The Number Behind the Simplest SIC-POVM

TL;DR

This paper explores the algebraic number theory challenges of symmetric informationally complete positive operator-valued measures (SIC-POVMs) in higher dimensions, providing a detailed example in the simplest case.

Contribution

It offers a detailed algebraic description of the simplest SIC-POVM, highlighting the deep number theory problems involved in higher dimensions.

Findings

Identifies key algebraic structures in the simplest SIC-POVM

Highlights complexity of extending to higher dimensions

Provides a concrete example for further theoretical exploration

Abstract

The simple concept of a SIC poses a very deep problem in algebraic number theory, as soon as the dimension of Hilbert space exceeds three. A detailed description of the simplest possible example is given.