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arXiv:quant-ph/0307180 · Physical Review Letters

Stability of Macroscopic Entanglement under Decoherence

Wolfgang Dür, Hans J. Briegel

Published May 6, 2004• 319 citations
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Abstract

We investigate the lifetime of macroscopic entanglement under the influence of decoherence. For Greenberger-Horne-Zeilinger-type superposition states, we find that the lifetime decreases with the size of the system (i.e., the number of independent degrees of freedom), and the effective number of subsystems that remain entangled decreases with time. For a class of other states (e.g., cluster states), however, we show that the lifetime of entanglement is independent of the size of the system.

DOI
10.1103/physrevlett.92.180403

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We investigate the lifetime of macroscopic entanglement under the influence of decoherence. For Greenberger-Horne-Zeilinger-type superposition states, we find that the lifetime decreases with the size of the system (i.e., the number of independent degrees of freedom), and the effective number of subsystems that remain entangled decreases with time. For a class of other states (e.g., cluster states), however, we show that the lifetime of entanglement is independent of the size of the system.
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.
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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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We study the robustness of the Greenberger-Horne-Zeilinger (``cat'') class of multipartite states under decoherence. The noise model is described by a general completely positive map for qubits independently coupled to the environment. In particular, the robustness of NNN-party entanglement is studied in the large NNN limit when (a) the number of spatially separated subsystems is fixed but the size of each subsystem becomes large and (b) the size of the subsystems is fixed while their number becomes arbitrarily large. We obtain conditions for entanglement in these two cases. Among our other results, we show that the parity of an entangled state (i.e., whether it contains an even or odd number of qubits) can lead to qualitatively different robustness of entanglement under certain conditions.