Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2205.06343 · Journal of Physics A Mathematical and Theoretical

Average capacity of quantum entanglement

Lu Wei

Published Jan 6, 2023• 11 citations
arXivHTMLPDF

Abstract

Abstract As an alternative to entanglement entropies, the capacity of entanglement becomes a promising candidate to probe and estimate the degree of entanglement of quantum bipartite systems. In this work, we study the statistical behavior of entanglement capacity over major models of random states. In particular, the exact and asymptotic formulas of average capacity have been derived under the Hilbert–Schmidt and Bures-Hall ensembles. The obtained formulas generalize some partial results of average capacity computed recently in the literature. As a key ingredient in deriving the results, we make use of techniques in random matrix theory and our previous results pertaining to the underlying orthogonal polynomials and special functions. Simulations have been performed to numerically verify the derived formulas.

DOI
10.1088/1751-8121/acb114

Related papers

Entanglement capacity of fermionic Gaussian states

Youyi Huang, Lu Wei

Abstract We study the capacity of entanglement as an alternative to entanglement entropies in estimating the degree of entanglement of quantum bipartite systems over fermionic Gaussian states. In particular, we derive the exact and asymptotic formulas of average capacity of two different cases—with and without particle number constraints. For the later case, the obtained formulas generalize some partial results of average capacity in the literature. The key ingredient in deriving the results is a set of new tools for simplifying finite summations developed very recently in the study of entanglement entropy of fermionic Gaussian states.

arXiv:2302.02229•Journal of Physics A Mathematical and Theoretical•Sep 21, 2023•1 citation

Quantum interpolating ensemble: Average entropies and orthogonal polynomials.

Lu Wei, Nicholas S. Witte

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 and physically relevant measures are the Hilbert-Schmidt ensemble and the Bures-Hall ensemble. In this work, we propose a generalized ensemble of density matrices, termed quantum interpolating ensemble, which is able to interpolate between these two seemingly unrelated ensembles. As a first step to understand the proposed ensemble, we derive the exact mean formulas of entanglement entropies over such an ensemble generalizing several recent results in the literature. We also derive some key properties of the corresponding orthogonal polynomials relevant to obtaining other statistical information of the entropies. Numerical results demonstrate the usefulness of the proposed ensemble in estimating the degree of entanglement of quantum states.

•Mar 7, 2021•2 citations

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

Lu Wei

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.

arXiv:2006.13746•Physical review. E•Dec 14, 2020•20 citations

Square Root Statistics of Density Matrices and Their Applications

Lyuzhou Ye, Youyi Huang, James C. Osborn +1 more

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.

Entropy•Jan 12, 2024•3 citations

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

Pierpaolo Vivo, Mauricio Porto Pato, Gleb Oshanin

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

arXiv:1602.01230•Physical review. E•May 2, 2016•63 citations

Kurtosis of von Neumann entanglement entropy

Youyi Huang, Lu Wei, Bjordis Collaku

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.

arXiv:2107.10978•Journal of Physics A Mathematical and Theoretical•Nov 4, 2021•12 citations
Abstract As an alternative to entanglement entropies, the capacity of entanglement becomes a promising candidate to probe and estimate the degree of entanglement of quantum bipartite systems. In this work, we study the statistical behavior of entanglement capacity over major models of random states. In particular, the exact and asymptotic formulas of average capacity have been derived under the Hilbert–Schmidt and Bures-Hall ensembles. The obtained formulas generalize some partial results of average capacity computed recently in the literature. As a key ingredient in deriving the results, we make use of techniques in random matrix theory and our previous results pertaining to the underlying orthogonal polynomials and special functions. Simulations have been performed to numerically verify the derived formulas.
Abstract We study the capacity of entanglement as an alternative to entanglement entropies in estimating the degree of entanglement of quantum bipartite systems over fermionic Gaussian states. In particular, we derive the exact and asymptotic formulas of average capacity of two different cases—with and without particle number constraints. For the later case, the obtained formulas generalize some partial results of average capacity in the literature. The key ingredient in deriving the results is a set of new tools for simplifying finite summations developed very recently in the study of entanglement entropy of fermionic Gaussian states.
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 and physically relevant measures are the Hilbert-Schmidt ensemble and the Bures-Hall ensemble. In this work, we propose a generalized ensemble of density matrices, termed quantum interpolating ensemble, which is able to interpolate between these two seemingly unrelated ensembles. As a first step to understand the proposed ensemble, we derive the exact mean formulas of entanglement entropies over such an ensemble generalizing several recent results in the literature. We also derive some key properties of the corresponding orthogonal polynomials relevant to obtaining other statistical information of the entropies. Numerical results demonstrate the usefulness of the proposed ensemble in estimating the degree of entanglement of quantum states.
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.
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.
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
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.