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arXiv:2006.13746 · Physical review. E

Exact variance of von Neumann entanglement entropy over the Bures-Hall measure

Lu Wei

Published Dec 14, 2020• 20 citations
arXivHTMLPDF

Abstract

The Bures-Hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing. In this work, we study the statistical behavior of quantum entanglement over the Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann entropy over such an ensemble has been recently obtained, whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average. The starting point of the calculations is the connection between correlation functions of the Bures-Hall ensemble and those of the Cauchy-Laguerre ensemble. The derived variance formula, together with the known mean formula, leads to a simple but accurate Gaussian approximation of the distribution of von Neumann entropy of finite-size systems. This Gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems.

DOI
10.1103/physreve.102.062128

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The Bures-Hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing. In this work, we study the statistical behavior of quantum entanglement over the Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann entropy over such an ensemble has been recently obtained, whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average. The starting point of the calculations is the connection between correlation functions of the Bures-Hall ensemble and those of the Cauchy-Laguerre ensemble. The derived variance formula, together with the known mean formula, leads to a simple but accurate Gaussian approximation of the distribution of von Neumann entropy of finite-size systems. This Gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems.
We study the degree of entanglement, as measured by von Neumann entropy, of bipartite systems over the Bures-Hall ensemble. Closed-form expressions of the first two cumulants of von Neumann entropy over the ensemble have been recently derived in the literature. In this paper, we focus on its skewness by calculating the third cumulant that describes the degree of asymmetry of the distribution. The main result is an exact closed-form formula of the third cumulant, which leads to a more accurate approximation to the distribution of von Neumann entropy. The key to obtaining the result lies on finding a dozen of new summation identities in simplifying a large number of finite summations involving polygamma functions.
We study the statistical behavior of quantum entanglement in bipartite systems over fermionic Gaussian states as measured by von Neumann entropy. The formulas of average von Neumann entropy with and without particle number constraints have been recently obtained, whereas the main results of this work are the exact yet explicit formulas of variances for both cases. For the latter case of no particle number constraint, the results resolve a recent conjecture on the corresponding variance. Different than the existing methods in computing variances over other generic state models, proving the results of this work relies on a new simplification framework. The framework consists of a set of new tools in simplifying finite summations of what we refer to as dummy summation and re-summation techniques.
Abstract In this work, we study the statistical behavior of entanglement in quantum bipartite systems under the Hilbert–Schmidt ensemble as assessed by the standard measure—the von Neumann entropy. Expressions of the first three exact cumulants of von Neumann entropy are known in the literature. The main contribution of the present work is the exact formula of the corresponding fourth cumulant that controls the tail behavior of the distribution. As a key ingredient in deriving the result, we make use of what we refer to as unsimplifiable summation bases leading to a complete cancellation. In addition to providing further evidence of the conjectured Gaussian limit of the von Neumann entropy, the obtained formula also provides an improved finite-size approximation to the distribution.
We study spectral moments of the Bures-Hall random matrices ensemble. The main result establishes a recurrence relation for the kkk-th spectral moment valid for a real-valued kkk, in contrast to prevailing results in the literature of different ensembles of assuming an integer kkk. 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.
To estimate the degree of quantum entanglement of random pure states, it is crucial to understand the statistical behavior of entanglement indicators such as the von Neumann entropy, quantum purity, and entanglement capacity. These entanglement metrics are functions of the spectrum of density matrices, and their statistical behavior over different generic state ensembles have been intensively studied in the literature. As an alternative metric, in this work, we study the sum of the square root spectrum of density matrices, which is relevant to negativity and fidelity in quantum information processing. In particular, we derive the finite-size mean and variance formulas of the sum of the square root spectrum over the Bures-Hall ensemble, extending known results obtained recently over the Hilbert-Schmidt ensemble.
Random states play a foundational role in different branches of modern quantum science. In this work, we study a recently proposed random state ensemble induced from von Neumann entropy through the Bogoliubov-Kubo-Mori (BKM) metric. In particular, we derive an exact yet explicit formula of average entanglement entropy over BKM ensemble. In obtaining the formula, we only make use of properties of normalization constant of the ensemble in the absence of its correlation kernel, contrary to average entropy computation of other ensembles. This new framework paves the way for calculating higher-order cumulants of BKM ensemble beyond the average.