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arXiv:2008.05153 · Physical Review A

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

Santosh Kumar

Published Jul 6, 2020• 18 citations
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Abstract

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.

DOI
10.1103/physreva.102.012405

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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.
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.
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 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.
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.
We discuss an application of the random matrix theory in the context of estimating the bipartite entanglement of a quantum system. We discuss how the Wishart ensemble (the earliest studied random matrix ensemble) appears in this quantum problem. The eigenvalues of the reduced density matrix of one of the subsystems have similar statistical properties as those of the Wishart matrices, except that their {\em trace is constrained to be unity}. We focus here on the smallest eigenvalue which serves as an important measure of entanglement between the two subsystems. In the hard edge case (when the two subsystems have equal sizes) one can fully characterize the probability distribution of the minimum eigenvalue for real, complex and quaternion matrices of all sizes. In particular, we discuss the important finite size effect due to the {\em fixed trace constraint}.