}(G)\) for \(\textrm{x}\in \{\textrm{loc}, \textrm{q},\textrm{qc}\}\).
Proof
(i) The inequalities follow from the fact that \(\mathcal{C}\mathcal{Q}_\textrm{loc}\subseteq \mathcal{C}\mathcal{Q}_\textrm{q} \subseteq \mathcal{C}\mathcal{Q}_\textrm{qc}\).
(ii) Let p be a synchronous NS correlation that is an \(\textrm{x}\)colouring of G by a set A. By Theorem 9.20, \(\mathcal {E}_p\in \mathcal{C}\mathcal{Q}_\textrm{x}\). By Remark 10.1, \(\mathcal {E}_p\) is Gproper. Thus, \(\xi _\textrm{x}(G)\le \chi _\textrm{x}(G)\).
Remarks. (i) There exist graphs G for which \(\xi (G) < \chi (G)\) (see e.g. [67]). By Proposition 10.3, for such G we have a strict inequality in Proposition 10.4 (ii) in the case \(\textrm{x} = \textrm{loc}\). In [52], an example of a graph G on 13 vertices was exhibited with the property that \(\xi (G) < \chi _\textrm{q}(G)\). By Proposition 10.4 (i), for this graph G, we have a strict inequality in Proposition 10.4 (ii) in the case \(\textrm{x} = \textrm{q}\). We do not know if a strict inequality can occur in the case \(\textrm{x} = \textrm{qc}\).
(ii) It was shown in [52] that there exists a graph G such that \(\chi _\textrm{q}(G) < \xi (G)\). By Proposition 10.4 (ii), this implies \(\xi _\textrm{q}(G) < \xi (G)\). We do not whether \(\xi _\textrm{qc}(G)\) can be strictly smaller than \(\xi _\textrm{q}(G)\).
We point out that there is another natural family of quantum orthogonal ranks, defined as follows. Call a CQNS correlation \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{AA}\) genuinely tracial if there exist a unital C*algebra \(\mathcal {A}\), a trace \(\tau : \mathcal {A}\rightarrow \mathbb {C}\) and a *homomorphism \(\pi : \mathcal {B}_{X,A}\rightarrow \mathcal {A}\) such that
We call a genuinely tracial CQNS correlation \(\mathcal {E}\) a genuine quantum \(\textrm{qc}\)colouring of G if \(\mathcal {E}\) is Gproper. In addition, we call \(\mathcal {E}\) a genuine quantum \(\textrm{q}\)colouring of G if the algebra \(\mathcal {A}\) for which (10.2) holds can be chosen to be finite dimensional, and \(\mathcal {E}\) is Gproper. Set
We have the inequalities \(\xi _\textrm{x}(G) \le \xi _\textrm{x}'(G)\), and we do not know if equality holds.
We exhibit a lower bound on \(\xi _\textrm{qc}'(G)\) in terms of the Lovász number \(\theta (G)\) 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 Sect. 10.2. If \(\kappa \subseteq X\times X\), let
thus, \(\mathcal {S}_{\kappa }\) is a linear subspace of \(M_X\) which is a bimodule over the diagonal algebra \(\mathcal {D}_X\). We write
and let \(\mathcal {S}_G:= \mathcal {S}_{E(G)}\) be the graph operator system of G [21], and \(\mathcal {S}_G^0:= \mathcal {S}_{E_0(G)}\) be the graph operator antisystem of G [70] (here we use the terminology of [7]).
Proposition 10.5
Let G be a graph with vertex set X. Then, \(\xi _\textrm{qc}' (G) \ge \sqrt{\frac{X}{\theta (G)}}\). Moreover, \(\xi _\textrm{q}'(K_{d^2}) = \xi _\textrm{qc}'(K_{d^2}) = d\).
Proof
Suppose that \(\mathcal {E}: \mathcal {D}_{XX}\rightarrow M_{AA}\) is a genuine quantum \(\textrm{qc}\)colouring of G and write \(\omega _{x,y} = \mathcal {E}(e_x e_x^* \otimes e_y e_y^*)\), \(x,y\in X\). We may assume that, in the definition of the correlation \(\mathcal {E}\), we have that \(\mathcal {A}=\mathcal {B}_{X,A}\) and \(\pi \) is the identity representation. Let H be a Hilbert space, \(\xi \in H\) a unit vector and \((E_{x,a,a'})_{a,a'}\) matrix unit systems in \(\mathcal {B}(H)\) such that if \(\rho : \mathcal {B}_{X,A}\rightarrow \mathcal {B}(H)\) is the *representation given by \(\rho (e_{x,a,a'}) = E_{x,a,a'}\), \(x\in X\), \(a,a'\in A\), then
Set \(\xi _{x,a,a'} = E_{x,a,a'}\xi \), \(x\in X\), \(a,a'\in A\); then
In addition, if \(x\sim y\), then
Let
and note that the matrix \(\left( Q_{a,a',b,b'}\right) _{a,a',b,b'}\) is positive. In addition, being a Gram matrix, \(Q_{a,b,a,b}\) is positive for all \(a,b\in A\). Write \(Q = \sum _{a,b\in A} Q_{a,b,a,b}\). By (10.4),
By Theorem 3.1, there exist operators \(V_{a,x}\) such that \((V_{a,x})_{a,x}\) is an isometry and \(E_{x,a,a'} = V_{a,x}^* V_{a',x}\), \(x\in X\), \(a,a'\in A\). Thus, if \(x\in X\), then
Write \(Q = D + T\), where D is diagonal and \(T \perp \mathcal {S}_{G}\). We have shown that \(D\le AI_X\); thus \(A I_X + T\in M_X^+\). It follows that
Let \((E_{a,b})_{a,b\in A}\) be a matrix unit system on H and \(U_x\) be a unitary, such that
(such unitaries exist because every two matrix unit systems on H are unitarily equivalent). Let \(\eta = (\overline{\lambda _x})_{x\in X}\in \mathbb {C}^X\). We have
where \(J_X\in M_X\) is the matrix all of whose entries are equal to 1. It follows that \(\frac{1}{A} J_X\le Q\); thus, \(\frac{X}{A} \le \Vert Q\Vert \). Together with (10.5), this implies \(\frac{X}{A} \le A\theta (G)\), completing the proof of the inequality.
