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

TL;DR

This paper derives a recurrence relation for spectral moments of the Bures-Hall ensemble, enabling new calculations of quantum entanglement measures like von Neumann entropy and purity.

Contribution

It introduces a novel recurrence relation for spectral moments valid for real-valued k, expanding beyond previous integer-only results, and applies it to quantum entanglement metrics.

Findings

Established a recurrence relation for spectral moments of the Bures-Hall ensemble.

Derived formulas for average von Neumann entropy and quantum purity.

Utilized Christoffel-Darboux formulas to simplify correlation kernel calculations.

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.