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Abstract:
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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 a certain product of contextuality
scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to
several graph invariants. 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 strictly dominated by a low level of
the Navascués–Pironio–Acín hierarchy of semidefinite programs, which we also apply
to contextuality scenarios.
We derive a wealth of results in our framework, many of these relating to quantum
and supraquantum contextuality and nonlocality, and state numerous open problems.
For example, we show that the set of quantum models on a con |
