Moment maps and Galois orbits in quantum information theory

TL;DR

This paper explores the geometric and algebraic structures underlying SIC-POVMs in quantum information, using moment maps and Galois theory to analyze their configurations, equations, and associated number fields.

Contribution

It introduces a new geometric framework for SIC-POVMs via moment maps and provides explicit descriptions of their equations and Galois orbit structures.

Findings

The image of SIC-POVMs under moment maps lies in an intersection of real quadrics.

Explicit descriptions of the equations defining SIC-POVMs are provided.

Conjectural insights into the number fields related to SIC-POVMs and their Galois orbits.

Abstract

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb C^d$. The resulting equations are interpreted in terms of moment maps by focussing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm U(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.