Equations for the overlaps of a SIC

TL;DR

This paper introduces a holomorphic quartic polynomial equation that precisely characterizes Weyl-Heisenberg SICs on the torus, offering a novel algebraic approach compared to existing methods involving conjugate variables.

Contribution

It presents a new holomorphic quartic polynomial equation for Weyl-Heisenberg SICs, contrasting with previous equations that involve complex conjugates, and explores properties of the projective Fourier transform.

Findings

The polynomial's zeros correspond exactly to Weyl-Heisenberg SICs.

All known equations for these SICs involve conjugate variables, unlike the new polynomial.

A related result on the powers of the Fourier transform of the group G is provided.

Abstract

We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other known systems of equations that determine a Weyl-Heisenberg SIC involve variables and their complex conjugates. We also give a related interesting result about the powers of the projective Fourier transform of the group G = Z d x Z d .