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Journal of Physics A Mathematical and Theoretical

Spectral moments of Bures–Hall ensemble and applications to entanglement entropy

Linfeng Wei, Youyi Huang, Lu Wei

Published Jun 5, 2026• 0 citations
Publisher

Abstract

Abstract We study spectral moments of the Bures–Hall random matrices ensemble. The main result establishes a recurrence relation for the k th spectral moment valid for a real-valued k , in contrast to prevailing results in the literature of different ensembles of assuming an integer k . The key to establish the recurrence relation is the obtained Christoffel–Darboux formulae of correlation kernels of the ensemble that avoid tedious summations. As an application of our spectral moment results, we re-derive the formulae of average von Neumann entropy and quantum purity of Bures–Hall ensemble conjectured by Ayana Sarkar and Santosh Kumar.

DOI
10.1088/1751-8121/ae791e

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Abstract We study spectral moments of the Bures–Hall random matrices ensemble. The main result establishes a recurrence relation for the k th spectral moment valid for a real-valued k , in contrast to prevailing results in the literature of different ensembles of assuming an integer k . The key to establish the recurrence relation is the obtained Christoffel–Darboux formulae of correlation kernels of the ensemble that avoid tedious summations. As an application of our spectral moment results, we re-derive the formulae of average von Neumann entropy and quantum purity of Bures–Hall ensemble conjectured by Ayana Sarkar and Santosh Kumar.
We study spectral moments of the Bures-Hall random matrices ensemble. The main result establishes a recurrence relation for the kkk-th spectral moment valid for a real-valued kkk, in contrast to prevailing results in the literature of different ensembles of assuming an integer kkk. The key to establish the recurrence relation is the obtained Christoffel-Darboux formulas of correlation kernels of the ensemble that avoid tedious summations. As an application of our spectral moment results, we re-derive the formulas of average von Neumann entropy and quantum purity of Bures-Hall ensemble conjectured by Ayana Sarkar and Santosh Kumar. This work is dedicated to the memory of Santosh Kumar.
Abstract The Bures–Hall ensemble is a unique measure of density matrices that satisfies various distinguished properties in quantum information processing. In this work, we study the statistical behavior of entanglement over the Bures–Hall ensemble as measured by the simplest form of an entanglement measure—the quantum purity. The main results of this work are the exact second and third moment expressions of quantum purity valid for any subsystem dimensions, where the corresponding results in the literature are limited to the scenario of equal subsystem dimensions. In obtaining the results, we have derived recurrence relations of the underlying integrals over the Cauchy–Laguerre biorthogonal polynomials that may be of independent interest.
Statistical ensembles of reduced density matrices of bipartite quantum systems play a central role in entanglement estimation, but do not capture the non-stationary nature of entanglement relevant to realistic quantum information processing. To address this limitation, we propose a dynamical extension of the Hilbert-Schmidt ensemble, a baseline statistical model for entanglement estimation, arising from non-intersecting squared Bessel processes and perform entanglement estimation via average entanglement entropy and quantum purity. The investigation is enabled by finding spectral moments of the proposed dynamical ensemble, which serves as a new approach for systematic computation of entanglement metrics. Along the way, we also obtain new results for the underlying multiple orthogonal polynomials of modified Bessel weights, including structure and recurrence relations, and a Christoffel-Darboux formula for the correlation kernels.
The Bures-Hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing. In this work, we study the statistical behavior of quantum entanglement over the Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann entropy over such an ensemble has been recently obtained, whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average. The starting point of the calculations is the connection between correlation functions of the Bures-Hall ensemble and those of the Cauchy-Laguerre ensemble. The derived variance formula, together with the known mean formula, leads to a simple but accurate Gaussian approximation of the distribution of von Neumann entropy of finite-size systems. This Gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems.
We study the degree of entanglement, as measured by von Neumann entropy, of bipartite systems over the Bures-Hall ensemble. Closed-form expressions of the first two cumulants of von Neumann entropy over the ensemble have been recently derived in the literature. In this paper, we focus on its skewness by calculating the third cumulant that describes the degree of asymmetry of the distribution. The main result is an exact closed-form formula of the third cumulant, which leads to a more accurate approximation to the distribution of von Neumann entropy. The key to obtaining the result lies on finding a dozen of new summation identities in simplifying a large number of finite summations involving polygamma functions.
Random states play a foundational role in different branches of modern quantum science. In this work, we study a recently proposed random state ensemble induced from von Neumann entropy through the Bogoliubov-Kubo-Mori (BKM) metric. In particular, we derive an exact yet explicit formula of average entanglement entropy over BKM ensemble. In obtaining the formula, we only make use of properties of normalization constant of the ensemble in the absence of its correlation kernel, contrary to average entropy computation of other ensembles. This new framework paves the way for calculating higher-order cumulants of BKM ensemble beyond the average.