Physics – Quantum Physics
Scientific paper
2012-03-06
Physics
Quantum Physics
12 pages
Scientific paper
There is a huge literature on Bell inequalities, with many ingenious derivations of families of inequalities. However, a unifying principle with a clear conceptual basis has proved elusive. In this paper, we introduce a form of Bell inequality based on logical consistency conditions, which we call logical Bell inequalities. This approach is both conceptually illuminating and technically powerful. We show that a rational inequality is satisfied by all non-contextual models if and only if it is equivalent to a logical Bell inequality. Thus quantitative tests for contextuality or non-locality always hinge on purely logical consistency conditions. We obtain explicit descriptions of complete sets of inequalities for the convex polytope of non-contextual probability models, and the derived polytope of expectation values for these models. Moreover, these results are obtained at a high level of generality: they apply not only to the familiar cases of Bell-type scenarios, for any number of parties, but to all Kochen-Specker configurations, and in fact to any family of sets of compatible measurements. This generality is achieved by working with measurement covers, following the sheaf-theoretic approach to non-locality and contextuality introduced by the first author and Adam Brandenburger. We also obtain results for a number of special cases. We show that a model achieves maximal violation of a logical Bell inequality if and only if it is strongly contextual. We show that all Kochen-Specker configurations lead to maximal violations of logical Bell inequalities in a state-independent fashion. We also derive specific violations of logical Bell inequalities for models which are possibilistically contextual, meaning that they admit logical proofs of contextuality.
Abramsky Samson
Hardy Lucien
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