Bures-Hall Ensemble: Spectral Densities and Average Entropies

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-Charvát-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.

Published version; 4 figures, 2 tables

References in corpus (8)

• Statistical distribution of quantum entanglement for a random bipartite state
• The Cauchy two-matrix model
• Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
• A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
• Distribution of G-concurrence of random pure states
• Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace
• Universal Spectral Correlations in Orthogonal-Unitary and Symplectic-Unitary Crossover Ensembles of Random Matrices
• Pfaffian Expressions for Random Matrix Correlation Functions

Cited by in corpus (3)

• Skewness of von Neumann entanglement entropy
• Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance
• Moments of quantum purity and biorthogonal polynomial recurrence