Geometry of contextuality from Grothendieck's coset space

TL;DR

This paper explores the geometric structures underlying quantum contextuality using Grothendieck's dessins d'enfants, revealing how certain coset geometries correspond to quantum commutation relations and contextuality signatures.

Contribution

It establishes a link between coset space geometries and quantum contextuality, introducing a geometric measure and identifying new contextual structures in multi-qubit systems.

Findings

Non-existence of specific dessins signals quantum contextuality.

Contextuality appears at index 9 in Mermin's square and 10 in Mermin's pentagram.

Most generalized polygons and n-qubit commuting sets are contextual.

Abstract

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli operators [Information 5, 209 (2014); 1310.4267 and 1404.6986 (quant-ph)]. Now we find that the non-existence of a dessin for which the commutator (a, b) = a^ (--1) b^( --1) ab precisely corresponds to the commutator of quantum observables [A, B] = AB -- BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index |G : H| = 9 in Mermin's square and at index 10 in Mermin's pentagram, as expected. Commuting sets of n-qubit observables with n \textgreater{} 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.