Contextuality, Cohomology and Paradox

TL;DR

This paper explores the deep connections between contextuality in quantum mechanics, logical paradoxes, and topological structures, revealing cohomological obstructions as witnesses to contextuality.

Contribution

It unifies contextuality across fields using sheaf theory, links it to logical paradoxes, and demonstrates topological origins via cohomology.

Findings

Contextuality relates to logical paradoxes.

Cohomological obstructions witness contextuality.

Topological methods reveal the structure of contextuality.

Abstract

Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.