Realise \(A = \mathbb Z_d = \{0,1,\ldots , d1\}\) and let \(X = A\times A\). Let \(\zeta \) be a primitive Ath root of unity. For \(x = (a',b')\) and \(y = (a'',b'')\in X\), let
and write \(\sigma _{x,y} = \xi _{x,y}\xi _{x,y}^*\). We have
Note that if \(\Theta = (\sigma _{x,y})_{x,y}\), then \(\mathcal {E}:= \mathcal {E}_{\Theta }\) is a CQNS correlation; indeed,
for all \(x,y\in X\). Since
we have that \(\mathcal {E}\) is \(K_{d^2}\)proper.
We claim that \(\mathcal {E}\) is genuinely tracial. To see this, let
and note that, for each \(x\in X\), the family \(\{E_{x,z,z'}: z,z'\in A\}\) is a matrix unit system. For \(x = (a',b'), y = (a'',b'')\in X\) and \(z,z',w,w'\in A\) we have
Therefore, \(\mathcal {E}\) is genuinely quantum tracial. It follows that \(\xi _\textrm{q}'(K_{d^2}) \le d\); On the other hand, \(\theta (K_{d^2}) = 1\) and hence \(\xi _\textrm{qc}'(K_{d^2}) \ge d\). Thus, \(\xi '_\textrm{qc}(K_{d^2}) = \xi '_\textrm{q}(K_{d^2}) = d\).
10.2 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 \(G\rightarrow H\) has \(Y = X\), \(B = A\), and \(\lambda (x,y,a,b) = 0\) if and only if, either \(x = y\) and \(a\ne b\), or \(x\sim y\) and \(a\not \sim b\). A synchronous NS correlation \(p = \left\{ \left( p(a,bx,y)\right) _{a,b\in A}: x,y\in X\right\} \) is thus called a perfect \(\textrm{x}\)strategy for the game \(G\rightarrow H\) if \(p\in \mathcal {C}_\textrm{x}\) and
For a subset \(\kappa \subseteq X\times X\), let \(P_{\kappa }: M_X\rightarrow M_X\) be the map given by
Thus, \(P_{\kappa }\) is the Schur projection onto \(\mathcal {S}_{\kappa }\); it can be canonically identified with the (positive) element \(\sum _{(x,y)\in \kappa } (e_x e_x^*) \otimes (e_y e_y^*)\) of \(\mathcal {D}_{XX}\). We set \((P_{\kappa })_{\perp } = P_{\kappa ^c}\). 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 \(p = \left\{ \left( p(a,bx,y)\right) _{a,b\in A}: x,y\in X\right\} \) be a synchronous NS correlation. The following are equivalent:

(i)
p is a perfect strategy for the homomorphism game \(G\rightarrow H\);

(ii)
\(\left\langle \mathcal {N}_p(P_G), (P_{H})_{\perp }\right\rangle = 0\).
Proof
(i)\(\Rightarrow \)(ii) We have
(ii)\(\Rightarrow \)(i) If \(x\sim y\) and \(a\not \sim b\), then \(e_x e_x^*\otimes e_y e_y^*\le P_G\) and \(e_a e_a^*\otimes e_b e_b^*\le (P_H)_{\perp }\). By the monotonicity of the pairing,
General operator systems in \(M_X\) were considered in [21] as a quantum versions of graphs (noting that \(\mathcal {S}_G\) is an operator system), while operator antisystems (that is, selfadjoint subspaces of \(M_X\) each of whose elements has trace zero [7]) were proposed as such a quantum version in [70] (noting that \(\mathcal {S}_G^0\) is an operator antisystem). 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 [22], employed also here, it will be convenient to use a slightly different perspective on noncommutative graphs, which we now describe. Let Z be a finite set, \(H = \mathbb {C}^Z\), \(H^\textrm{d}\) be its dual space and \(\textrm{d}: H\rightarrow H^\textrm{d}\) be the map given by \(\textrm{d}(\xi ) (\eta ) = \langle \eta ,\xi \rangle \); we write \(\xi ^{\textrm{d}} = \textrm{d}(\xi )\). Note that if \(T\in \mathcal {L}(H)\), then
Let \(\theta : H\otimes H\rightarrow \mathcal {L}(H^{\textrm{d}},H)\) be the linear map given by
By (10.6),
We denote by \(\textrm{m}: H\otimes H\rightarrow \mathbb {C}\) the map given by
Let also \(\mathfrak {f}: H\otimes H \rightarrow H\otimes H\) be the flip operator given by \(\mathfrak {f}(\xi \otimes \eta ) = \eta \otimes \xi \). Note that if \(\xi ,\eta , \zeta _1,\zeta _2 \in H\), then
and hence
thus,
In addition,
and hence
Definition 10.7
A linear subspace \(\mathcal {U}\subseteq H\otimes H\) is called skew if \(\textrm{m}(\mathcal {U}) = \{0\}\) and symmetric if \(\mathfrak {f}(\mathcal {U}) = \mathcal {U}\).
