# Bures-Hall Ensemble: Spectral Densities and Average Entropies

## TL;DR

This paper analyzes the spectral properties of the Bures-Hall ensemble of random density matrices, deriving exact formulas for spectral densities and average entropies, and compares these with known ensembles.

## Contribution

It provides the first exact spectral density formulas and average entropy calculations for the Bures-Hall ensemble, linking fixed trace and unrestricted trace cases.

## Key findings

- Exact Pfaffian formulas for level density
- Finite sum expression for average Havrda-Charvát-Tsallis entropy
- Conjectured simple formulas for von Neumann entropy and purity

## Abstract

We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures-Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Havrda-Charv\'at-Tsallis (HCT) entropy as a finite sum. Averages of von Neumann entropy, linear entropy and purity follow by considering appropriate limits in the average HCT expression. Based on exact evaluations of the average von Neumann entropy and the average purity, we also conjecture very simple formulae for these, which are similar to those in the Hilbert-Schmidt ensemble.