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Physical Review A

Dynamical entanglement versus symmetry and dynamics of classical approximations

Nikola Burić

Published May 24, 2006• 12 citations
Publisher

Abstract

It is shown that dynamical entanglement between two qubits depends on the symmetry of the quantum model. On the other hand, the latter is reflected in the qualitative properties of the dynamics of a classical approximation of the quantum system. For generic separable pure initial states, the dynamical entanglement is larger if the system is less symmetric and its classical approximation is chaotic. The influence of different types of Markov environments on the established relation between the dynamical entanglement, symmetry and the classical dynamics is also studied.

DOI
10.1103/physreva.73.052111

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It is shown that dynamical entanglement between two qubits depends on the symmetry of the quantum model. On the other hand, the latter is reflected in the qualitative properties of the dynamics of a classical approximation of the quantum system. For generic separable pure initial states, the dynamical entanglement is larger if the system is less symmetric and its classical approximation is chaotic. The influence of different types of Markov environments on the established relation between the dynamical entanglement, symmetry and the classical dynamics is also studied.
Entanglement dynamics in representative examples of Markov open quantum systems with qualitatively different dynamics are studied. Rings of qubits with thermal or dephasing local environment are used to study the qualitative properties of the entanglement dynamics depending on the interqubit interaction, type of environment and the initial state. It is demonstrated that the effect of the local environment is manifested as an exponential decrease of the entanglement superimposed on the entanglement dynamics in the isolated system.
Echo dynamics and fidelity are often used to discuss stability in quantum-information processing and quantum chaos. Yet fidelity yields no information about entanglement, the characteristic property of quantum mechanics. We study the evolution of entanglement in echo dynamics. We find qualitatively different behavior between integrable and chaotic systems on one hand and between random and coherent initial states for integrable systems on the other. For the latter the evolution of entanglement is given by a classical time scale. Analytic results are illustrated numerically in a Jaynes-Cummings model.
Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.
We analyze numerically the dynamical generation of quantum entanglement in a system of two interacting particles, started in a coherent separable state, for decreasing values of ℏ. As ℏ→0 the entanglement entropy, computed at any finite time, converges to a finite nonzero value. The limit law that rules the time dependence of entropy is well reproduced by purely classical computations. Its general features can be explained by simple classical arguments, which expose the different ways entanglement is generated in systems that are classically chaotic or regular.
The entanglement evolution of the coupled qubits interacting with a non-Markov environment is investigated in terms of concurrence. The results show that the entanglement of the quantum systems depends not only on the initial state of the system but also on the coupling between the qubit and the environment. For the initial state , the coupled qubits will always been in the maximum entangled state under an asymmetric coupling. For the initial state , in contrast, the entangling degree of the coupled qubits is always equal to unity and does not depend on the evolving time under the symmetric coupling. We find that the stronger the interaction between the qubits is, the better the struggle against the entanglement sudden death is.