Commutation Groups and State-Independent Contextuality
This work provides a foundational algebraic framework for analyzing quantum non-classicality, which is incremental but essential for theoretical quantum information science.
The paper tackles the problem of understanding state-independent contextuality in quantum mechanics by introducing commutation groups as an algebraic structure, and it characterizes when contextual words can arise and constructs non-contextual value assignments in other cases.
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.