Some geometric interpretations of quantum fidelity

TL;DR

This paper explores the geometric interpretation of quantum fidelity, analyzing its minimal values under various quantum channels and relating it to subspace angles, providing new insights into quantum state similarity.

Contribution

It introduces a geometric perspective on quantum fidelity, deriving minimal fidelity results under different channels and connecting fidelity to subspace angles.

Findings

Minimal fidelity characterized for unital, mixed unitary, and arbitrary channels.

Fidelity interpreted as a geometric measure between subspaces.

Connection established between fidelity and principal angles.

Abstract

We consider quantum fidelity between two states $\rho$ and $\sigma$, where we fix $\rho$ and allow $\sigma$ to be sent through a quantum channel. We determine the minimal fidelity where one minimizes over (a) all unital channels, (b) all mixed unitary channels, and (c) arbitrary channels. We derive results involving the minimal eigenvalue of $\rho$, which we can interpret as a convex combination coefficient. As a consequence, we give a new geometric interpretation of the minimal fidelity with respect to the closed, convex set of density matrices and with respect to the closed, convex set of quantum channels. We further investigate the geometric nature of fidelity by considering density matrices arising as normalized projections onto subspaces; in this way, fidelity can be viewed as a geometric measure of distance between two spaces. We give a connection between fidelity and the canonical (principal) angles between the subspaces.