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

We study the classical and quantumvalues of a class of oneand two-party unique games, that generalizes thewell-knownXORgames to the case of non-binary outcomes. In the bipartite case the generalizedXOR (XOR-d) gameswe study are a subclass of thewell-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 offinding equivalent games to the notion of signed graphs and switching equivalence from graph theory.We relate the problemof computing the classical value of single-party anti-correlation XORgames tofinding the edge bipartization number of a graph, which is known to beMaxSNPhard, 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 quantumalgebraic violation of these games, and show that nofinite 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 formulti-party XOR-type inequalities involving two-body correlation functions.