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:2211.05587 · Physical review. E

Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance

Aritra Laha, Santosh Kumar

Published Mar 14, 2023• 5 citations
arXivHTMLPDF

Abstract

One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.

DOI
10.1103/physreve.107.034206

Related papers

Random density matrices: Analytical results for mean root fidelity and the mean-square Bures distance

Aritra Laha, Agrim Aggarwal, Santosh Kumar

The Bures distance holds a special place among various distance measures due to its several distinguishing features and has applications in diverse problems in quantum information theory. It is related to fidelity, and among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean-square Bures distance between a fixed density matrix and a random density matrix and also between two random density matrices. In the course of derivation, we also obtain the spectral density for the product of the above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean-square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.

arXiv:2105.02743•Physical Review A•Aug 31, 2021•9 citations

Exact mean and variance of the squared Hellinger distance for random density matrices

Vinay Kumar, Kaushik Vasan, Santosh Kumar

The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is easier to compute than the Bures distance, another measure that shares these properties, particularly in information-theoretic applications. Furthermore, in these applications, random quantum states are crucial for securing communication, studying entanglement, benchmarking quantum systems, aiding in quantum state tomography, and understanding how quantum systems behave in noisy environments. Recent works have computed exact results for distance measures such as Bures and Hilbert-Schmidt for random density matrices; however, no such results exist for the Hellinger distance. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Our derivation utilizes Weingarten functions to perform the necessary unitary group integrals and is supported by existing results for eigenvalue moments of random density matrices. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.

arXiv:2409.14560•Physical review. E•May 5, 2025•1 citation

Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

Santosh Kumar

Hilbert-Schmidt distance is one of the prominent distance measures in quantum information theory which finds applications in diverse problems, such as construction of entanglement witnesses, quantum algorithms in machine learning, and quantum-state tomography. In this work, we calculate exact and compact results for the mean-square Hilbert-Schmidt distance between a random density matrix and a fixed density matrix, and also between two random density matrices. In the course of derivation, we also obtain corresponding exact results for the distance between a Wishart matrix and a fixed Hermitian matrix, and two Wishart matrices. We verify all our analytical results using Monte Carlo simulations. Finally, we apply our results to investigate the Hilbert-Schmidt distance between reduced density matrices generated using coupled kicked tops.

arXiv:2008.05153•Physical Review A•Jul 6, 2020•18 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

Estimating Distances between Quantum Gaussian States

Alexander Semenovich Holevo

Abstract Upper bounds for the Bures distance as well as for the trace-norm distance between quantum Gaussian states of CCR in terms of the mean vectors and covariance matrices are obtained, basing on a fidelity-like quantity which we call the states overlap. We argue that these bounds are better adapted to the Bures distance and hence to the corresponding methods of state estimation. We also briefly recall our results concerning bounds for the states overlap for fermionic Gaussian states of CAR. The problems studied in this paper can be considered as a noncommutative analog of estimation of the Hellinger distance and the total variance distance between Gaussian probability distributions in the classical probability theory.

Lobachevskii Journal of Mathematics•Jun 1, 2025•1 citation

Generation of Bures-Hall mixed states using coupled kicked tops

Ayana Sarkar, Santosh Kumar

We simulate a system of coupled kicked tops to generate random density matrices distributed according to the Bures-Hall measure, which has an important role in quantum information theory. We study the effects of stochasticity and coupling parameters of the coupled-kicked-top system on the behavior of the associated bipartite pure state, eigenvalues and eigenvectors of the reduced state, and average entropies. For strongly chaotic phase space and adequate coupling between the constituting tops (subsystems), we find that the results of simulation agree with analytical results of random matrix theory. We also examine local fluctuation properties of the Bures-Hall eigenvalues using the distribution of nearest-neighbor spacing ratios. In the limit of high-dimensional density matrices, the empirical ratio distribution is found to approach the Wigner-surmise-like result for the Gaussian unitary ensemble. Additionally, we present some closed-form results for the Bures-Hall spectral density and corresponding moments using Meijer $G$ functions.

Physical Review A•Mar 18, 2021•9 citations
One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.
The Bures distance holds a special place among various distance measures due to its several distinguishing features and has applications in diverse problems in quantum information theory. It is related to fidelity, and among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean-square Bures distance between a fixed density matrix and a random density matrix and also between two random density matrices. In the course of derivation, we also obtain the spectral density for the product of the above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean-square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.
The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is easier to compute than the Bures distance, another measure that shares these properties, particularly in information-theoretic applications. Furthermore, in these applications, random quantum states are crucial for securing communication, studying entanglement, benchmarking quantum systems, aiding in quantum state tomography, and understanding how quantum systems behave in noisy environments. Recent works have computed exact results for distance measures such as Bures and Hilbert-Schmidt for random density matrices; however, no such results exist for the Hellinger distance. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Our derivation utilizes Weingarten functions to perform the necessary unitary group integrals and is supported by existing results for eigenvalue moments of random density matrices. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.
Hilbert-Schmidt distance is one of the prominent distance measures in quantum information theory which finds applications in diverse problems, such as construction of entanglement witnesses, quantum algorithms in machine learning, and quantum-state tomography. In this work, we calculate exact and compact results for the mean-square Hilbert-Schmidt distance between a random density matrix and a fixed density matrix, and also between two random density matrices. In the course of derivation, we also obtain corresponding exact results for the distance between a Wishart matrix and a fixed Hermitian matrix, and two Wishart matrices. We verify all our analytical results using Monte Carlo simulations. Finally, we apply our results to investigate the Hilbert-Schmidt distance between reduced density matrices generated using coupled kicked tops.
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.
Abstract Upper bounds for the Bures distance as well as for the trace-norm distance between quantum Gaussian states of CCR in terms of the mean vectors and covariance matrices are obtained, basing on a fidelity-like quantity which we call the states overlap. We argue that these bounds are better adapted to the Bures distance and hence to the corresponding methods of state estimation. We also briefly recall our results concerning bounds for the states overlap for fermionic Gaussian states of CAR. The problems studied in this paper can be considered as a noncommutative analog of estimation of the Hellinger distance and the total variance distance between Gaussian probability distributions in the classical probability theory.
We simulate a system of coupled kicked tops to generate random density matrices distributed according to the Bures-Hall measure, which has an important role in quantum information theory. We study the effects of stochasticity and coupling parameters of the coupled-kicked-top system on the behavior of the associated bipartite pure state, eigenvalues and eigenvectors of the reduced state, and average entropies. For strongly chaotic phase space and adequate coupling between the constituting tops (subsystems), we find that the results of simulation agree with analytical results of random matrix theory. We also examine local fluctuation properties of the Bures-Hall eigenvalues using the distribution of nearest-neighbor spacing ratios. In the limit of high-dimensional density matrices, the empirical ratio distribution is found to approach the Wigner-surmise-like result for the Gaussian unitary ensemble. Additionally, we present some closed-form results for the Bures-Hall spectral density and corresponding moments using Meijer GGG functions.