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

Kumar, Santosh

doi:10.1007/s10955-019-02245-z

Cited by 16 publications

(4 citation statements)

References 85 publications

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“…The above described three level nested loop algorithm may be contrasted with that for the smallest eigenvalue [44], which involves a two level nested loop only due to a less involved mathematical structure there.…”

Section: Appendices Appendix I Refinement Of Upper Limit Of Inner Sum...mentioning

confidence: 99%

“…This has turned out to be very effective considering the availability of modern software packages which can handle recursive symbolic computation very efficiently. For the smallest eigenvalue of real Wishart-Laguerre matrices, in [38,39] Edelman provided a recursion scheme which has subsequently been generalised to other ensembles and symmetry classes [1,10,19,[40][41][42][43][44]. However, extending this to the largest eigenvalue distribution has remained elusive due to its comparatively more convoluted mathematical structure [1].…”

Section: Introductionmentioning

confidence: 99%

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Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

Forrester¹,

Kumar²

2019

J. Phys. A: Math. Theor.

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The largest eigenvalue distribution of the Wishart-Laguerre ensemble, indexed by Dyson parameter β and Laguerre parameter a, is fundamental in multivariate statistics and finds applications in diverse areas. Based on a generalization 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, β non-negative integers and finite matrix size. For β = 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.

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Section: Appendices Appendix I Refinement Of Upper Limit Of Inner Sum...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

Forrester¹,

Kumar²

2019

J. Phys. A: Math. Theor.

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“…However, as far as random mixed states are concerned there is no such unique measure to describe their statistics [26][27][28][29]. A widely used probability measure is the Hilbert-Schmidt measure on the set of the finite-dimensional mixed-sates [27][28][29][30][31][34][35][36][37][38][39][40][41][42][43][44][45]. Statistical investigation of various distance measures involving these Hilbert-Schmidt random states has become an active area of research due to its fundamental as well as applied aspects [21,[46][47][48][49][50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

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

2021

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One of the key issues in quantum information theory related problems concerns with that of 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 supplement the recently obtained results for the mean root fidelity and mean of squared Bures distance [Phys. Rev. A 104, 022438 ( 2021)]. 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.

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“…The classical β ensembles have been studied extensively throughout the last couple of decades, leading to a vast collection of results: see [222], [80], [280], [43], [319], [218] and references therein for a diverse but non-exhaustive sample. A result that is of great use to us is that the spectral moments of the classical β ensembles can be expanded in either N or 1/N, in line with equation (1.1.32).…”

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confidence: 99%

Recursive Characterisations of Random Matrix Ensembles and Associated Combinatorial Objects

Rahman

2023

Bull. Aust. Math. Soc.

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

Cited by 16 publications

References 85 publications

Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

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

Recursive Characterisations of Random Matrix Ensembles and Associated Combinatorial Objects

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