Suppose that \(\mathcal {U}\) is a symmetric skew subspace of \(H\otimes H\). Let \(\mathcal {S}_{\mathcal {U}} = \theta (\mathcal {U})\); by (10.8) and (10.9), the subspace \(\mathcal {S}_{\mathcal {U}}\) of \(\mathcal {L}(H^{\textrm{d}},H)\) satisfies

\(T\in \mathcal {S}_{\mathcal {U}}\) \(\Longrightarrow \) \(\textrm{d}^{1}\circ T^* \circ \textrm{d}^{1} \in \mathcal {S}_{\mathcal {U}}\), and

\(T\in \mathcal {S}_{\mathcal {U}}\) \(\Longrightarrow \) \(\sum _{z\in Z} \langle (T\circ \textrm{d}) (e_z),e_z\rangle = 0\).
We call a subspace of \(\mathcal {L}(H^{\textrm{d}},H)\) satisfying these properties a twisted operator antisystem. Conversely, given a twisted operator antisystem \(\mathcal {S}\subseteq \mathcal {L}(H^{\textrm{d}},H)\), (10.8) and (10.9) imply that the subspace \(\mathcal {U}_{\mathcal {S}} = \theta ^{1}(\mathcal {S})\) of \(H\otimes H\) is symmetric and skew. Given a graph G, let
it is clear that \(\mathcal {U}_G\) is a symmetric skew subspace of \(\mathbb {C}^X\otimes \mathbb {C}^X\). We thus consider symmetric skew subspaces of \(\mathbb {C}^X\otimes \mathbb {C}^X\) as a noncommutative version of graphs. We note that if \(\partial : H^{\textrm{d}}\rightarrow H\) is the unitary operator given on the standard basis \(\{e_z\}_{z\in Z}\) in H by \(\partial (e_z^{\textrm{d}})\rightarrow e_z\), \(z\in Z\), then \(\mathcal {S}\subseteq \mathcal {L}( H^{\textrm{d}},H)\) is a twisted operator antisystem if and only if the subspace \(\mathcal {S}\partial ^{1}\subseteq \mathcal {L}(H)\) is closed under transposition and traceless, see [10, p. 36].
We write \(P_{\mathcal {U}}\) for the orthogonal projection from \(\mathbb {C}^X\otimes \mathbb {C}^X\) onto \(\mathcal {U}\). Let \(\mathcal {U}_{\perp }\subset \left( \mathbb {C}^X\otimes \mathbb {C}^X\right) ^{\textrm{d}}\) be the annihilator of \(\mathcal {U}\) and write \(P_{\mathcal {U}_\perp }\in \mathcal {L}\left( (\mathbb {C}^X\otimes \mathbb {C}^X)^{\textrm{d}}\right) \) for the orthogonal projection onto \(\mathcal {U}_{\perp }\). Observe that \(\zeta ^{\textrm{d}}\in \mathcal {U}_{\perp }\) if and only if \(\zeta \) belongs to the orthogonal complement \(\mathcal {U}^{\perp }\) of \(\mathcal {U}\) in \(\mathbb {C}^X\otimes \mathbb {C}^X\). Thus, for \(\zeta \in H\otimes H\) we have
and hence
Let A be a finite set and \(\omega \in M_A\). Writing \(f_\omega \) for the functional on \(M_A\) given by \(f_\omega (\rho ) = \textrm{Tr}(\rho \omega ^\textrm{t})\), we have that the map \(\omega \rightarrow f_{\omega }\) is a complete order isomorphism from \(M_A\) onto \(M_A^\textrm{d}\) (see e.g. [63, Theorem 6.2]). On the other hand, the map \(\omega ^{\textrm{d}}\mapsto \omega ^\textrm{t}\) is a *isomorphism from \(\mathcal {L}\left( (\mathbb {C}^A)^{\textrm{d}}\right) \) onto \(M_A\). The composition of these maps, \(\omega ^{\textrm{d}}\mapsto f_{w^t}\), is thus a complete order isomorphism from \(\mathcal {L}\left( (\mathbb {C}^A)^{\textrm{d}}\right) \) onto \(M_A^\textrm{d}\). In the sequel, we identify these two spaces; note that, via this identification,
Definition 10.8
Let X and A be finite sets and \(\mathcal {U}\subseteq \mathbb {C}^X\otimes \mathbb {C}^X\), \(\mathcal {V}\subseteq \mathbb {C}^A\otimes \mathbb {C}^A\) be symmetric skew subspaces. A QNS correlation \(\Gamma : M_{XX}\rightarrow M_{AA}\) is called

(i)
a quantum commuting homomorphism from \(\mathcal {U}\) to \(\mathcal {V}\) (denoted \(\mathcal {U}{\mathop {\rightarrow }\limits ^\textrm{qc}}\mathcal {V}\)) if \(\Gamma \) is tracial and
$$\begin{aligned} \left\langle \Gamma \left( P_{\mathcal {U}}\right) ,P_{\mathcal {V}_{\perp }}\right\rangle = 0. \end{aligned}$$(10.13) 
(ii)
a quantum homomorphism from \(\mathcal {U}\) to \(\mathcal {V}\) (denoted \(\mathcal {U}{\mathop {\rightarrow }\limits ^\textrm{q}}\mathcal {V}\)) if \(\Gamma \) is quantum tracial and (10.13) holds;

(iii)
a local homomorphism from \(\mathcal {U}\) to \(\mathcal {V}\) (denoted \(\mathcal {U}{\mathop {\rightarrow }\limits ^\textrm{loc}}\mathcal {V}\)) if \(\Gamma \) is locally tracial and (10.13) holds.
