The Cohomology of Non-Locality and Contextuality

TL;DR

This paper applies sheaf cohomology to analyze non-locality and contextuality in quantum models, providing cohomological obstructions that serve as witnesses for contextuality in key quantum phenomena.

Contribution

It introduces a cohomological framework using sheaf theory to identify obstructions to global sections, offering a new mathematical approach to study contextuality.

Findings

Cohomological obstructions vanish for non-contextual models.

Obstructions are non-zero for models like PR boxes and GHZ states.

Provides cohomological witnesses for contextuality in quantum models.

Abstract

In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual. We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.