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arXiv:0806.4228 · Physical Review A

Evolution equation of entanglement for bipartite systems

Zong-Guo Li, Shao-Ming Fei, Z. D. Wang, W. M. Liu

Published Feb 6, 2009• 76 citations
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Abstract

We explore how entanglement of a bipartite system evolves when one subsystem undergoes the action of an arbitrary noisy channel. It is found that the dynamics of entanglement of such system is determined by the channel's action on the maximally entangled state, which includes as a special case the results for two-qubit systems [Konrad et al., Nat. Phys. 4, 99 (2008)]. In particular, for multiqubit or qubit-qudit systems, we get a general factorization law for the evolution equation of entanglement, with one qubit being subject to a noisy channel.

DOI
10.1103/physreva.79.024303

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We explore how entanglement of a bipartite system evolves when one subsystem undergoes the action of an arbitrary noisy channel. It is found that the dynamics of entanglement of such system is determined by the channel's action on the maximally entangled state, which includes as a special case the results for two-qubit systems [Konrad et al., Nat. Phys. 4, 99 (2008)]. In particular, for multiqubit or qubit-qudit systems, we get a general factorization law for the evolution equation of entanglement, with one qubit being subject to a noisy channel.
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We discuss entanglement evolution of a multiqubit system in which one of its qubits is subjected to a general noisy channel. For such a system, an evolution equation of entanglement for a lower bound for multiqubit concurrence is derived. Using this evolution equation, the entanglement dynamics of an initially mixed three-qubit state composed of a Greenberger-Horne-Zeilinger (GHZ) and a WWW state is analyzed if one of the qubits is affected by a phase, an amplitude, or a generalized amplitude damping channel.
The entanglement evolution of two qubits under local, single-, and two-sided noisy channels is investigated. It is found that for all pure initial states, the entanglement under a one-sided noisy channel is completely determined by the maximal trace distance which is the main element to construct the measure of non-Markovianity. For the two-sided noisy channel case, when the qubits are initially prepared in a general class of states, either pure or mixed, the entanglement can be expressed as the product of the initial entanglement and the channels' action on the maximally entangled state.
We investigate the possibility and conditions to factorize the entanglement evolution of a multiqubit system passing through multi-sided noisy channels. By means of a lower bound of concurrence (LBC) as entanglement measure, we derive an explicit formula of LBC evolution of the N-qubit generalized Greenberger-Horne-Zeilinger (GGHZ) state under some typical noisy channels, based on which two kinds of factorizing conditions for the LBC evolution are presented. In this case, the time-dependent LBC can be determined by a product of initial LBC of the system and the LBC evolution of a maximally entangled GGHZ state under the same multi-sided noisy channels. We analyze the realistic situations where these two kinds of factorizing conditions can be satisfied. In addition, we also discuss the dependence of entanglement robustness on the number of the qubits and that of the noisy channels.