Given operator antisystems \(\mathcal {S}\subseteq M_X\) and \(\mathcal {T}\subseteq M_A\), Stahlke [70] defines a noncommutative graph homomorphism from \(\mathcal {S}\) to \(\mathcal {T}\) to be a quantum channel \(\Phi : M_X\rightarrow M_A\) with family \(\{M_i\}_{i=1}^m\) of Kraus operators, such that \(M_i \mathcal {S} M_j^*\subseteq \mathcal {T}\), \(i,j = 1,\dots ,m\); if such \(\Phi \) exists, he writes \(\mathcal {S}\rightarrow \mathcal {T}\). The appropriate version of this notion for twisted operator antisystems—directly modelled on Stahlke’s definition—is as follows. For \(T\in M_Z\), we write \(\overline{T} = T^{* \textrm{t}}\) for the conjugated matrix of T.
Definition 10.9
Let X and A be finite sets, and \(\mathcal {S}\subseteq \mathcal {L}\left( (\mathbb {C}^{X})^{\textrm{d}},\mathbb {C}^X\right) \) and \(\mathcal {T}\subseteq \mathcal {L}\big ((\mathbb {C}^{A})^{\textrm{d}},\mathbb {C}^A\big )\) be twisted operator antisystems. A homomorphism from \(\mathcal {S}\) into \(\mathcal {T}\) is a quantum channel
such that
If \(\mathcal {S}\) and \(\mathcal {T}\) are twisted operator antisystems, we write \(\mathcal {S}\rightarrow \mathcal {T}\) as in [70] to denote the existence of a homomorphism from \(\mathcal {S}\) to \(\mathcal {T}\).
Proposition 10.10
Let X and A be finite sets and \(\mathcal {U}\subseteq \mathbb {C}^X\otimes \mathbb {C}^X\), \(\mathcal {V}\subseteq \mathbb {C}^A\otimes \mathbb {C}^A\) be symmetric skew spaces. Then, \(\mathcal {U}{\mathop {\rightarrow }\limits ^\textrm{loc}}\mathcal {V}\) if and only if \(\mathcal {S}_{\mathcal {U}}\rightarrow \mathcal {S}_{\mathcal {V}}\).
Proof
Suppose that \(\mathcal {U}{\mathop {\rightarrow }\limits ^\textrm{loc}}\mathcal {V}\) and let \(\Gamma \) be a locally tracial QNS correlation for which (10.13) holds. By Theorem 9.9, there exist quantum channels \(\Phi _j: M_X\rightarrow M_A\), \(j = 1,\dots ,k\), such that \(\Gamma = \sum _{j=1}^k \lambda _j \Phi _j\otimes \Phi _j^{\sharp }\) as a convex combination. We have
since each of the terms in the sum on the left hand side is nonnegative, selecting j with \(\lambda _j > 0\) and setting \(\Phi = \Phi _j\), we have
Let \(\Phi (\omega ) = \sum _{i=1}^m M_i \omega M_i^*\), \(\omega \in M_X\), be a Kraus representation of \(\Phi \). For \(\omega \in M_X\), we have
It follows that
Let \(\xi \in \mathcal {U}\) and \(\eta \in \mathcal {V}^{\perp }\) be unit vectors; then, \(\xi \xi ^*\le P_{\mathcal {U}}\). In addition, \(\eta ^\textrm{d} = P_{\mathcal {V}_{\perp }}\eta ^\textrm{d}\) and hence \((\eta \eta ^*)^\textrm{d}=\eta ^\textrm{d}(\eta ^\textrm{d})^* \le P_{\mathcal {V}_{\perp }}\); thus, (10.14) implies
By (10.15) and positivity,
which, by (10.12), means that
Thus, \((M_i \otimes \overline{M}_j)\xi \in \mathcal {V}\) for every \(\xi \in \mathcal {U}\) and, by (10.7),
that is, \(\mathcal {S}_{\mathcal {U}}\rightarrow \mathcal {S}_{\mathcal {V}}\).
Conversely, suppose that \(\Phi : M_X\rightarrow M_A\) is a quantum channel with a family of Kraus operators \((M_i)_{i=1}^m\subseteq \mathcal {L}(\mathbb {C}^X,\mathbb {C}^A)\) such that \(\overline{M}_j \mathcal {S}_{\mathcal {U}} M_i^\textrm{d}\subseteq \mathcal {S}_{\mathcal {V}}\), \(i,j = 1,\dots ,m\). The previous paragraphs show that
for all unit vectors \(\xi \in \mathcal {U}\), \(\eta \in \mathcal {V}^{\perp }.\) It follows that
for all unit vectors \(\eta \in \mathcal {V}^{\perp }\). Taking infimum over all such \(\eta \), we obtain
for all unit vectors \(\xi \in \mathcal {U}\). Thus, by (10.12) and (10.11),
for all unit vectors \(\xi \in \mathcal {U}\). Writing \(P_{\mathcal {U}} = \sum _{i=1}^l \xi _i\xi _i^*\), where \((\xi _i)_{i=1}^l\) is an orthonormal basis of \(\mathcal {U}\), we obtain \(\left\langle (\Phi \otimes \Phi ^{\sharp })(P_{\mathcal {U}}),P_{\mathcal {V}_{\perp }}\right\rangle = 0\).
