The Poincar\'e half-plane for informationally complete POVMs

TL;DR

This paper explores the geometric and group-theoretic foundations of informationally complete POVMs using the Poincaré half-plane model, linking them to modular groups, permutation gates, and the Kochen-Specker theorem.

Contribution

It introduces a novel geometric approach to constructing IC-POVMs via the Poincaré upper half-plane and connects their structure to modular groups and quantum contextuality.

Findings

IC-POVMs relate to modular group subgroups

Fiducial states derived from Poincaré model

Connections to Kochen-Specker theorem

Abstract

It has been shown that classes of (minimal asymmetric) informationally complete POVMs in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states [M. Planat and Z. Gedik, R. Soc. open sci. 4, 170387 (2017)]. The latter states may also be derived starting from the Poincar\'e upper half-plane model H. For doing this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates whose some of the eigenstates are the seeked fiducials. The structure of some IC-POVMs is found to be intimately related to the Kochen-Specker theorem.