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Chinese Physics Letters

Inseparability of Phase-Damped Bipartite Gaussian States

Li-zhen Jiang

Published Sep 29, 2009• 0 citations
Publisher

Abstract

For bipartite non-Gaussian states which are prepared by applying the phase damping to Gaussian states, we use Fock subspace inseparable criterion and the Shchukin–Vogel inseparable criterion to analyze their inseparability. The lowest order of the two criteria is obtained. Fock subspace criterion is more efficient in detecting the entanglement of damped Gaussian state by numerical results. Two photon Fock subspace criterion does not improve the entanglement detecting condition.

DOI
10.1088/0256-307x/26/10/100303

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For bipartite non-Gaussian states which are prepared by applying the phase damping to Gaussian states, we use Fock subspace inseparable criterion and the Shchukin–Vogel inseparable criterion to analyze their inseparability. The lowest order of the two criteria is obtained. Fock subspace criterion is more efficient in detecting the entanglement of damped Gaussian state by numerical results. Two photon Fock subspace criterion does not improve the entanglement detecting condition.
We prove that previous various forms of the inseparability of a Gaussian state are equivalent to the negativity of a partial transposed density matrix in the one-photon Fock subspace. We give an infinite series of inequalities; the fulfillment of each inequality shows the presence of entanglement in bipartite systems. For many non-Gaussian states, we show that Fock-space inseparability criteria are theoretically more efficient and simpler than Shchukin-Vogel criteria, which are based on the moments of the states.
We investigate the efficiency of inseparability criteria in detecting the entanglement properties of two-mode non-Gaussian states of the electromagnetic field. We focus our study on the relevant class of two-mode squeezed number states. These states combine the entangling capability of two-mode squeezers with the non-Gaussian character (nonclassicality) of number states. They allow for some exact analytical treatments, and include as a particular case the two-mode Gaussian squeezed vacuum. We show that the generalized PPT criterion recently proposed by Shchukin and Vogel, based on higher order statistical moments, is very efficient in detecting the entanglement for this class of non-Gaussian states.
In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion.
We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. For bipartite system, we transform the validity of a Gaussian witness to a bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of bosonic Gaussian channels. We demonstrate necessary and sufficient criteria of separability for two-mode and some four-mode Gaussian quantum states. We also propose multipartite Gaussian witnesses based on numeric evidence. The entanglement and multipartite entanglement detecting powers of the Gaussian witness criteria are illustrated by the mixture of two Gaussian states. We apply the Gaussian witness criterion to a Gaussian Fock state, which is prepared by passing a product Fock state through bosonic Gaussian channel. We show that the necessary criterion of separability for a Gaussian Fock state does not rely on the initial product Fock state, it relies on the channel.
Inseparability criteria for continuous and discrete bipartite quantum states based on moments of annihilation and creation operators are studied by developing the idea of Shchukin-Vogel criterion [Phys. Rev. Lett. 95, 230502 (2005)]. If a state is separable, then the corresponding matrix of moments is separable too. Thus, we derive generalized criteria based on the separability properties of the matrix of moments. In particular, a criterion based on realignment of moments in the matrix is proposed as an analog of the standard realignment criterion for density matrices. Other inseparability inequalities are obtained by applying positive maps to the matrix of moments. Usefulness of the Shchukin-Vogel criterion to describe bipartite-entanglement of more than two modes is demonstrated: we obtain various three-mode inseparability criteria, including some previously known ones, which were originally derived from the Cauchy-Schwarz inequality.