For graphs G and H, write \(G\rightarrow H\) if there exists a homomorphism from G to H. The next corollary justifies viewing the symmetric skew spaces as noncommutative graphs.
Corollary 10.11
Let G and H be graphs. We have that \(G\rightarrow H\) if and only if \(\mathcal {U}_G {\mathop {\rightarrow }\limits ^{\textrm{loc}}} \mathcal {U}_H\).
Proof
Write X and A for the vertex sets of G and H, respectively. Assume that \(G\rightarrow H\). By [70], \(\mathcal {S}_G^0\rightarrow \mathcal {S}_H^0\). Write \(\{M_i\}_{i=1}^m\) for the set of Kraus operators such that \(M_i \mathcal {S}_G^0 M_j^*\subseteq \mathcal {S}_H^0\), \(i,j = 1,\dots ,m\). Let \(J_X: \mathbb {C}^X\rightarrow \mathbb {C}^X\) be the map given by \(J_X(\eta ) = \bar{\eta }\). Then, \(\theta (e_x\otimes e_y) = J_X\circ e_ye_x^*\circ \textrm{d}^{1}\), \(x,y\in X\). Therefore,
implying \(\overline{M}_i\mathcal {S}_{\mathcal {U}_G}M_j^{\textrm{d}}\subseteq \mathcal {S}_{\mathcal {U}_H}\); by Proposition 10.10, \(\mathcal {U}_G {\mathop {\rightarrow }\limits ^{\textrm{loc}}} \mathcal {U}_H\). The converse follows after reversing the arguments.
10.3 General quantum nonlocal games
We write \(\mathcal {P}_{\mathcal {M}}\) for the projection lattice of a von Neumann algebra \(\mathcal {M}\), and denote as usual by \(\vee \) (resp. \(\wedge \)) the join (resp. the wedge) operation in \(\mathcal {P}_{\mathcal {M}}\); thus, for \(P_1,P_2\in \mathcal {P}_{\mathcal {M}}\), the projection \(P_1\vee P_1\) (resp. \(P_1\wedge P_2\)) has range the closed span (resp. the intersection) of the ranges of \(P_1\) and \(P_2\). If \(\mathcal {M}\) and \(\mathcal {N}\) are von Neumann algebras, a map \({\varphi }: \mathcal {P}_{\mathcal {M}}\rightarrow \mathcal {P}_{\mathcal {N}}\) is called join continuous if \({\varphi }\left( \vee _{i\in \mathbb {I}} P_i\right) = \vee _{i\in \mathbb {I}} {\varphi }(P_i)\) for any family \(\{P_i\}_{i\in \mathbb {I}}\subseteq \mathcal {P}_{\mathcal {M}}\). Note that if \(\mathcal {M}\) 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 \(\mathcal {U}\). As in Sect. 10.2, we denote by \(\mathcal {U}_{\perp }\) the annihilator of \(\mathcal {U}\) in the space \(H^\textrm{d}\), and by \(P_{\perp }\)—the orthogonal projection on \(H^\textrm{d}\) with range \(\mathcal {U}_{\perp }\).
Definition 10.12
Let X, Y, A and B be finite sets.

(i)
A map \({\varphi }: \mathcal {P}_{M_{XY}}\rightarrow \mathcal {P}_{M_{AB}}\) (resp. \({\varphi }: \mathcal {P}_{\mathcal {D}_{XY}}\rightarrow \mathcal {P}_{M_{AB}}\), \({\varphi }: \mathcal {P}_{\mathcal {D}_{XY}}\rightarrow \mathcal {P}_{\mathcal {D}_{AB}}\)) is called a quantum nonlocal game (resp. a classicaltoquantum nonlocal game, a classical nonlocal game) if \({\varphi }\) is join continuous and \({\varphi }(0) = 0\). We say that such \({\varphi }\) is a game from XY to AB.

(ii)
A QNS (resp. CQNS, NS) correlation \(\Lambda \) is called a perfect strategy for the quantum (resp. classicaltoquantum, classical) nonlocal game \({\varphi }\) if
$$\begin{aligned} \left\langle \Lambda (P),{\varphi }(P)_{\perp }\right\rangle = 0, \ \ \ P\in \mathcal {P}_{M_{XY}} \ (\text{ resp. } P\in \mathcal {P}_{\mathcal {D}_{XY}}). \end{aligned}$$(10.16)
Remark 10.13
(i) Join continuous zeropreserving maps \({\varphi }: \mathcal {P}_{\mathcal {B}(H)} \rightarrow \mathcal {P}_{\mathcal {B}(K)}\), where H and K are Hilbert spaces, were first considered by J. A. Erdos in [26]. They are equivalent to bilattices introduced in [68]—that is, subsets \(\mathfrak {B}\subseteq \mathcal {P}_{\mathcal {B}(H)} \times \mathcal {P}_{\mathcal {B}(K)}\) such that \((P,0), (0,Q)\in \mathfrak {B}\) for all \(P\in \mathcal {P}_{\mathcal {B}(H)}\), \(Q\in \mathcal {P}_{\mathcal {B}(K)}\), and \((P_1,Q_1), (P_2,Q_2)\in \mathfrak {B}\) \(\Rightarrow \) \((P_1\vee P_2,Q_1\wedge Q_2)\in \mathfrak {B}\) and \((P_1\wedge P_2,Q_1\vee Q_2)\in \mathfrak {B}\). Thus, quantum nonlocal games (resp. classicaltoquantum nonlocal games, classical nonlocal 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 (10.16) are reminiscent of J. A. Erdos’ characterisation [26] of reflexive spaces of operators, introduced by L. N. Loginov and V. S. Shulman in [47]. As shown in [26], a subspace \(\mathcal {S}\subseteq \mathcal {B}(H,K)\) (H and K being Hilbert spaces) is reflexive in the sense of [47] if and only if there exists a join continuous zeropreserving map \({\varphi }: \mathcal {P}_{\mathcal {B}(H)} \rightarrow \mathcal {P}_{\mathcal {B}(K)}\) such that \(\mathcal {S}\) coincides with the space
(ii) The quantum (resp. classicaltoquantum, classical) nonlocal game \({\varphi }\) with \({\varphi }(P) = I_{AB}\) for every nonzero \(P \in \mathcal {P}_{M_{XY}}\) (resp. \(P \in \mathcal {P}_{\mathcal {D}_{XY}}\)) 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 nosignalling correlations.
