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arXiv:quant-ph/0410103 · Physical Review A

Semiclassical limit of the entanglement in closed pure systems

R. M. Angelo, K. Furuya

Published Apr 13, 2005• 28 citations
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Abstract

We discuss the semiclassical limit of the entanglement for the class of closed pure systems. By means of analytical and numerical calculations, we obtain two main results: (i) the short-time entanglement does not depend on Planck's constant and (ii) the long-time entanglement increases as more semiclassical regimes are attained. On one hand, this result is in contrast with the idea that the entanglement should be destroyed when the macroscopic limit is reached. On the other hand, it emphasizes the role played by decoherence in the process of emergence of the classical world. We also found that, for Gaussian initial states, the entanglement dynamics may be described by an entirely classical entropy in the semiclassical limit.

DOI
10.1103/physreva.71.042321

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We discuss the semiclassical limit of the entanglement for the class of closed pure systems. By means of analytical and numerical calculations, we obtain two main results: (i) the short-time entanglement does not depend on Planck's constant and (ii) the long-time entanglement increases as more semiclassical regimes are attained. On one hand, this result is in contrast with the idea that the entanglement should be destroyed when the macroscopic limit is reached. On the other hand, it emphasizes the role played by decoherence in the process of emergence of the classical world. We also found that, for Gaussian initial states, the entanglement dynamics may be described by an entirely classical entropy in the semiclassical limit.
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Entanglement is considered as a purely quantum effect. The present paper, however, joins many other works that pursue the idea of describing it using classical quantities. In particular, we are interested in the entanglement dynamics of a pure state composed of two parts, initially prepared in a product of coherent states and governed by a generic Hamiltonian. In this scenario, the linear entropy of the reduced state, our entanglement quantifier, was shown to be written, in the short-time regime, in terms of a real trajectory of the underlying classical dynamics. We extend this semiclassical result by demonstrating that it is possible to include complex trajectories in the calculation. This strategy contributes to improving the previous approximation, extending its accuracy for longer values of time. We also show, for a particular Hamiltonian, a first application of the achieved formula, attesting to its efficiency.
We analyze the dynamical generation of entanglement in systems of two interacting spins initially prepared in a product of spin coherent states. For arbitrary time-independent Hamiltonians, we derive a semiclassical expression for the purity of the reduced density matrix as a function of time. The final formula, subsidiary to the linear entropy, shows that the short-time dynamics of entanglement depends exclusively on the stability of trajectories governed by the underlying classical Hamiltonian. In addition, this semiclassical measure is shown to reproduce the general properties of its quantum counterpart and give the expected result in the large spin limit. The accuracy of the semiclassical formula is further illustrated in a problem of phase exchange for two particles of spin jjj.
We study dynamical generation of entanglement in bipartite quantum systems, characterized by purity (or linear entropy), and caused by the coupling between the two subsystems. The explicit semiclassical theory of purity decay is derived for integrable classical dynamics of the uncoupled system and for localized (general Gaussian wave packet) initial states. Purity decays as an algebraic function of (time) $\ifmmode\times\else\texttimes\fi{}$ (strength of perturbation), independently of the Planck's constant.
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