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arXiv:1310.7511

A Criterion for the Bipartite Separability of Bell Diagonal States

Ming-Chung Tsai, Po-Chung Chen, Wei-Chi Su, Zheng-Yao Su

Published Oct 28, 2013• 0 citations
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Abstract

A decomposition form is introduced in this report to establish a criterion for the bi-partite separability of Bell diagonal states. A such criterion takes a quadratic form of the coefficients of a given Bell diagonal states and can be derived via a simple algorithmic calculation of its invariants. In addition, the criterion can be extended to a quantum system of higher dimension.

DOI
10.48550/arxiv.1310.7511

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A decomposition form is introduced in this report to establish a criterion for the bi-partite separability of Bell diagonal states. A such criterion takes a quadratic form of the coefficients of a given Bell diagonal states and can be derived via a simple algorithmic calculation of its invariants. In addition, the criterion can be extended to a quantum system of higher dimension.
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 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.
We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in C d ⊗ C d (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for d ≤ 4 . Furthermore, for d ≥ 5 , we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on N -partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.