Linear game non-contextuality and Bell inequalities—a graph-theoretic approach

Abstract

We study the classical and quantum values of a class of one-and two-party unique games, that generalizes the well-known XOR games to the case of non-binary outcomes. In the bipartite case the generalized XOR (XOR-d) games we study are a subclass of the well-known linear games. We introduce a 'constraint graph' associated to such a game, with the constraints defining the game represented by an edge-coloring of the graph. We use the graph-theoretic characterization to relate the task of finding equivalent games to the notion of signed graphs and switching equivalence from graph theory. We relate the problem of computing the classical value of single-party anti-correlation XOR games to finding the edge bipartization number of a graph, which is known to be MaxSNP hard, and connect the computation of the classical value of XOR-d games to the identification of specific cycles in the graph. We construct an orthogonality graph of the game from the constraint graph and study its Lovász theta number as a general upper bound on the quantum value even in the case of single-party contextual XOR-d games. XOR-d games possess appealing properties for use in deviceindependent applications such as randomness of the local correlated outcomes in the optimal quantum strategy. We study the possibility of obtaining quantum algebraic violation of these games, and show that no finite XOR-d game possesses the property of pseudo-telepathy leaving the frequently used chained Bell inequalities as the natural candidates for such applications. We also show this lack of pseudo-telepathy for multi-party XOR-type inequalities involving two-body correlation functions.

Focus on device independent quantum information

“…This issue contains a number of results in this direction, painting a diverse picture of the 'nonlocality landscape', now often reformulated as multiplayer games. The authors of [30] study linear games, a generalization of XOR games, which correspond to correlation inequalities. [24] study a different variation of XOR games, so-called CHSH q games.…”

Randomness in quantum mechanics: philosophy, physics and technology

“…Any hyperplane in the space of probabilities that separates the classical polytope from the rest determines a Bell inequality: everything that is above the upper horizontal dashed In general (n, m, d) scenarios, the complexity of characterizing the corresponding classical polytope is enormous. It is fairly easy to see that, even for (n, 2, 2), the number of its vertices (extremal points) is equal to 2 2n , hence it grows exponentially with n. Nevertheless, a considerable effort has been made in recent time to characterize multi-party nonlocality (Brunner et al, 2014;Liang et al, 2015;Rosicka et al, 2016;Tura et al, 2014aTura et al, , 2015Tura et al, , 2014b.…”

“…its walls of maximal dimensions (lower horizontal dashed line).In general (n, m, d) scenarios, the complexity of characterizing the corresponding classical polytope is enormous. It is fairly easy to see that, even for (n, 2, 2), the number of its vertices (extremal points) is equal to 2 2n , hence it grows exponentially with n. Nevertheless, a considerable effort has been made in recent time to characterize multi-party nonlocality(Brunner et al, 2014;Liang et al, 2015;Rosicka et al, 2016;Tura et al, 2014aTura et al, , 2015Tura et al, , 2014b.Among the many other device independent applications, the nonlocality appears to be a valuable resource in random number generation, certification, expansion and amplification, which we outline in the following subsections. In fact, it has been shown that Bell nonlocal correlation is a genuine resource, in the framework of a resource theory, where the allowed operations are restricted to device independent local operations(Gallego et al, 2012;Vicente, 2014).…”

Permutation graphs and unique games

“…It is clear that every consistent vertexlabeling is optimal. For a given graph G and an edge-labeling K : E → S n , the assignment number, β ′ C (G, K), is the number of consistent vertex-labelings possible for G and K. In [8], labeled graphs are used in the study of contextuality. When investigating contextuality, we attempt to quantify how much the outcome of a measurement of a physical observable depends on the context in which it is measured.…”

“…While this has not immediate practical interest, it is a playground to introduce new mathematical notions and gain a better understanding of special cases. In particular, we consider, the set L n = {π i ∈ S n : π i (x) ≡ i − x (mod n) for x, i ∈ [n]}, x + y ≡ i (mod 2), which corresponds to the class of GXOR games, studied in [8], and S 2 , which we compare to signed graphs.…”