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

TL;DR

This work analyzes the spectral moments of the Bures-Hall ensemble by deriving a three-term recurrence for the real-order moments $\kappa(R_k)$, $R_k=\sum_{i=1}^m x_i^k$, using Christoffel-Darboux formulas for the associated Cauchy-Laguerre biorthogonal kernels. It builds the necessary kernel machinery by establishing CD formulas and derivative relations that yield a summation-free representation of the kernels, enabling a clean derivation of the moment recurrence with explicit coefficient functions $g_1(k)$, $g_2(k)$ and $g_3(k)$. As an application, the recurrence is employed to compute higher-order cumulants of entanglement statistics, in particular reproducing the mean von Neumann entropy and quantum purity for the Bures-Hall ensemble, matching conjectured formulas. The approach provides a systematic framework for higher-order entanglement cumulants via derivatives of spectral moments and suggests avenues for extending these results to other ensembles and cumulants with potential impact on quantum information measures. $\kappa(R_k)$, $T_k=\sum x_i^k\ln x_i$, and $S=-\sum_i \lambda_i\ln\lambda_i$ are interconnected through the derived recurrences, enabling closed-form expressions for key entanglement descriptors.

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 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.

Theorems & Definitions (9)

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• Theorem 1
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• Proposition 4