1212.4084 (Tobias Fritz et al.)

A Combinatorial Approach to Nonlocality and Contextuality    [PDF]

Tobias Fritz, Anthony Leverrier, Ana Belén Sainz

So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from the Foulis-Randall product. In the spirit of CSW, we find close relationships to various invariants studied in combinatorics. The recently proposed Local Orthogonality Principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is dominated by a low level of the Navascu\'es-Pironio-Ac\'in hierarchy of semidefinite programs, which we apply to contextuality scenarios. We hope that our approach may also serve as an introduction for combinatorialists to the subject of nonlocality and contextuality. Our conjectures on graphs whose Shannon capacity coincides with their independence number may be of particular interest.

View original: http://arxiv.org/abs/1212.4084