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Commutation Groups and State-Independent Contextuality

Samson Abramsky, Serban-Ion Cercelescu, Carmen-Maria Constantin

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.

Commutation Groups and State-Independent Contextuality

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 -groups.
Paper Structure (17 sections, 22 theorems, 16 equations)

This paper contains 17 sections, 22 theorems, 16 equations.

Table of Contents

  1. Commutation groups
  2. Commutation groups by generators and relations
  3. Comparison with solution groups
  4. Linear algebraic construction of commutation groups
  5. Contextuality arguments in commutation groups
  6. Contextual words
  7. Comparison with other Heisenberg groups
  8. No state-independent contextuality in odd characteristics
  9. Contextuality in even characteristics
  10. Positive results
  11. The $\mathbb{Z}_2$ case
  12. Beyond $\mathbb{Z}_2$: padding, splitting and variable changes
  13. Classification of contextuality for matrices in Darboux normal form
  14. Reduction to Darboux normal form
  15. Unitary representation
  16. ...and 2 more sections

Key Result

Theorem 1

The rewrite system $\rightarrow_{\mathsf{G}}$ is confluent and normalizing. The set of normal forms is $\mathscr{N}$ (up to identification of $J_0$ and $1$).

Theorems & Definitions (25)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Example 6
  • Proposition 7
  • Proposition 8
  • Theorem 9
  • Example 10
  • ...and 15 more