Random density matrices: Analytical results for mean root fidelity and mean square Bures distance

TL;DR

This paper derives exact analytical formulas for mean root fidelity and mean square Bures distance between random quantum states, with applications in quantum information theory and validation through simulations.

Contribution

It provides new analytical results for mean root fidelity and Bures distance between random density matrices, including spectral density derivations and comparisons with physical models.

Findings

Analytical formulas for mean root fidelity and Bures distance derived.

Spectral density for products of density matrices obtained.

Results validated with Monte Carlo simulations and physical model comparisons.

Abstract

Bures distance holds a special place among various distance measures due to its several distinguished features and finds applications in diverse problems in quantum information theory. It is related to fidelity and, among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean square Bures distance between a fixed density matrix and a random density matrix, and also between two random density matrices. In the course of derivation, we also obtain spectral density for product of above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.