Exploring the geometry of qutrit state space using symmetric informationally complete probabilities

TL;DR

This paper investigates the geometric structure of qutrit state space by analyzing SIC measurement probabilities, categorizing SICs into families, and deriving geometric properties and invariants that describe the convex body of qutrits.

Contribution

It introduces a classification of qutrit SICs into eight families and derives geometric invariants and formulas characterizing the state space's structure.

Findings

Identified eight SIC families related to the extended Clifford group

Derived a formula for the extreme points of the qutrit state space

Established a polar equation for boundary states

Abstract

We examine the geometric structure of qutrit state space by identifying the outcome probabilities of symmetric informationally complete (SIC) measurements with quantum states. We categorize the infinitely many qutrit SICs into eight SIC families corresponding to independent orbits of the extended Clifford group. Every SIC can be uniquely identified from a set of geometric invariants that we use to establish several properties of the convex body of qutrits, which include a simple formula describing its extreme points, an expression for the rotation between the probability vectors for distinct qutrit SICs, and a polar equation for its boundary states.