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Journal of Physics A Mathematical and Theoretical

The separability of tripartite Gaussian states with amplification and amplitude damping

Xiaoyu Chen

Published Apr 2, 2008• 0 citations
Publisher

Abstract

The most general evolution solution of multi-mode continuous variable states is given when it undergoes amplification, amplitude damping and thermal noise. In the case of some kind of initial Gaussian state totally symmetrically interacting with the environment, the conditions for full separability and full entanglement of the final tripartite three-mode Gaussian state are worked out.

DOI
10.1088/1751-8113/41/15/155301

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The most general evolution solution of multi-mode continuous variable states is given when it undergoes amplification, amplitude damping and thermal noise. In the case of some kind of initial Gaussian state totally symmetrically interacting with the environment, the conditions for full separability and full entanglement of the final tripartite three-mode Gaussian state are worked out.
We give the time evolution solution of simultaneous amplitude and phase damping for any continuous variable state. For the simultaneous amplitude and phase damping of two-mode $x\text{\ensuremath{-}}p$ symmetric entangled Gaussian states, two analytical conditions of the separability are given. One is the sufficient condition of separability. The other is the condition of positive partial transpose (PPT) separability where the Peres-Horodecki criterion is applied. Between the two conditions there may exist bound entanglement. The simplest example is the simultaneous amplitude and phase damping of a two-mode squeezed vacuum state. The damped state is non-Gaussian.
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.