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A necessary and sufficient criterion of separability for multipartite quantum states

Shengjun Wu, Yongde Zhang

Published Jun 13, 2000• 1 citation
OpenAlex

Abstract

We present a necessary and sufficient criterion for the separability of multipartite quantum states, this criterion also tell us how to write a separable multipartite state as a convex sum of separable pure states. To work out this criterion, we need to solve a set of equations, it is easy to solve these quations analytically if the density matrix of the given quantum state has few nonzero eigenvalues.

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We present a necessary and sufficient criterion for the separability of multipartite quantum states, this criterion also tell us how to write a separable multipartite state as a convex sum of separable pure states. To work out this criterion, we need to solve a set of equations, it is easy to solve these quations analytically if the density matrix of the given quantum state has few nonzero eigenvalues.
Recently, a new and powerful separability criterion was introduced in [O. Rudolph, quant-ph/0202121] and [Chen {\it et al.}, quant-ph/0205017]. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely triparite entanglement. We also show that for multipartite system any permutation of indices of density matrix leads to separability criterion.
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given mmm-partite subsystem with mmm qubits and the purity of states of this mmm-partite subsystem for a pure state in multipartite systems with arbitrary finite n(>m)n(>m)n(>m) qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite systems.
The kkk-separability of mixed nnn-partite (n≥3n\geq 3n≥3) quantum states is mainly discussed in this paper. We obtain practical kkk-separability criteria to identify kkk-nonseparable mixed multipartite states in arbitrary dimensional quantum systems and detect genuine nnn-partite entanglement (k=2k=2k=2) and entanglement (k=3k=3k=3) which have previously not been identified. No optimization or eigenvalue evaluation is needed, our criteria can be evaluated by simple computations involving components of the density matrix and can be implemented in today's experiments.
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.