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

Youyi Huang, Lu Wei

TL;DR

The paper develops a time-dependent random-matrix-inspired model for entanglement using non-intersecting squared Bessel processes to extend the Hilbert-Schmidt ensemble. It derives recurrence relations for real-order spectral moments $m_k$, enabling systematic computation of entanglement cumulants and linear statistics, including $T_k=\sum_i x_i^k\ln x_i$. Exact finite-$n$ formulae are obtained for the average quantum purity $\mathbb{E}[P]$ and the von Neumann entropy $\mathbb{E}[S]$, with $\mathbb{E}[P]$ given by a closed rational form in $\alpha$, $c$, and $n$, and $\mathbb{E}[S]$ expressed via $\mathbb{E}[T]$ and $m_1$; in the limit $a\to0$ the results recover the Hilbert-Schmidt/Page statistics. The work also develops new biorthogonal polynomial structures, including four-point and structure relations for type I and II multiple orthogonal polynomials, and a Christoffel-Darboux formula for the correlation kernel, enriching the mathematical toolbox for non-intersecting Bessel processes and their spectral properties.

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.

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

TL;DR

The paper develops a time-dependent random-matrix-inspired model for entanglement using non-intersecting squared Bessel processes to extend the Hilbert-Schmidt ensemble. It derives recurrence relations for real-order spectral moments , enabling systematic computation of entanglement cumulants and linear statistics, including . Exact finite- formulae are obtained for the average quantum purity and the von Neumann entropy , with given by a closed rational form in , , and , and expressed via and ; in the limit the results recover the Hilbert-Schmidt/Page statistics. The work also develops new biorthogonal polynomial structures, including four-point and structure relations for type I and II multiple orthogonal polynomials, and a Christoffel-Darboux formula for the correlation kernel, enriching the mathematical toolbox for non-intersecting Bessel processes and their spectral properties.

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.
Paper Structure (14 sections, 13 theorems, 140 equations, 1 figure)

This paper contains 14 sections, 13 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-intersecting squared Bessel process
  3. Spectral Moments of Non-intersecting Squared Bessel Process
  4. Determinantal process of non-intersecting squared Bessel paths: Existing and new results
  5. Existing results on biorthogonal polynomials
  6. New results on biorthogonal polynomials of non-intersecting squared Bessel process
  7. Recurrence relations of spectral moments
  8. Dynamical Entanglement Estimation
  9. Computation of average quantum purity
  10. Computation of average entanglement entropy
  11. Coefficients and identities
  12. Coefficients in Proposition \ref{['prop:r1']}
  13. Relations of integrals $I_{s,t}$
  14. Coefficients in Proposition \ref{['prop:r2']}

Key Result

Lemma 2.1

The type I function $\widetilde{Q}_n^{(\alpha,c)}$ admits the four-point rule

Figures (1)

  • Figure 1: Numerical simulation of 20 rescaled non-intersecting squared Bessel paths with $a = 5$, $T=1$.

Theorems & Definitions (13)

  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Corollary 3.1
  • ...and 3 more