Coming full circle -- A unified framework for Kochen-Specker contextuality

TL;DR

This paper unifies various frameworks of Kochen-Specker contextuality using a new approach called 'context connections' and develops a comprehensive algebraic framework to characterize contextuality in finite-dimensional quantum systems.

Contribution

It introduces the concept of 'context connections' and an 'observable algebras' framework to unify and generalize existing approaches to Kochen-Specker contextuality.

Findings

Provides a complete characterization of KS contextuality for finite-dimensional systems.

Shows how the new framework subsumes marginal and graph-theoretic approaches.

Establishes precise relationships between different notions of contextuality.

Abstract

Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, when understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the mathematics of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often is unclear. In fact, there is a glaring discrepancy already between the original notion of contextuality introduced by Kochen and Specker on the one side [J. Math. Mech., 17, 59, (1967)], and the modern approach of studying contextual correlations on the other [Rev. Mod. Phys., 94, 045007 (2022)]. In a companion paper [arXiv:2408.16764], we introduce the conceptually new tool called ``context connections'', which allows to cast and analyse Kochen-Specker (KS) contextuality in new form. Here, we generalise this notion, and based on it prove a complete characterisation of KS contextuality for finite-dimensional systems. To this end, we develop the framework of ``observable algebras". We show in detail how this framework subsumes the marginal and graph-theoretic approaches to contextuality, and thus that it offers a unified perspective on KS contextuality. In particular, we establish the precise relationships between the various notions of ``contextuality" used in the respective settings, and in doing so, generalise a number of results on the characterisation of the respective notions in the literature.