(iii) Let G be a graph with vertex set X and A be a finite set. The quantum graph colouring game considered in Sect. 10.1 is the classicaltoquantum nonlocal game \({\varphi }: \mathcal {P}_{\mathcal {D}_{XX}}\rightarrow \mathcal {P}_{M_{AA}}\), given by
Similarly, letting \(\mathcal {U}\subseteq \mathbb {C}^X\otimes \mathbb {C}^X\) and \(\mathcal {V}\subseteq \mathbb {C}^A\otimes \mathbb {C}^A\) be symmetric skew spaces, we define the homomorphism game \(\mathcal {U}\rightarrow \mathcal {V}\) to be the quantum nonlocal game \(\psi \), given by
For \(\textrm{x}\in \{\textrm{loc}, \textrm{q}, \textrm{qc}\}\), we have that \(\mathcal {U}{\mathop {\rightarrow }\limits ^{\textrm{x}}}\mathcal {V}\) if and only if the game \(\mathcal {U}\rightarrow \mathcal {V}\) has a perfect strategy of class \(\mathcal {Q}_\textrm{x}\).
Let \((X,Y,A,B,\lambda )\) be a nonlocal game. For a subset \(\alpha \subseteq X\times Y\), let \(P_{\alpha } \in \mathcal {P}_{\mathcal {D}_{XY}}\) be the projection with range \(\textrm{span}\{e_x\otimes e_y: (x,y)\in \alpha \}\). For \((x,y)\in X\times Y\), let
We associate with \(\lambda \) the (unique) classical nonlocal game \({\varphi }_{\lambda }: \mathcal {P}_{\mathcal {D}_{XY}} \rightarrow \mathcal {P}_{\mathcal {D}_{AB}}\) determined by the requirement
Proposition 10.14
An NS correlation p is a perfect strategy for the nonlocal game (with rule function) \(\lambda \) if and only if \(\mathcal {N}_p\) is a perfect strategy for \({\varphi }_{\lambda }\).
Proof
Note that if \((x,y)\in X\times Y\), then \(\left( P_{\beta _{x,y}(\lambda )}\right) _{\perp }\) has range \(\textrm{span}\{e_a e_a^*\otimes e_b e_b^*: \lambda (x,y,a,b) = 0\}\). As in Proposition 10.6, it is thus easily seen that p is a perfect strategy for \(\lambda \) if and only if
Assume that p is a perfect strategy for \(\lambda \). For a projection \(P\in \mathcal {D}_{XY}\), write \(P = \vee \{P_{\{(x,y)\}}: P(e_x\otimes e_y) = e_x\otimes e_y\}\); then,
Thus, \(\left\langle \mathcal {N}_p(P_{\{(x,y)\}}),{\varphi }_\lambda (P)_{\perp }\right\rangle = 0\) for all pairs (x, y) with \(P(e_x\otimes e_y) = e_x\otimes e_y\). Taking the join over all those (x, y), we conclude that \(\left\langle \mathcal {N}_p(P),{\varphi }_\lambda (P)_{\perp }\right\rangle = 0\). The converse is direct from the first paragraph.
Definition 10.15
Let X, Y, A, B, Z and W be finite sets and \({\varphi }_1\) (resp. \({\varphi }_2\)) be a game from XY to AB (resp. from AB to ZW). The composition of \({\varphi }_1\) and \({\varphi }_2\) is the game \({\varphi }_2\circ {\varphi }_1\) from XY to ZW.
It is clear that \({\varphi }_2\circ {\varphi }_1\) is welldefined in all cases except when \({\varphi }_1\) is a quantum game, while \({\varphi }_2\) is a classicaltoquantum game.
Lemma 10.16
Let X, A and Z be finite sets, H and K be Hilbert spaces and \(E\in M_X\otimes M_A\otimes \mathcal {B}(H)\) and \(F\in M_A\otimes M_Z\otimes \mathcal {B}(K)\) be stochastic operator matrices. Set
Then, \(G = (G_{x,x',z,z'})_{x,x',z,z'}\) is a stochastic operator matrix in \(M_X\otimes M_Z\otimes \mathcal {B}(K\otimes H)\).
Proof
Let \(V = (V_{a,x})_{a,x}\) (resp. \(W = (W_{z,a})_{z,a}\)) be an isometry from \(H^X\) (resp. \(K^A\)) to \(\tilde{H}^A\) (resp. \(\tilde{K}^Z\)) for some Hilbert space \(\tilde{H}\) (resp. \(\tilde{K}\)), such that
for all \(x,x'\in X\), \(a,a'\in A\) and \(z,z'\in Z\). Set
For \(x,x'\in X\), we have
thus, \((U_{z,x})_{z,x}\) is an isometry from \((K\otimes H)^X\) into \((\tilde{K}\otimes \tilde{H})^Z\). In addition, for \(x,x'\in X\) and \(z,z'\in Z\), we have
By Theorem 3.1, G is a stochastic operator matrix acting on \(K\otimes H\).
