Equations for the overlaps of a SIC

Len Bos and Shayne Waldron


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