Random density matrices: analytical results for mean root fidelity and mean square Bures distance

Laha, Aritra; Aggarwal, Agrim; Kumar, Santosh

doi:10.48550/arxiv.2105.02743

Cited by 2 publications

(2 citation statements)

References 65 publications

(126 reference statements)

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“…We note that an exact expression was recently found for the fidelity of two random density matrices, consistent with our large-N results[65]. Our result is complementary as the exact expression is very complicated and not tractable at large-N .…”

supporting

confidence: 90%

Distinguishing Random and Black Hole Microstates

Kudler-Flam,

Narovlansky,

Ryu

2021

Preprint

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This is an expanded version of the short report [Phys. Rev. Lett. 126, 171603 (2021)], where the relative entropy was used to distinguish random states drawn from the Wishart ensemble as well as black hole microstates. In this work, we expand these ideas by computing many generalizations including the Petz Rényi relative entropy, sandwiched Rényi relative entropy, fidelities, and trace distances. These generalized quantities are able to teach us about new structures in the space of random states and black hole microstates where the von Neumann and relative entropies were insufficient. We further generalize to generic random tensor networks where new phenomena arise due to the locality in the networks. These phenomena sharpen the relationship between holographic states and random tensor networks. We discuss the implications of our results on the black hole information problem using replica wormholes, specifically the state dependence (hair) in Hawking radiation. Understanding the differences between Hawking radiation of distinct evaporating black holes is an important piece of the information problem that was not addressed by entropy calculations using the island formula. We interpret our results in the language of quantum hypothesis testing and the subsystem eigenstate thermalization hypothesis (ETH), deriving that chaotic (including holographic) systems obey subsystem ETH for all subsystems less than half the total system size.

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confidence: 90%

Distinguishing Random and Black Hole Microstates

Kudler-Flam,

Narovlansky,

Ryu

2021

Preprint

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“…A number of statistics have been calculated for the Bures-Hall ensemble which are global in nature and only involve the one or two-point correlations such as fidelity [50], quantum purity E[Tr(ρ 2 )] [50], [34], [14], [28] and averages of the von Neumann entropy E[−Tr(ρ log ρ)] [35] and its higher moments [39], [52], [51], [30]. In [16] the smallest eigenvalue distribution of the fixed-trace Laguerre beta-ensemble, i.e.…”

Section: Volume Measures Of Quantum Statesmentioning

confidence: 99%

Gap probabilities for the Bures-Hall Ensemble and the Cauchy-Laguerre Two-Matrix Model

Witte¹,

Wei²

2022

Preprint

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The Bures metric and the associated Bures-Hall measure is arguably the best choice for studying the spectrum of the quantum mechanical density matrix with no apriori knowledge of the system. We investigate the probability of a gap in the spectrum of this model, either at the bottom [0, s) or at the top (s, 1], utilising the connection of this Pfaffian point-process with the allied problem in the determinantal point-process of the two-dimensional Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two variables s, t. To this end we develop new general results about Cauchy bi-orthogonal polynomial system for a more general class of weights than the Laguerre densities: in particular a new Christoffel-Darboux formula, reproducing kernels and differential equations for the polynomials and their associated functions. This system is most simply expressed as rank-3 matrix variables and possesses an associated cubic bilinear form. Furthermore under specialisation to truncated Laguerre type densities for the weight, of direct relevance to the Cauchy-Laguerre system, we construct a closed system of constrained, nonlinear differential equations in two deformation variables s, t, and observe that the recurrence, spectral and deformation derivative structures form a compatible and integrable triplet of Lax equations.

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Random density matrices: analytical results for mean root fidelity and mean square Bures distance

Cited by 2 publications

References 65 publications

Distinguishing Random and Black Hole Microstates

Distinguishing Random and Black Hole Microstates

Gap probabilities for the Bures-Hall Ensemble and the Cauchy-Laguerre Two-Matrix Model

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