Quantum interpolating ensemble: Biorthogonal polynomials and average entropies

TL;DR

This paper introduces a new quantum ensemble that interpolates between well-known density matrix ensembles, providing explicit formulas for average quantum purity and von Neumann entropy, and explores associated bi-orthogonal polynomials.

Contribution

It presents the first explicit calculations of average quantum purity and entropy for an interpolating ensemble, extending the theory of bi-orthogonal polynomials in the $ heta$-deformed Cauchy-Laguerre model.

Findings

Explicit formulas for average quantum purity and von Neumann entropy.

Recurrence relations for bi-orthogonal polynomials at positive integer $ heta$.

General results for the $ heta$-deformed Cauchy-Laguerre two-matrix model.

Abstract

The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known measures are the Hilbert-Schmidt and Bures-Hall ensembles. In this work, the averages of quantum purity and von Neumann entropy for an ensemble that interpolates between these two major ensembles are explicitly calculated for finite-dimensional systems. The proposed interpolating ensemble is a specialization of the $\theta$-deformed Cauchy-Laguerre two-matrix model and new results for this latter ensemble are given in full generality, including the recurrence relations satisfied by their associated bi-orthogonal polynomials when $\theta$ assumes positive integer values.