# Equations for the overlaps of a SIC

## Len Bos and Shayne Waldron

## 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 $G=\Zd\times\Zd$.

**Keywords:**
finite tight frames,
SIC (symmetric informationally complete positive operator valued measure),
Heisenberg group,
Clifford group,
complex equiangular lines,

**Math Review Classification:**
Primary 20F70, 81P15, 81Q10, 81R05;
Secondary 05B30, 42C15, 51F25.

**Length:** 19 Pages

**Last Updated:** 14 February 2023

## Availability: