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arXiv:0812.3959 · Physical Review A

Evolution of entanglement for quantum mixed states

Chang‐shui Yu, X. X. Yi, He‐Shan Song

Published Dec 18, 2008• 25 citations
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Abstract

A simple relation is introduced for concurrence to describe how much the entanglement of a bipartite system is at least left if either (or both) subsystem undergoes an arbitrary physical process. This provides a lower bound for concurrence of mixed states (pure states are included) in contrast to the upper bound given by Konrad et al. [Nat. Phys. 4, 99 (2008)]. Our results are also suitable for general high-dimensional bipartite quantum systems.

DOI
10.1103/physreva.78.062330

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A simple relation is introduced for concurrence to describe how much the entanglement of a bipartite system is at least left if either (or both) subsystem undergoes an arbitrary physical process. This provides a lower bound for concurrence of mixed states (pure states are included) in contrast to the upper bound given by Konrad et al. [Nat. Phys. 4, 99 (2008)]. Our results are also suitable for general high-dimensional bipartite quantum systems.
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We present a lower bound of concurrence for arbitrary-dimensional bipartite quantum states. This lower bound may be used to improve all the known lower bounds of concurrence. Moreover, the lower bound gives rise to an operational sufficient criterion of distillability of quantum entanglement. The significance of our result is illustrated by quantitative evaluation of entanglement for entangled states that fail to be identified by the usual concurrence estimation method and by showing the distillability of mixed states that cannot be recognized by other distillability criteria.
The dynamics of the quantum entanglement is a fundamental characteristic for various quantum systems. Since the computable entanglement measure for higher dimensional quantum states itself is absent, the dynamics of the entanglement expressed in an operational method will be of interest. We study the dynamics of $\ensuremath{\tau}$, an analytical lower bound of squared concurrence, of a bipartite $d\ensuremath{\bigotimes}d$ quantum state when one party goes through an arbitrary noisy channel. For a pure input state, the range of $\ensuremath{\tau}$ is obtained explicitly. For a mixed input state, an upper bound of $\ensuremath{\tau}$ is found. Interestingly, the tangle ${\ensuremath{\tau}}^{\ensuremath{'}}$, as an upper bound of squared concurrence, also has a similar dynamical property. Our results are similar to that of Konrad et al. and can help the estimation of high-dimension bipartite entanglement in experiments.
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.