Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance

TL;DR

This paper derives exact analytical results for the average and variance of the squared Bures distance between quantum states, enhancing understanding of quantum state distinguishability and providing practical approximation methods.

Contribution

It provides new exact formulas for the mean and variance of the squared Bures distance between random density matrices, extending previous results and enabling probability density approximations.

Findings

Analytical formulas for mean fidelity and variance of squared Bures distance.

Good agreement between analytical results and Monte Carlo simulations.

Validation with reduced density matrices from physical quantum systems.

Abstract

One of the key issues in quantum information theory related problems concerns with that of distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work, we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix, and also between two independent random density matrices. These results supplement the recently obtained results for the mean root fidelity and mean of squared Bures distance [Phys. Rev. A 104, 022438 (2021)]. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops, and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.