Distances of qubit density matrices on Bloch sphere

TL;DR

This paper explores the mathematical relationship between qubit density matrices and the Einstein gyrogroup, establishing an isomorphism and linking their respective metrics to deepen understanding of quantum state geometry.

Contribution

It introduces an isomorphism between the Einstein gyrogroup and qubit density matrices, connecting their geometric structures and metrics.

Findings

Established an isomorphism between Einstein gyrogroup and qubit density matrices.

Linked the trace metric on density matrices to the rapidity metric on the gyrogroup.

Provided a geometric interpretation of qubit states in relation to relativistic velocity addition.

Abstract

We recall the Einstein velocity addition on the open unit ball $\B$ of $\R^{3}$ and its algebraic structure, called the Einstein gyrogroup. We establish an isomorphism between the Einstein gyrogroup on $\B$ and the set of all qubit density matrices representing mixed states endowed with an appropriate addition. Our main result establishes a relation between the trace metric for the qubit density matrices and the rapidity metric on the Einstein gyrogroup $\B$.