The Sheaf-Theoretic Structure Of Non-Locality and Contextuality

TL;DR

This paper employs sheaf theory to unify the understanding of non-locality and contextuality in quantum mechanics, providing a systematic algebraic approach and new insights into their hierarchy and implications.

Contribution

It introduces a sheaf-theoretic framework that generalizes probability tables, establishes a hierarchy of no-go theorems, and connects non-locality with contextuality and no-signalling in a unified manner.

Findings

Contextuality and non-locality correspond to obstructions to global sections.

A linear algebraic method systematically computes these obstructions.

Quantum mechanics obeys an extended no-signalling principle for all commuting observables.

Abstract

We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting which 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 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.