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:1807.02212 · Entropy

On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State

Lu Wei

Published May 27, 2019• 19 citations
arXivHTMLPDF

Abstract

The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.

DOI
10.3390/e21050539

Related papers

Exact variances of von Neumann entropy over fermionic Gaussian states

Youyi Huang, Lu Wei

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.

IEEE International Symposium on Information Theory (ISIT)•Jun 25, 2023•0 citations

Skewness of von Neumann entanglement entropy

Lu Wei

Abstract We study quantum bipartite systems in a random pure state, where von Neumann entropy is considered as a measure of the entanglement. Expressions of the first and second exact cumulants of von Neumann entropy, relevant respectively to the average and fluctuation behavior, are known in the literature. The focus of this paper is on its skewness that specifies the degree of asymmetry of the distribution. Computing the skewness requires additionally the third cumulant, an exact formula of which is the main result of this work. In proving the main result, we obtain as a byproduct various summation identities involving polygamma and related functions. The derived third cumulant also leads to an improved approximation to the distribution of von Neumann entropy.

arXiv:1910.01199•Journal of Physics A Mathematical and Theoretical•Dec 18, 2019•23 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

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

Second-order statistics of fermionic Gaussian states

Youyi Huang, Lu Wei

Abstract We study the statistical behavior of entanglement in quantum bipartite systems over fermionic Gaussian states as measured by von Neumann entropy and entanglement capacity. The focus is on the variance of von Neumann entropy and the mean entanglement capacity that belong to the so-defined second-order statistics. The main results are the exact yet explicit formulas of the two considered second-order statistics for fixed subsystem dimension differences. We also conjecture the exact variance of von Neumann entropy valid for arbitrary subsystem dimensions. Based on the obtained results, we analytically study the numerically observed phenomena of Gaussianity of von Neumann entropy and linear growth of average capacity.

arXiv:2111.08216•Journal of Physics A Mathematical and Theoretical•Jan 24, 2022•12 citations

A bound on expectation values and variances of quantum observables via Rényi entropy and Tsallis entropy

Sujuan Zhang, Jing Li

Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for [Formula: see text] observables respectively.

International Journal of Quantum Information•Apr 1, 2021•2 citations
The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.
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 We study quantum bipartite systems in a random pure state, where von Neumann entropy is considered as a measure of the entanglement. Expressions of the first and second exact cumulants of von Neumann entropy, relevant respectively to the average and fluctuation behavior, are known in the literature. The focus of this paper is on its skewness that specifies the degree of asymmetry of the distribution. Computing the skewness requires additionally the third cumulant, an exact formula of which is the main result of this work. In proving the main result, we obtain as a byproduct various summation identities involving polygamma and related functions. The derived third cumulant also leads to an improved approximation to the distribution of von Neumann entropy.
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 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 We study the statistical behavior of entanglement in quantum bipartite systems over fermionic Gaussian states as measured by von Neumann entropy and entanglement capacity. The focus is on the variance of von Neumann entropy and the mean entanglement capacity that belong to the so-defined second-order statistics. The main results are the exact yet explicit formulas of the two considered second-order statistics for fixed subsystem dimension differences. We also conjecture the exact variance of von Neumann entropy valid for arbitrary subsystem dimensions. Based on the obtained results, we analytically study the numerically observed phenomena of Gaussianity of von Neumann entropy and linear growth of average capacity.
Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for [Formula: see text] observables respectively.