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arXiv:quant-ph/0209086 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Saturation of the production of quantum entanglement between weakly coupled mapping systems in a strongly chaotic region

Atushi Tanaka, Hiroshi Fujisaki, Takayuki Miyadera

Published Oct 10, 2002• 60 citations
arXivHTMLPDF

Abstract

The production of quantum entanglement between weakly coupled mapping systems, whose classical counterparts are both strongly chaotic, is investigated. In the weak-coupling regime, it is shown that time-correlation functions of the unperturbed systems determine the entanglement production. In particular, we elucidate that the increment of the nonlinear parameter of coupled kicked tops does not accelerate the entanglement production in the strongly chaotic region. An approach to the dynamical inhibition of entanglement is suggested.

DOI
10.1103/physreve.66.045201

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The production of quantum entanglement between weakly coupled mapping systems, whose classical counterparts are both strongly chaotic, is investigated. In the weak-coupling regime, it is shown that time-correlation functions of the unperturbed systems determine the entanglement production. In particular, we elucidate that the increment of the nonlinear parameter of coupled kicked tops does not accelerate the entanglement production in the strongly chaotic region. An approach to the dynamical inhibition of entanglement is suggested.
We investigate how the dynamical production of quantum entanglement for weakly coupled mapping systems is influenced by the chaotic dynamics of the corresponding classical system. We derive a general perturbative formula for the entanglement production rate which is defined by using the linear entropy of the subsystem. This formula predicts that {\it the increment of the strength of chaos does not enhance the production rate of entanglement} when the coupling is weak enough and the subsystems are strongly chaotic. The prediction is confirmed by numerical experiments for coupled kicked tops and rotors. We also discuss the entanglement production using the Husimi representation of the reduced density matrix.
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.
Entanglement production in coupled chaotic systems is studied with the help of kicked tops. Deriving the correct classical map, we have introduced and used the reduced Husimi function to visualize the possible behaviors of a wavepacket. 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.
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.
We study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled tops, where chaos and entanglement arise from the same physical mechanism. The long-time-averaged entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section--it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudorandom states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. For a map with global chaos, we derive that value based on analytic results for the entropy of random states. For a mixed phase space, we use the Percival conjecture to identify a "chaotic subspace" of the Hilbert space. The typical entanglement, averaged over the unitarily invariant Haar measure in this subspace, agrees with the long-time-averaged entanglement for initial states in the chaotic sea. In all cases the dynamically generated entanglement is that of a random complex vector, even though the system is time-reversal invariant, and the Floquet operator is a member of the circular orthogonal ensemble.