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Journal of Xi'an Technological University

Separability Criterion of Bipartite Qubit States

BI Hong-mei

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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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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.
The “separability problem” in quantum information theory is a quite important and well-known hard problem. The low-dimensional system satisfies the PPT criterion. However, the high-dimensional system problem has been shown to be NP-hard problem. In general, it is very difficult to find the analytic solution of the density matrix for the high-dimensional system. Therefore, getting an analytic solution for two-qubit system is an interesting and useful problem. We propose a novel criterion for separability and entanglement-verification of two-qubit system. We expressed the density matrix by a sum of a principal density matrix and six separable density matrices. The necessary and sufficient conditions for the two-qubit system include that if the four involved coefficients [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and the principal density matrix [Formula: see text] are separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, our criterion results in a totally different conclusion compared to Horodecki’s criterion. We believe that the new criterion is more stringent than existing PPT methods.
In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.