Contextuality and the fundamental theorems of quantum mechanics

TL;DR

This paper explores the deep connection between contextuality and fundamental quantum theorems, reformulating key results like Kochen-Specker, Wigner's, Gleason's, and Bell's theorems within a presheaf framework to highlight the central role of contextuality in quantum structure.

Contribution

It broadens the perspective on contextuality by relating it to multiple core theorems and provides presheaf-based reformulations applicable to von Neumann algebras, emphasizing contextuality's foundational importance.

Findings

Reformulation of the Kochen-Specker theorem using presheaves.

Extension of the presheaf approach to Wigner's, Gleason's, and Bell's theorems.

Demonstration that quantum structure is largely encoded by contextuality.

Abstract

Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr and later realised more technically by Kochen and Specker. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen-Specker theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.