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arXiv:1005.3837 · International Journal of Quantum Information

DECAY OF COHERENCE AND ENTANGLEMENT OF A SUPERPOSITION STATE FOR A CONTINUOUS VARIABLE SYSTEM IN AN ARBITRARY HEAT BATH

G. W. Ford, Robert F. O’Connell

Published Aug 1, 2010• 1 citation
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Abstract

We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. In contrast to a previous related work, we avoid a master equation approach and we calculate exactly the time development of this non-Gaussian state due to interaction with an arbitrary heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.

DOI
10.1142/s0219749910006587

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We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. In contrast to a previous related work, we avoid a master equation approach and we calculate exactly the time development of this non-Gaussian state due to interaction with an arbitrary heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.
We outline an exact approach to decoherence and entanglement problems for continuous variable systems. The method is based on a construction of quantum distribution functions introduced by Ford and Lewis \cite{ford86} in which a system in thermal equilibrium is placed in an initial state by a measurement and then sampled by subsequent measurements. With the Langevin equation describing quantum Brownian motion, this method has proved to be a powerful tool for discussing such problems. After reviewing our previous work on decoherence and our recent work on disentanglement, we apply the method to the problem of a pair of particles in a correlated Gaussian state. The initial state and its time development are explicitly exhibited. For a single relaxation time bath at zero temperature exact numerical results are given. The criterion of Duan et al. \cite{duan00} for such states is used to prove that the state is initially entangled and becomes separable after a finite time (entanglement sudden death).
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we present a description of continuous-variable entanglement for a system consisting of two non-interacting modes embedded in a thermal environment. By using the Peres–Simon necessary and sufficient criterion for the separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place for non-zero temperatures of the environment. Only for zero temperature of the thermal bath, the initial entangled state remains entangled for finite times. We also show that, independent of its type, namely separable or entangled, the initial state evolves asymptotically to an equilibrium state that is always separable.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous variable entanglement for a system consisting of two non-interacting bosonic modes embedded in a thermal environment. The calculated measure of entanglement is entanglement of formation. We describe the evolution of entanglement in terms of the covariance matrix for symmetric Gaussian input states. In the case of an entangled initial squeezed thermal state, entanglement suppression (entanglement sudden death) takes place, for all non-zero temperatures of the thermal bath. After that, the system remains for all times in a separable state. For a zero temperature of the thermal bath, the system remains entangled for all finite times, but in the limit of asymptotic large times the state becomes separable.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable entanglement for a system consisting of two noninteracting modes embedded in a thermal environment. By using the Peres-Simon necessary and sufficient criterion for separability, we show that for all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place, for non-zero temperatures of the environment. Only for a zero temperature of the thermal bath, the initial entangled state remains entangled for finite times. We analyze also the time evolution of logarithmic negativity, which characterizes the degree of entanglement of the state, and of Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement.
We consider the decay of quantum entanglement quantified by the concurrence of a pair of two-level systems, each of which is interacting with a reservoir at finite temperature TTT. For a broad class of initially entangled states, we demonstrate that the system always becomes disentangled in a finite time, i.e., ``entanglement sudden death'' occurs. This class includes all states which previously had been found to have long-lived entanglement in zero-temperature reservoirs. Our general result is illustrated by an example.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous‐variable entanglement for a system consisting of two noninteracting modes embedded in a thermal environment. By using the Peres‐Simon necessary and sufficient criterion for separability of two‐mode Gaussian states, we describe the evolution of entanglement for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place, for non‐zero temperatures of the environment. Only for a zero temperature the initial entangled state remains entangled for finite times. We also show that, independent of its type—separable or entangled, the initial state evolves asymptotically to an equilibrium state which is always separable.