SICs: Extending the list of solutions

TL;DR

This paper extends the known list of symmetric informationally complete positive operator-valued measures (SIC-POVMs) in various dimensions, providing new solutions with specific symmetries up to dimension 323, advancing the understanding of Zauner's conjecture.

Contribution

It offers a comprehensive list of Weyl-Heisenberg covariant SICs with known symmetries in dimensions up to 323, expanding the numerical evidence for Zauner's conjecture.

Findings

Complete list of SICs in dimensions ≤90

Single solution with Zauner's symmetry for all dimensions ≤121

Solutions with higher symmetry in select dimensions

Abstract

Zauner's conjecture asserts that $d^2$ equiangular lines exist in all $d$ complex dimensions. In quantum theory, the $d^2$ lines are dubbed a SIC, as they define a favoured standard informationally complete quantum measurement called a SIC-POVM. This note supplements A. J. Scott and M. Grassl [J. Math. Phys. 51 (2010), 042203] by extending the list of published numerical solutions. We provide a putative complete list of Weyl-Heisenberg covariant SICs with the known symmetries in dimensions $d\leq 90$, a single solution with Zauner's symmetry for every $d\leq 121$ and solutions with higher symmetry for $d=124,143,147,168,172,195,199,228,259$ and $323$.