Moments of quantum purity and biorthogonal polynomial recurrence

TL;DR

This paper derives exact second and third moments of quantum purity for the Bures-Hall ensemble, revealing new insights into entanglement statistics in quantum information, especially for unequal subsystem dimensions.

Contribution

It provides the first exact formulas for moments of quantum purity for arbitrary subsystem dimensions and introduces recurrence relations for biorthogonal polynomial integrals.

Findings

Exact second and third moments of quantum purity derived.

Recurrence relations for Cauchy-Laguerre biorthogonal polynomials established.

Results applicable to unequal subsystem dimensions in quantum systems.

Abstract

The Bures-Hall ensemble is a unique measure of density matrices that satisfies various distinguished properties in quantum information processing. In this work, we study the statistical behavior of entanglement over the Bures-Hall ensemble as measured by the simplest form of an entanglement entropy - the quantum purity. The main results of this work are the exact second and third moment expressions of quantum purity valid for any subsystem dimensions, where the corresponding results in the literature are limited to the scenario of equal subsystem dimensions. In obtaining the results, we have derived recurrence relations of the underlying integrals over the Cauchy-Laguerre biorthogonal polynomials that may be of independent interest.