Asymmetric SICs over finite fields

TL;DR

This paper constructs infinitely many new symmetric informationally complete measurements over finite fields, including some with trivial symmetry groups, and conjectures their infinite existence in various dimensions.

Contribution

It introduces the first infinite family of SICs over finite fields, including totally asymmetric cases with trivial automorphism groups.

Findings

Existence of infinitely many new SICs over finite fields.

Some SICs have trivial automorphism groups.

Conjecture of infinite existence of totally asymmetric SICs.

Abstract

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.