The Cohomology of Non-Locality and Contextuality

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 PeresMermin 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.

Contextual Semantics: From Quantum Mechanics to Logic, Databases, Constraints, and Complexity

“…This geometric language is substantiated by the results in [7], which show that sheaf cohomology can be used to characterise these obstructions, and to witness contextuality in a wide range of cases.…”

“…• The obstructions to global sections witnessing contextuality are characterised in terms of sheaf cohomology in [7] with Shane Mansfield and Rui Barbosa, and a range of examples are treated in this fashion.…”

“…We use the notation D B (X) for the set of boolean distributions on a set X. 7 In quantum mechanics, these correspond to compatible families of observables. Note that, if we use the semiring of non-negative reals instead, we obtain probability distributions with finite support.…”

A complete characterization of all-versus-nothing arguments for stabilizer states

“…Our computations provide a very large number of quantum-realisable strongly contextual empirical models admitting AvN arguments. These new models could potentially find applications in quantum information and computation, as well as contributing to the ongoing theoretical study of strong contextuality as a key feature of quantum mechanics [10,12,32,41].…”

On the Cohomology of Contextuality

“…This rigorous mathematical formulation has allowed the application of powerful methods of sheaf cohomology to the study of the topological structure of contextuality [2,5]. Central to this approach is the notion of cohomological obstruction to the existence of global sections, i.e.…”