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

2026-02-01

AI Summary

This study addresses the lack of efficient computational methods for non-integer spectral moments in Bures–Hall random matrix ensembles by establishing, for the first time, a recurrence relation valid for arbitrary real-order moments \(k\), thereby overcoming the traditional restriction to integer orders. By integrating orthogonal polynomial theory with Christoffel–Darboux kernel techniques, the approach circumvents cumbersome summations and enables efficient analytical computation of key quantum information quantities, such as the average von Neumann entropy and quantum purity. The work not only rigorously confirms several conjectured formulas but also provides a robust mathematical foundation for entanglement entropy research, significantly broadening the applicability of random matrix theory in quantum information science.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• spectral moments
• Bures-Hall ensemble
• entanglement entropy
• von Neumann entropy
• quantum purity

Innovation

• spectral moments
• Bures-Hall ensemble
• Christoffel-Darboux formula
• von Neumann entropy
• quantum purity