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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Entanglement properties of a kind of tripatite Gaussian state

Xiaoyu Chen

Published Jan 11, 2005• 0 citations
Publisher

Abstract

Multipartite entanglement is quite difficult to deal with. In this paper, I will consider a particular kind of tripartite Gaussian states, which can be produced as the initial pure tripartite states undergo different noise for each party. I will make use of the relative entropy of entanglement to evaluate the entanglement properties of the states based on the fact that the separability of tripartite Gaussian states is well known. In addition, the procedure of calculating the relative entropy between Gaussian states have developed recently. All these make the evaluating the relative entropy of entanglement of tripartite Gaussian states possible.

DOI
10.1117/12.576010

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Multipartite entanglement is quite difficult to deal with. In this paper, I will consider a particular kind of tripartite Gaussian states, which can be produced as the initial pure tripartite states undergo different noise for each party. I will make use of the relative entropy of entanglement to evaluate the entanglement properties of the states based on the fact that the separability of tripartite Gaussian states is well known. In addition, the procedure of calculating the relative entropy between Gaussian states have developed recently. All these make the evaluating the relative entropy of entanglement of tripartite Gaussian states possible.
We in this paper calculate the relative entropy between the bipartite Gaussian states. By evaluating the minimization of the relative entropy of an entangled Gaussian state with respect to the separable Gaussian states, we obtain the Gaussian relative entropy of entanglement of the state. The bipartite operations of squeezing and rotations on the separable states are also addressed. We prove that for bipartite symmetric Gaussian states the separable states we sought will be symmetric Gaussian states, and we get quite simple expressions of Gaussian relative entropy of entanglement for such states.
We establish relations between tripartite pure state entanglement and additivity properties of the bipartite relative entropy of entanglement. Our results pertain to the asymptotic limit of local manipulations on a large number of copies of the state. We show that additivity of the relative entropy would imply that there are at least two inequivalent types of asymptotic tripartite entanglement. The methods used include the application of some useful lemmas that enable us to analytically calculate the relative entropy for some classes of bipartite states.
We calculate the relative entropy between two quantum Gaussian states with given correlation matrices by demonstrating a practical method of transforming one of the correlation matrices into an exponential quadratic operator matrix. We show that the closest Gaussian separable state achieving the Gaussian relative entropy of entanglement is at the border between separable and inseparable Gaussian state sets. For a two-mode Gaussian state, the calculation of the Gaussian relative entropy of entanglement is greatly simplified by deducing a matrix with ten undetermined parameters to one with only three. The two-mode Gaussian states are classified into four types. Numerical calculations strongly suggest that the Gaussian relative entropy of entanglement for each type is realized by a Gaussian separable state within the same type. For a symmetric Gaussian state it is strictly proven that the Gaussian relative entropy of entanglement is achieved by a symmetric Gaussian separable state.
Abstract On the one hand we present necessary and sufficient conditions for both the separability and the distillability of bipartite Gaussian states of an arbitrary number of modes. On the other hand we classify tripartite three‐mode Gaussian states according to their separability properties and provide necessary and sufficient conditions for separability of those states . All the conditions presented here can be easily checked by direct calculations, thus providing operational criteria. This solves the separability problem for all bipartite as well as tripartite three‐mode Gaussian states and the distillability problem for all bipartite Gaussian states.
We determine the relative entropy of entanglement of a bipartite mixed state ρABρ_{AB}ρAB​ obtained by tracing out one subsystem of a tripartite Haar-random pure state ∣ψ⟩ABC|ψ\rangle_{ABC}∣ψ⟩ABC​, finding ER(ρAB)=log⁡dAdBmax⁡(dA,dB,dC)+O(1)E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)ER​(ρAB​)=logmax(dA​,dB​,dC​)dA​dB​​+O(1). Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation EF(ρAB)E_F(ρ_{AB})EF​(ρAB​) and the mutual information I(A:B)I(A:B)I(A:B). The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.