Entanglement properties of multipartite entangled states under the influence of decoherence

Abstract

We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for 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 non-distillable 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 re-scaling and in the limit of large number of particles N → ∞. 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.

Quantum-information splitting using multipartite cluster states

“…They show a strong violation of local reality and are shown to be robust against decoherence [25,26]. Much work has been done in trying to characterize the entanglement exhibited by these states, owing to their promising usefulness in Quantum information theory [27,28].…”

Bipartite Bell Inequalities for Hyperentangled States

“…By "size" we mean either the number of particles or the number of internal degrees of freedom. However, Mermin [18] showed that the correlations found by n spacelike separated observers that share n qubits in a GHZ state violate a n-party (with n ≥ 3) BI by a factor that increases exponentially with n. An experimental verification of this exponentially growing nonlocality using GHZ states is difficult because it requires n spacelike separated measurements, and because n-party GHZ states' sensitivity to decoherence also grows with n [19].…”

Two destructive effects of decoherence on Bell inequality violation

“…This interaction can cause the two-particle system to undergo decoherence [5,6,7], and the system will practically lose its quantum properties. The decay of Bell violation and entanglement measures of a decohering quantum system was studied recently in a number of works (see, e.g., [8,9,10,11,12]). In this paper we show that in the model of decoherence we use, there are two ways in which partial decoherence affects the ability of our two-particle system to violate the CHSH inequality.…”

“…For state (11), the reduced density matrix of S has form (10), with r replaced by r * 1 r 2 . We see that the two-particle system with decoherence due to interactions of both particles is described by the same density matrix as the system in which only one particle interacts with its environment.…”