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arXiv:0805.1694 · Physical Review A

Dynamics of global entanglement under decoherence

Afshin Montakhab, Ali Asadian

Published Jun 13, 2008• 25 citations
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Abstract

We investigate the dynamics of global entanglement, the Meyer-Wallach measure, under decoherence, analytically. We study two important classes of multipartite entangled states, the Greenberger-Horne-Zeilinger and the $\mathit{W}$ state. We obtain exact results for various models of system-environment interactions (decoherence). Our results shows distinctly different scaling behavior for these initially entangled states indicating a relative robustness of the $\mathit{W}$ state, consistent with previous studies.

DOI
10.1103/physreva.77.062322

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We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for Greenberger-Horne-Zeilinger (GHZ) -type superposition states decreases with the size of the system, while for a class of other states---namely, all graph states with constant degree---the lifetime is independent of the system size. We show that these results are largely independent of the specific decoherence model and are in particular valid for all models which deal with individual couplings of particles to independent environments, described by some quantum optical master equation of Lindblad form. For GHZ states, we derive analytic expressions for the lifetime of distillable entanglement and determine when the state becomes fully separable. For all graph states, we derive lower and upper bounds on the lifetime of entanglement. The lower bound is based on a specific distillation protocol, while upper bounds are obtained by showing that states resulting from decoherence in general become nondistillable or even separable after a finite time. This is done using different methods, namely, (i) the map describing the decoherence process (e.g., the action of a thermal bath on the system) becomes entanglement breaking, (ii) the resulting state becomes separable, and (iii) the partial transposition with respect to certain partitions becomes positive. To this aim, we establish a method to calculate the spectrum of the partial transposition for all mixed states which are diagonal in a graph-state basis. We also consider entanglement between different groups of particles and determine the corresponding lifetimes as well as the change of the kind of entanglement with time. This enables us to investigate the behavior of entanglement under rescaling and in the limit of large number of particles $N\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}$. Finally we investigate the lifetime of encoded quantum superposition states and show that one can define an effective time in the encoded system which can be orders of magnitude smaller than the physical time. This provides an alternative view on quantum error correction and examples of states whose lifetime of entanglement (between groups of particles) in fact increases with the size of the system.
We consider the collision model of Ziman et al. and study the robustness of NNN-qubit Greenberger-Horne-Zeilinger (GHZ), WWW, and linear cluster states. Our results show that NNN-qubit entanglement of GHZ states would be extremely fragile under collisional decoherence, and that of WWW states could be more robust than that of linear cluster states. We indicate that the collision model of Ziman et al. could provide a physical mechanism for some known results in this area of investigation. More importantly, we show that it could give a clue as to how NNN-partite distillable entanglement would be relatively rare in our macroscopic classical world.
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