Quaternionic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>transform and non-Hermitian random matrices

Burda, Z.; Święch, Artur

doi:10.1103/physreve.92.052111

Cited by 7 publications

(11 citation statements)

References 56 publications

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“…where G is a GinUE matrix, U a CUE matrix, and a, b real constants. To this end, we employ quaternionic free probability [17,[51][52][53][54][55][56][57][58][59][60][61][62][63], which we start by briefly reviewing below.…”

Section: Discussionmentioning

confidence: 99%

Spectral transitions and universal steady states in random Kraus maps and circuits

et al. 2020

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The study of dissipation and decoherence in generic open quantum systems recently led to the investigation of spectral and steady-state properties of random Lindbladian dynamics. A natural question is then how realistic and universal those properties are. Here, we address these issues by considering a different description of dissipative quantum systems, namely, the discrete-time Kraus map representation of completely positive quantum dynamics. Through random matrix theory (RMT) techniques and numerical exact diagonalization, we study random Kraus maps, allowing for a varying dissipation strength, and their local circuit counterpart. We find the spectrum of the random Kraus map to be either an annulus or a disk inside the unit circle in the complex plane, with a transition between the two cases taking place at a critical value of dissipation strength. The eigenvalue distribution and the spectral transition are well described by a simplified RMT model that we can solve exactly in the thermodynamic limit, by means of non-Hermitian RMT and quaternionic free probability. The steady state, on the contrary, is not affected by the spectral transition. It has, however, a perturbative crossover regime at small dissipation, inside which the steady state is characterized by uncorrelated eigenvalues. At large dissipation (or for any dissipation for a large-enough system), the steady state is well described by a random Wishart matrix. The steady-state properties thus coincide with those already observed for random Lindbladian dynamics, indicating their universality. Quite remarkably, the statistical properties of the local Kraus circuit are qualitatively the same as those of the nonlocal Kraus map, indicating that the latter, which is more tractable, already captures the realistic and universal physical properties of generic open quantum systems.

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Section: Discussionmentioning

confidence: 99%

Spectral transitions and universal steady states in random Kraus maps and circuits

et al. 2020

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“…We establish that it actually holds for every word of Ginibre matrices, and universally (that is, beyond the Gaussian case). By a similar proof technique, we show that the limit of all the mixed matrix moments of G w depends only on the length of w. Such mixed moments are deeply connected with both eigenvalues and eigenvectors; they appear naturally in the moments of Girko's hermitized form, or in the expansion of the quaternionic resolvent (see [8]), and their connection with eigenvector overlaps has been studied by Walters and Starr in [44]. The fact that the first order asymptotics of these mixed moments only depends on the length suggests that the limit of empirical measure of eigenvalues of G w only depends on the length, as is the case for singular values.…”

Section: Introductionmentioning

confidence: 88%

“…These mixed moments have been studied by Starr and Walters [44]. They appear naturally in the moments of Girko's hermitized form (G − z)(G − z) * , as well as in the quaternionic resolvent (see [8,33]). They are known to be linked to relevant statistics of eigenvalues and eigenvectors of random matrices.…”

Section: Mixed Moments Of Wordsmentioning

confidence: 99%

On Words of non-Hermitian Random Matrices

Dubach¹,

Peled²

2019

Preprint

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We consider words G i 1 • • • G im involving i.i.d. complex Ginibre matrices, and study their singular values and eigenvalues. We show that the limit distribution of the squared singular values of every word of length m is a Fuss-Catalan distribution with parameter m + 1. This generalizes previous results concerning powers of a complex Ginibre matrix, and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, sparse matrices and band matrices.These results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically.

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“…The elements of this matrix are related to each other, g 22 (z) =ḡ 11 (z) and g 21 (z) = −ḡ 12 (z) [44], so we have…”

Section: Large N Limitmentioning

confidence: 99%

Eigenvector statistics of the product of Ginibre matrices

2017

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We develop a method to calculate left-right eigenvector correlations of the product of m independent N × N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N . We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O = 1 + (m/2)(N − 1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N → ∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

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QuaternionicRtransform and non-Hermitian random matrices

Cited by 7 publications

References 56 publications

Spectral transitions and universal steady states in random Kraus maps and circuits

Spectral transitions and universal steady states in random Kraus maps and circuits

On Words of non-Hermitian Random Matrices

Eigenvector statistics of the product of Ginibre matrices

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