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arXiv:1212.0082 · Chinese Physics Letters

Separability Criterion for Bipartite States and Its Generalization to Multipartite Systems

Jie‐Hui Huang, Liyun Hu, Lei Wang, Shi‐Yao Zhu

Published Apr 1, 2014• 2 citations
arXivHTMLPDF

Abstract

A group of symmetric operators are introduced to carry out the separability criterion for bipartite and multipartite quantum states. All the symmetric operators, represented by a symmetric matrix with only two nonzero elements, and their arbitrary linear combinations are found to be entanglement witnesses. By using these symmetric operators, Wootters' separability criterion for two-qubit states can be generalized to bipartite and multipartite systems in arbitrary dimensions.

DOI
10.1088/0256-307x/31/4/040303

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A group of symmetric operators are introduced to carry out the separability criterion for bipartite and multipartite quantum states. All the symmetric operators, represented by a symmetric matrix with only two nonzero elements, and their arbitrary linear combinations are found to be entanglement witnesses. By using these symmetric operators, Wootters' separability criterion for two-qubit states can be generalized to bipartite and multipartite systems in arbitrary dimensions.
Generally,to decide a given quantum state to be entangled or not,or to be useful for quantum information processing purposes is a difficult question.In this note,the separability of quantum states in bipartite quantum systems is investigated.According to the analysis of the expanded form of density matrix in the separable quantum state and the relationship of expansion coefficient,a new separability criterion of bipartite quantum states is obtained: if a quantum state in a bipartite quantum system of arbitrary dimension is separable,the expansion coefficient satisfies certain inequality.This method requires only the dimensions of systems and the expansion coefficient of density matrix which can be easily got,and the computation is easy,so it has its application value.
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We study the genuine multipartite entanglement of arbitrary n-partite quantum states by representing the density matrices in terms of the generalized Pauli operators. We introduce a general framework for detecting genuine multipartite entanglement and non full-separability of multipartite quantum states with arbitrary dimensions based on correlation tensors. Effective criterion is derived to verify the genuine multipartite entanglement. Detailed examples are given to show that the criterion detects more genuine multipartite entanglement than the existing criteria.
We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.