The sheaf-theoretic structure of non-locality and contextuality

By Samson Abramsky, Adam Brandenburger. 2011

606 citations recorded by the source corpus.

We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting that generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen–Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples—due to Bell, Hardy and Greenberger, Horne and Zeilinger, respectively—occupy successively higher levels of this hierarchy. A general correspondence is shown between the existence of local hidden-variable realizations using negative probabilities, and no-signalling; this is based on a result showing that the linear subspaces generated by the non-contextual and no-signalling models, over an arbitrary measurement cover, coincide. Maximal non-locality is generalized to maximal contextuality, and characterized in purely qualitative terms, as the non-existence of global sections in the support. A general setting is developed for the Kochen–Specker-type results, as generic, model-independent proofs of maximal contextuality, and a new combinatorial condition is given, which generalizes the ‘parity proofs’ commonly found in the literature. We also show how our abstract setting can be represented in quantum mechanics. This leads to a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4a37d06bfde9aa08f13333e89bb60af067b665e9

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Citations in the corpus

Snow Queen Is Evil and Beautiful: Experimental Evidence for Probabilistic Contextuality in Human Choices (2017, methodology): “The expectation in this experiment was that the responses to αi and βj could be treated as random variables Ai and Bj , respectively, and the CHSH inequality (1) could then be used to reveal the presence or absence of contextuality.”
A Combinatorial Approach to Nonlocality and Contextuality (2012, methodology): “Some notable exceptions are the following: (a) The study of test spaces in quantum logic [31,100], (b) Spekkens’ work on measurement and preparation contextuality [65,90], (c) The graph-theoretic approach of Cabello, Severini and Winter (CSW) [26], (d) The sheaf-theoretic approach pioneered by Abram”