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International Joint Conference on Computational Sciences and Optimization (CSO)

An O(N) Algorithm of Separability for Two-Partite Arbitrarily Dimensional Pure States

Yinxiang Long, Daowen Qiu, Dongyang Long

Published Apr 1, 2009• 2 citations
Publisher

Abstract

How to discriminate entanglement state and separable state rapidly is a key task in quantum information theory. In this paper, we will give a simple separability criterion and a fast algorithm for bipartite pure state systems based on the two order minors of the coefficient matrices of quantum states. By our algorithm, it only needs at most O(d) times operations of multiplication and comparison to judge separability for two-partite pure states in a d dimensional Hilbert space. Furthermore, our algorithm can be easily generalized to multi-partite system. For n-partite pure states with dimension d, our algorithm only needs at most O(d\ln(d)) times operations of multiplication and comparison.

DOI
10.1109/cso.2009.339

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How to discriminate entanglement state and separable state rapidly is a key task in quantum information theory. In this paper, we will give a simple separability criterion and a fast algorithm for bipartite pure state systems based on the two order minors of the coefficient matrices of quantum states. By our algorithm, it only needs at most O(d) times operations of multiplication and comparison to judge separability for two-partite pure states in a d dimensional Hilbert space. Furthermore, our algorithm can be easily generalized to multi-partite system. For n-partite pure states with dimension d, our algorithm only needs at most O(d\ln(d)) times operations of multiplication and comparison.
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.
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.
In this paper, we give out some effective criterions which can be used to judge the separability of multipartite pure states. We obtain the relationship between separability and Schmidt decomposable of multipartite pure states in Theorem1. The first criterion derived from Theorem2 dose not need the Schmidt decomposition which is hard to find for multipartite states. Theorem3 is more profound which can be used to deduce Corollary1 which is one of the main results in [1]. Finally, we give out an algorithm which can be used to judge the separability of multipartite pure states effectively.
It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.