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Physical Review Letters

Exact Formula of the Distribution of Schmidt Eigenvalues for Dynamical Formation of Entanglement in Quantum Chaos

Hiroto Kubotani, Satoshi Adachi, Mikito Toda

Published Jun 17, 2008• 35 citations
Publisher

Abstract

The exact formula of the one-level distribution of the Schmidt eigenvalues is obtained for dynamical formation of entanglement in quantum chaos. The formula is based on the random matrix theory of the fixed-trace ensemble, and is derived using the theory of the holonomic system of differential equations. We confirm that the formula describes the universality of the distribution of the Schmidt eigenvalues in quantum chaos.

DOI
10.1103/physrevlett.100.240501

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Exact Formula of the Distribution of Schmidt Eigenvalues for Dynamical Formation of Entanglement in Quantum Chaos

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The exact formula of the one-level distribution of the Schmidt eigenvalues is obtained for dynamical formation of entanglement in quantum chaos. The formula is based on the random matrix theory of the fixed-trace ensemble, and is derived using the theory of the holonomic system of differential equations. We confirm that the formula describes the universality of the distribution of the Schmidt eigenvalues in quantum chaos.
Abstract We derive exact analytical formulae for the distribution of the largest Schmidt eigenvalue as an explicit piecewise polynomial with rational coefficients and for its moments as rational numbers by using random matrix theory based on the fixed-trace ensemble in order to study the quantum entanglement. The derivation utilizes a new connection between the multivariate hypergeometric functions and the Painlevé systems. The formulae are compared to numerical experiments performed in the coupled-kicked top system to reveal their sensitivity to the type of underlying dynamics, regular or chaos.
We study statistics of entanglement generated by quantum chaotic dynamics. Using an ensemble of the very large number (>/~10(7)) of quantum states obtained from the temporally evolving coupled kicked tops, we verify that the estimated one-body distribution of the squared Schmidt eigenvalues for the quantum chaotic dynamics can agree surprisingly well with the analytical one for the universality class of the random matrices described by the fixed trace ensemble (FTE). In order to quantify this agreement, we introduce the L(1) norm of the difference between the one-body distributions for the quantum chaos and FTE and use it as an indicator of the dynamical randomness. As we increase the scaled coupling constant, the L(1) difference decreases. When the effective Planck constant is not small enough, the decrease saturates, which implies quantum suppression of dynamical randomness. On the other hand, when the effective Planck constant is small enough, the decrease of the L(1) difference continues until it is masked by statistical fluctuation due to finiteness of the ensemble. Furthermore, we carry out two statistical analyses, the χ(2) goodness of fit test and an autocorrelation analysis, on the difference between the distributions to seek for dynamical remnants buried under the statistical fluctuation. We observe that almost all fluctuating deviations are statistical. However, even for well-developed quantum chaos, unexpectedly, we find a slight nonstatistical deviation near the largest Schmidt eigenvalue. In this way, the statistics of Schmidt eigenvalues enables us to measure dynamical randomness of quantum chaos with reference to the random matrix theory of FTE.
We study statistics of entanglement generated by quantum chaotic dynamics. Using an ensemble of the very large number (≳107) of quantum states obtained from the temporally evolving coupled kicked tops, we verify that the estimated one-body distribution of the squared Schmidt eigenvalues for the quantum chaotic dynamics can agree surprisingly well with the analytical one for the universality class of the random matrices described by the fixed trace ensemble (FTE). In order to quantify this agreement, we introduce the L1 norm of the difference between the one-body distributions for the quantum chaos and FTE and use it as an indicator of the dynamical randomness. As we increase the scaled coupling constant, the L1 difference decreases. When the effective Planck constant is not small enough, the decrease saturates, which implies quantum suppression of dynamical randomness. On the other hand, when the effective Planck constant is small enough, the decrease of the L1 difference continues until it is masked by statistical fluctuation due to finiteness of the ensemble. Furthermore, we carry out two statistical analyses, the χ2 goodness of fit test and an autocorrelation analysis, on the difference between the distributions to seek for dynamical remnants buried under the statistical fluctuation. We observe that almost all fluctuating deviations are statistical. However, even for well-developed quantum chaos, unexpectedly, we find a slight nonstatistical deviation near the largest Schmidt eigenvalue. In this way, the statistics of Schmidt eigenvalues enables us to measure dynamical randomness of quantum chaos with reference to the random matrix theory of FTE.
Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.