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. © 2020 American Physical Society.

Santosh Kumar

Physics

(SoNS) School of Natural Sciences

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Physical Review A

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean. © 2020 IOP Publishing Ltd.

Journal of Physics A: Mathematical and Theoretical

Spectral statistics for the difference of two Wishart matrices

The largest eigenvalue distribution of the Wishart?Laguerre ensemble, indexed by Dyson parameter and Laguerre parameter a, is fundamental in multivariate statistics and finds applications in diverse areas. Based on a generalisation of the Selberg integral, we provide an effective recursion scheme to compute this distribution explicitly in both the original model, and a fixed-trace variant, for a, non-negative integers and finite matrix size. For = 2 this circumvents known symbolic evaluation based on determinants which become impractical for large dimensions. Our exact results have immediate applications in the areas of multiple channel communication and bipartite entanglement. Moreover, we are also led to the exact solution of a long standing problem of finding a general result for Landauer conductance distribution in a chaotic mesoscopic cavity with two ideal leads. Thus far, exact closed-form results for this were available only in the Fourier?Laplace space or could be obtained on a case-by-case basis. © 2019 IOP Publishing Ltd.

Recursion scheme for the largest-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport

We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures-Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Havrda-Charvát-Tsallis (HCT) entropy as a finite sum. Averages of von Neumann entropy, linear entropy and purity follow by considering appropriate limits in the average HCT expression. Based on exact evaluations of the average von Neumann entropy and the average purity, we also conjecture very simple formulae for these, which are similar to those in the Hilbert-Schmidt ensemble. © 2019 IOP Publishing Ltd.

Bures-Hall ensemble: Spectral densities and average entropies

The statistics of the smallest eigenvalue of Wishart–Laguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost significance in understanding the spectral behavior of Wishart–Laguerre ensembles and, among other things, unveil the underlying universality aspects in the asymptotic limits. However, obtaining exact and explicit expressions by expanding determinants or Pfaffians becomes impractical if large dimension matrices are involved. For the real matrices (β= 1) Edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices. Very recently, an analogous recurrence scheme has been obtained for the complex matrices (β= 2). In the present work we extend this to β-Wishart–Laguerre ensembles for the case when exponent α in the associated Laguerre weight function, λ α e - β λ / 2 , is a non-negative integer, while β is positive real. This also gives access to the smallest eigenvalue density of fixed trace β-Wishart–Laguerre ensemble, as well as moments for both cases. Moreover, comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly. Exact evaluations for large values of n (the matrix dimension) and α also enable us to compare with Tracy–Widom density and large deviation results of Katzav and Castillo. We also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the Wigner–Smith matrix and are relevant to the problem of quantum chaotic scattering. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Journal of Statistical Physics

Recursion for the Smallest Eigenvalue Density of β -Wishart–Laguerre Ensemble

The statistical behaviour of the smallest eigenvalue has important implications for systems which can be modeled using a Wishart-Laguerre ensemble, the regular one or the fixed trace one. For example, the density of the smallest eigenvalue of the Wishart-Laguerre ensemble plays a crucial role in characterizing multiple channel telecommunication systems. Similarly, in the quantum entanglement problem, the smallest eigenvalue of the fixed trace ensemble carries information regarding the nature of entanglement. For real Wishart-Laguerre matrices, there exists an elegant recurrence scheme suggested by Edelman to directly obtain the exact expression for the smallest eigenvalue density. In the case of complex Wishart-Laguerre matrices, for finding exact and explicit expressions for the smallest eigenvalue density, existing results based on determinants become impractical when the determinants involve large-size matrices. In this work, we derive a recurrence scheme for the complex case which is analogous to that of Edelman's for the real case. This is used to obtain exact results for the smallest eigenvalue density for both the regular, and the fixed trace complex Wishart-Laguerre ensembles. We validate our analytical results using Monte Carlo simulations. We also study scaled Wishart-Laguerre ensemble and investigate its efficacy in approximating the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace ensemble to investigate the behaviour of the smallest eigenvalue in the paradigmatic system of coupled kicked tops. © 2017 IOP Publishing Ltd.