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The simultaneous amplitude and phase damping of two mode squeezed vacuum state

Xiaoyu Chen

Published Aug 28, 2005• 0 citations
OpenAlex

Abstract

The Peres-Horodecki criterion of positivity under partial transpose is studied for the simultaneous amplitude and phase damping of a two mode squeezed vacuum state. The criterion turns out to be a necessary and sufficient condition for separability, although the damped state is no longer gaussian. The proof is at the sense of algebraic programming and also perturbation theory. The criterion is stronger than that obtained by the uncertainty principle on the second moments of the partial transposed density matrix.

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The Peres-Horodecki criterion of positivity under partial transpose is studied for the simultaneous amplitude and phase damping of a two mode squeezed vacuum state. The criterion turns out to be a necessary and sufficient condition for separability, although the damped state is no longer gaussian. The proof is at the sense of algebraic programming and also perturbation theory. The criterion is stronger than that obtained by the uncertainty principle on the second moments of the partial transposed density matrix.
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.
The Peres-Horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states. The partial transpose operation admits, in the continuous case, a geometric interpretation as mirror reflection in phase space. This recognition leads to uncertainty principles, stronger than the traditional ones, to be obeyed by all separable states. For all bipartite Gaussian states, the Peres-Horodecki criterion turns out to be a necessary and sufficient condition for separability.
A general separability condition on the second moment (covariance matrix) for continuous-variable two-party systems is derived by an analysis analogous to the derivation of Kennard's uncertainty relation without referring to the non-negativity of the partially transposed density matrix. This separability criterion is generally more stringent than that used by Simon which is based on the non-negativity of partially transposed density matrix, and thus this criterion may be useful in the analysis of general continuous two-party systems. Another separability criterion used by Duan et al. is shown to be generally weaker than that of Simon. We thus have a hierarchy of separability criterions, but all these criterions when combined with suitable squeezing become equivalent at the boundary of the PPP-representation condition and thus turned out to be sufficient to analyze the separability of two-party Gaussian systems.
We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.
I investigate the decoherence of two-mode squeezed vacuum states by analyzing the relative entropy of entanglement. I consider two sources of decoherence: (i) the phase damping and (ii) the amplitude damping due to the coupling to the thermal environment. In particular, I give the exact value of the relative entropy of entanglement for the phase damping model. For the amplitude damping model, I give an upper bound for the relative entropy of entanglement, which turns out to be a good approximation for the entanglement measure in the usual experimental situations.