Sporadic SICs and the Normed Division Algebras

TL;DR

This paper explores the connection between sporadic symmetric informationally complete measurements (SICs) and normed division algebras, highlighting how integer lattices in complex numbers, quaternions, and octonions underpin their symmetry groups.

Contribution

It reveals the role of normed division algebras in classifying certain SIC-POVMs with doubly transitive symmetry groups.

Findings

Normed division algebras relate to SIC symmetry groups

Classification of SIC-POVMs with specific symmetry properties

Identification of lattice structures in complex, quaternion, and octonion numbers

Abstract

Recently, Zhu classified all the SIC-POVMs whose symmetry groups act doubly transitively. Lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups.