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Physical Review E

Measuring dynamical randomness of quantum chaos by statistics of Schmidt eigenvalues

Hiroto Kubotani, Satoshi Adachi, Mikito Toda

Published Jun 26, 2013• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.87.062921

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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.
The entanglement of eigenstates in two coupled, classically chaotic kicked tops is studied in dependence of their interaction strength. The transition from the noninteracting and unentangled system toward full random matrix behavior is governed by a universal scaling parameter. Using suitable random matrix transition ensembles we express this transition parameter as a function of the subsystem sizes and the coupling strength for both unitary and orthogonal symmetry classes. The universality is confirmed for the level spacing statistics of the coupled kicked tops and a perturbative description is in good agreement with numerical results. The statistics of Schmidt eigenvalues and entanglement entropies of eigenstates is found to follow a universal scaling as well. Remarkably, this is not only the case for large subsystems of equal size but also if one of them is much smaller. For the entanglement entropies a perturbative description is obtained, which can be extended to large couplings and provides very good agreement with numerical results. Furthermore, the transition of the statistics of the entanglement spectrum toward the random matrix limit is demonstrated for different ratios of the subsystem sizes.
Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random matrix theory modeling of composite quantum systems, investigated recently, entails a universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, which is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.