Non-intersecting Squared Bessel Process: Spectral Moments and Dynamical Entanglement Entropy

TL;DR

This paper introduces a dynamical extension of the Hilbert-Schmidt ensemble based on non-intersecting squared Bessel processes, enabling systematic entanglement estimation through spectral moments and revealing new mathematical properties.

Contribution

It develops a novel dynamical ensemble model for entanglement estimation and derives spectral moments, orthogonal polynomial relations, and correlation kernel formulas.

Findings

Spectral moments of the dynamical ensemble computed.

New structure and recurrence relations for orthogonal polynomials.

Christoffel-Darboux formula derived for correlation kernels.

Abstract

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.