Commutation Groups and State-Independent Contextuality

TL;DR

This paper develops an algebraic framework using commutation groups to analyze state-independent contextuality in quantum mechanics, providing new tools for understanding quantum advantage and non-classicality.

Contribution

It introduces commutation groups and a linear algebraic construction to study contextuality, characterizes when contextual words can arise, and constructs non-contextual value assignments.

Findings

Characterization of when contextual words can arise in commutation groups

Construction of non-contextual value assignments in certain cases

Representation of commutation groups as subgroups of generalized Pauli groups

Abstract

We introduce an algebraic structure for studying state-independent contextuality arguments, a key form of quantum non-classicality exemplified by the well-known Peres-Mermin magic square, and used as a source of quantum advantage. We introduce \emph{commutation groups} presented by generators and relations, and analyse them in terms of a string rewriting system. There is also a linear algebraic construction, a directed version of the Heisenberg group. We introduce \emph{contextual words} as a general form of contextuality witness. We characterise when contextual words can arise in commutation groups, and explicitly construct non-contextual value assignments in other cases. We give unitary representations of commutation groups as subgroups of generalized Pauli $n$-groups.