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arXiv:quant-ph/0203117 · Physical Review Letters

Testing Statistical Bounds on Entanglement Using Quantum Chaos

Jayendra N. Bandyopadhyay, Arul Lakshminarayan

Published Jul 22, 2002• 150 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.89.060402

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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.
Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, presence of chaos in the system produces more entanglement. However, coupling strength between two subsystems is also very important parameter for the entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed state entanglement production in chaotic systems.
Entanglement production in coupled chaotic systems is studied with the help of kicked tops. Deriving the correct classical map, we have used the reduced Husimi function, the Husimi function of the reduced density matrix, to visualize the possible behaviors of a wave packet. We have studied a phase-space based measure of the complexity of a state and used random matrix theory (RMT) to model the strongly chaotic cases. Extensive numerical studies have been done for the entanglement production in coupled kicked tops corresponding to different underlying classical dynamics and different coupling strengths. An approximate formula, based on RMT, is derived for the entanglement production in coupled strongly chaotic systems. This formula, applicable for arbitrary coupling strengths and also valid for long time, complements and extends significantly recent perturbation theories for strongly chaotic weakly coupled systems.
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.
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.
We investigate how the dynamical production of quantum entanglement for weakly coupled, composite quantum systems is influenced by the chaotic dynamics of the corresponding classical system, using coupled kicked tops. The linear entropy for the subsystem (a kicked top) is employed as a measure of entanglement. A perturbative formula for the entanglement production rate is derived. The formula contains a correlation function that can be evaluated only from the information of uncoupled tops. Using this expression and the assumption that the correlation function decays exponentially which is plausible for chaotic tops, it is shown that the increment in the strength of chaos does not enhance the production rate of entanglement when the coupling is weak enough and the subsystems (kicked tops) are strongly chaotic. The result is confirmed by numerical experiments. The perturbative approach is also applied to a weakly chaotic region, where tori and chaotic sea coexist in the corresponding classical phase space, to reexamine a recent numerical study that suggests an intimate relationship between the linear stability of the corresponding classical trajectory and the entanglement production rate.