We call the stochastic operator matrix G from Lemma 10.16 the composition of F and E and denote it by \(F\circ E\).
Theorem 10.17
Let \({\varphi }_1\) (resp. \({\varphi }_2\)) be a quantum game from XY to AB (resp. from AB to ZW) and \(\textrm{x}\in \{\textrm{loc},\textrm{q},\textrm{qa},\textrm{qc},\textrm{ns}\}\).

(i)
If \(\Gamma _i\) is a perfect strategy for \({\varphi }_i\) from the class \(\mathcal {Q}_\textrm{x}\), \(i = 1,2\), then \(\Gamma _2\circ \Gamma _1\) is a perfect strategy for \({\varphi }_2\circ {\varphi }_1\) from the class \(\mathcal {Q}_\textrm{x}\).

(ii)
Assume that \(X = Y\), \(A = B\) and \(Z = W\). If \(\Gamma _i\) is a perfect tracial (resp. quantum tracial, locally tracial) strategy for \({\varphi }_i\), \(i = 1,2\), then \(\Gamma _2\circ \Gamma _1\) is a perfect tracial (resp. quantum tracial, locally tracial) strategy for \({\varphi }_2\circ {\varphi }_1\).
Proof
First note that if \(\Gamma _i\) is a QNS correlation, then so is \(\Gamma _2\circ \Gamma _1\). Indeed, suppose that \(\rho \in M_{XY}\) is such that \(\textrm{Tr}_X\rho = 0\). By Remark 2.1, \(\textrm{Tr}_A\Gamma _1(\rho ) = 0\), and hence, again by Remark 2.1, \(\textrm{Tr}_Z(\Gamma _2(\Gamma _1(\rho )) = 0\).
Suppose that \(\Gamma _i\in \mathcal {Q}_\textrm{qc}\), \(i = 1,2\). Let \((E^{(i)},F^{(i)})\) be a commuting pair of stochastic operator matrices acting on a Hilbert space \(H_i\), and \(\sigma _i\) be a normal state on \(\mathcal {B}(H_i)\), such that \(\Gamma _i = \Gamma _{E^{(i)}\cdot F^{(i)},\sigma _i}\), \(i = 1,2\). Write \(E^{(1)} = \left( E^{(1)}_{x,x',a,a'}\right) \), \(F^{(1)} = \left( F^{(1)}_{y,y',b,b'}\right) \), \(E^{(2)} = \left( E^{(2)}_{a,a',z,z'}\right) \) and \(F^{(2)} = \left( F^{(2)}_{b,b',w,w'}\right) \). Set \(H = H_2 \otimes H_1\), \(\sigma = \sigma _2\otimes \sigma _1\), \(E = E^{(2)}\circ E^{(1)}\) and \(F = F^{(2)}\circ F^{(1)}\); note that, by Lemma 10.16, E and F are stochastic operator matrices. It is straightforward that (E, F) is a commuting pair. Write \(E = (E_{x,x',z,z'})\) and \(F = (F_{y,y',w,w'})\). Note that
and hence
thus, \(\Gamma _2\circ \Gamma _1 = \Gamma _{E\cdot F,\sigma }\).
If \(\Gamma _i\in \mathcal {Q}_\textrm{q}\), \(i = 1,2\), then the arguments in the previous paragraph—replacing operator products by tensor products as necessary—show that \(\Gamma _2\circ \Gamma _1\in \mathcal {Q}_\textrm{q}\). By the continuity of the composition, the assumptions \(\Gamma _i\in \mathcal {Q}_\textrm{qa}\), \(i = 1,2\), imply that \(\Gamma _2\circ \Gamma _1\in \mathcal {Q}_\textrm{qa}\). Finally, assume that \(\Gamma _i\in \mathcal {Q}_\textrm{loc}\), \(i = 1,2\), and write \(\Gamma _i = \sum _{k=1}^{m_i} \lambda _k^{(i)} \Phi _k^{(i)} \otimes \Psi _k^{(i)}\) as a convex combination, where \(\Phi _k^{(i)}: M_X\rightarrow M_A\) and \(\Psi _k^{(i)}: M_Y\rightarrow M_B\) are quantum channels, \(i = 1,2\). Then,
as a convex combination, and hence \(\Gamma _2\circ \Gamma _1\in \mathcal {Q}_\textrm{loc}\).
Suppose that \(\Gamma _i\) is a tracial QNS correlation; thus, there exist unital C*algebras \(\mathcal {A}_1\) and \(\mathcal {A}_2\), traces \(\tau _1\) and \(\tau _2\) on \(\mathcal {A}_1\) and \(\mathcal {A}_2\), respectively, and stochastic matrices \(E^{(1)}\in M_X\otimes M_A\otimes \mathcal {A}_1\) and \(E^{(2)}\in M_A\otimes M_Z\otimes \mathcal {A}_2\), such that \(\Gamma _i = \Gamma _{E^{(i)}, \tau _i}\), \(i = 1,2\). The arguments given for (i) show that
where \(\tau _2\otimes \tau _1\) is the product trace on \(\mathcal {A}_2\otimes _{\min }\mathcal {A}_1\); \(E^{(2)}\circ E^{(1)}\) is considered as a stochastic \(\mathcal {A}_2\otimes _{\min }\mathcal {A}_1\)matrix (note that we identify \(\left( E^{(2)}\circ E^{(1)}\right) ^\textrm{op}\) with \(E^{(2)\textrm{op}}\circ E^{(1)\textrm{op}}\) in the natural way).
