Characterising the Performance of XOR Games and the Shannon Capacity of Graphs

In this Letter we give a set of necessary and sufficient conditions such that quantum players of a two-party XOR game cannot perform any better than classical players. With any such game, we associate a graph and examine its zero-error communication capacity. This allows us to specify a broad new class of graphs for which the Shannon capacity can be calculated. The conditions also enable the parametrization of new families of games that have no quantum advantage for arbitrary input probability distributions, up to certain symmetries. In the future, these might be used in information-theoretic studies on reproducing the set of quantum nonlocal correlations.