Commutation Groups and State-Independent Contextuality

2026-03-12

AI Summary

This work addresses the systematic characterization and verification of state-independent contextuality in quantum mechanics by proposing an algebraic framework based on abelian groups. Integrating string rewriting systems with the linear-algebraic structure of the discrete Heisenberg group, the study introduces “contextual words” as algebraic witnesses of contextuality. By establishing necessary and sufficient conditions for the emergence of contextual words within abelian groups, the paper presents the first explicit construction of non-contextual value assignments and realizes a unitary representation of abelian groups within generalized Pauli groups. This approach offers a novel algebraic characterization of state-independent contextuality, combining theoretical clarity with computational tractability.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services
• Security and Privacy: Large-scale security measurements

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.

Problem

• state-independent contextuality
• commutation groups
• contextual words
• quantum non-classicality
• Peres-Mermin magic square

Innovation

• commutation groups
• state-independent contextuality
• contextual words
• string rewriting system
• generalized Pauli groups