It remains to show that if \(\Gamma _i\) is a perfect strategy for \({\varphi }_i\), \(i = 1,2\), then \(\Gamma _2\circ \Gamma _1\) is a perfect strategy for \({\varphi }_2\circ {\varphi }_1\). Let \(P\in \mathcal {P}_{M_{XY}}\) and \(\omega \) be a pure state with \(\omega \le P\). Then, \(\Gamma _1(\omega ) = {\varphi }_1(P)\Gamma _1(\omega ){\varphi }_1(P)\) and hence \(\Gamma _1(\omega )\le {\varphi }_1(P)\). Similarly, for any pure state \(\sigma \) with \(\sigma \le {\varphi }_1(P)\) we have \(\Gamma _2(\sigma )={\varphi }_2({\varphi }_1(P))\Gamma _2(\sigma ){\varphi }_2({\varphi }_1(P))\), giving \(\left\langle (\Gamma _2(\sigma ),({\varphi }_2\circ {\varphi }_1)(P)_{\perp } \right\rangle = 0\). In particular,
As in the proof of Proposition 10.10, this yields
establishing the claim. \(\square \)
Suppose that \(p_1\) (resp. \(p_2\)) is an NS correlation from XY to AB (resp. from AB to ZW). It is straightforward to verify that the correlation p with \(\mathcal {N}_p = \mathcal {N}_{p_2}\circ \mathcal {N}_{p_1}\) is given by
we write \(p = p_2\circ p_1\). Such compositions were first studied in [58]. For a nonlocal game from XY to AB (resp. from AB to ZW) with rule function \(\lambda _1\) (resp. \(\lambda _2\)), let \(\lambda _2 \circ \lambda _1: X\times Y \times Z\times W\rightarrow \{0,1\}\) be given by
Combining Theorem 10.17 with classical reduction and Proposition 10.14, we obtain the following perfect strategy version of [58, Proposition 3.5], which simultaneously extends the graph homomorphism transitivity results contained in [58, Theorem 3.7].
Corollary 10.18
Let \(\lambda _1\) (resp. \(\lambda _2\)) be the rule functions of nonlocal games from XY to AB (resp. from AB to ZW) and \(\textrm{x}\in \{\textrm{loc},\textrm{q},\textrm{qa},\textrm{qc},\textrm{ns}\}\). If \(p_i\) is a perfect strategy for \(\lambda _i\) from the class \(\mathcal {C}_\textrm{x}\), \(i = 1,2\), then \(p_2\circ p_1\) is a perfect strategy for \(\lambda _2\circ \lambda _1\) from the class \(\mathcal {C}_\textrm{x}\).
Combining Theorem 10.17 with Remark 10.13 (iii) yields the following transitivity result; in view of Proposition 10.10, it extends [70, Proposition 9].
Corollary 10.19
Let X, A and Z be finite sets, \(\mathcal {U}\subseteq \mathbb {C}^X\otimes \mathbb {C}^X\), \(\mathcal {V}\subseteq \mathbb {C}^A\otimes \mathbb {C}^A\) and \(\mathcal {W}\subseteq \mathbb {C}^Z\otimes \mathbb {C}^Z\) be symmetric skew spaces, and \(\textrm{x}\in \{\textrm{loc},\textrm{q},\textrm{qc}\}\). If \(\mathcal {U}{\mathop {\rightarrow }\limits ^{\textrm{x}}}\mathcal {V}\) and \(\mathcal {V} {\mathop {\rightarrow }\limits ^{\textrm{x}}}\mathcal {W}\), then \(\mathcal {U}{\mathop {\rightarrow }\limits ^{\textrm{x}}}\mathcal {W}\).
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Acknowledgements
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. We thank the anonymous referee for their detailed reading of the manuscript and their valuable remarks and suggestions. The first named author was supported by NSF grants CCF2115071 and DMS2154459, and Simons Foundation grant 708084. The authors acknowledge the support of AIM through their participation of the AIM workshop on “Nonlocal games in quantum information theory” in May 2021.
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Notes. (i)
After the paper was completed, we became aware of the work [9], in which the authors define quantumtoclassical nosignalling correlations and study a version of the homomorphism game from a noncommutative to a classical graph. Although these can be viewed as dual to our classicaltoquantum correlations (Sect. 7), there is no direct duplication of results in the current paper with those in [9]. (ii) The article [10], published before the acceptance of the present paper, studies in detail quantum graph homomorphisms of different types, providing operatoralgebraic characterisations thereof. In particular, in [10, Theorem 5.5], we obtain a version of Proposition 10.10 for homomorphisms of quantum type, showing that the existence of such a homomorphism is equivalent to the existence of a local homomorphism from a quantum graph ampliated by a maximally entangled state. This implies that homomorphisms of quantum type constitute a special class of the entanglementassisted noncommutative graph homomorphisms defined in [70, Definition 15]. The identification of an operational characterisation of the latter notion remains an open problem.
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Todorov, I.G., Turowska, L. Quantum Nosignalling Correlations and Nonlocal Games. Commun. Math. Phys. 405, 141 (2024). https://doi.org/10.1007/s0022002405001x
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DOI: https://doi.org/10.1007/s0022002405001x