A Conjecture on Almost Flat SIC-POVMs

TL;DR

This paper explores a conjecture relating SIC-POVMs with anti-unitary symmetry to number theory, specifically Stark units, and investigates whether certain identities can determine these units, finding the answer is generally negative but with potentially mild failure.

Contribution

It analyzes the conjecture connecting SIC-POVMs and Stark units, examining if the identities can uniquely determine Stark units, revealing limitations of this approach.

Findings

The identity does not fully determine Stark units.

The failure to determine Stark units might be mild.

The conjecture relates SIC-POVMs to number theory through Stark units.

Abstract

A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.