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AIP conference proceedings

Entanglement, Symmetry and Qualitative Properties of Dynamics

Nikola Burić

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We analyze relations between dynamical entanglement on one hand and symmetry of a quantum system and the qualitative properties of the dynamics of a classical approximation of the quantum system on the other, using examples of coupled qubits. For generic separable initial states, the dynamical entanglement is larger if the system is less symmetric and its classical approximation is chaotic, rather than integrable.

DOI
10.1063/1.2713466

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We analyze relations between dynamical entanglement on one hand and symmetry of a quantum system and the qualitative properties of the dynamics of a classical approximation of the quantum system on the other, using examples of coupled qubits. For generic separable initial states, the dynamical entanglement is larger if the system is less symmetric and its classical approximation is chaotic, rather than integrable.
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.
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 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.
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.
The dynamical properties of quantum entanglement in the Dicke model without rotating-wave approximation are investigated in terms of the reduced-density linear entropy. The characteristic time of decoherence process in the early-time evolution is numerically obtained and it is shown that the characteristic time decreases as the coupling parameter increases. The mean entanglement, which is defined to be averaged over time, is employed to describe the influences of both quantum phase transition and corresponding classical chaos on the behavior of entanglement. For a given energy, initial conditions are taken to be minimum uncertainty wave packets centered at regular and chaotic regions of the classical phase space. It is shown that the entanglement has a distinct change at the quantum phase transition, and that the entanglement for regular initial conditions is smaller than that for chaotic ones in the case of weak coupling, while it fluctuates with small amplitude in strong coupling and for chaotic initial conditions.
The relation between the dynamical properties of a coupled quasiparticle-oscillator system in the mixed quantum-classical and fully quantized descriptions is investigated. The system is considered as a model for applying a stepwise quantization. Features of the nonlinear dynamics in the mixed description such as the presence of a separatrix structure or regular and chaotic motion are shown to be reflected in the evolu- tion of the quantum state vector of the fully quantized system. In particular, it is demonstrated how wave packets propagate along the separatrix structure of the mixed description, and that chaotic dynamics leads to a strongly entangled quantum state vector. Special emphasis is given to viewing the system from a dyn- amical Born-Oppenheimer approximation defining integrable reference oscillators, and elucidating the role of the nonadiabatic couplings which complement this approximation into a rigorous quantization scheme.