Geometric contextuality from the Maclachlan-Martin Kleinian groups

TL;DR

This paper explores how certain Kleinian groups, specifically non-compact arithmetic subgroups generated by elliptic isometries, can serve as filters to identify geometric contextuality in quantum systems, connecting group theory with quantum contextuality.

Contribution

It introduces a novel approach using Maclachlan-Martin Kleinian groups to detect geometric contextuality in quantum observables, linking hyperbolic geometry with quantum contextuality.

Findings

Kleinian subgroups serve as effective contextuality filters.

Standard quantum contextual geometries are encompassed within this framework.

Bianchi groups over imaginary quadratic fields are particularly relevant.

Abstract

There are contextual sets of multiple qubits whose commutation is parametrized thanks to the coset geometry $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=\left\langle x,y\right\rangle$. One defines geometric contextuality from the discrepancy between the commutativity of cosets on $\mathcal{G}$ and that of quantum observables.It is shown in this paper that Kleinian subgroups $K=\left\langle f,g\right\rangle$ that are non-compact, arithmetic, and generated by two elliptic isometries $f$ and $g$ (the Martin-Maclachlan classification), are appropriate contextuality filters. Standard contextual geometries such as some thin generalized polygons (starting with Mermin's $3 \times 3$ grid) belong to this frame. The Bianchi groups $PSL(2,O\_d)$, $d \in \{1,3\}$ defined over the imaginary quadratic field $O\_d=\mathbb{Q}(\sqrt{-d})$ play a special role.