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arXiv:quant-ph/0206008

Separability of mixed quantum states: linear contractions approach

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

Published Jun 2, 2002• 17 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0206008

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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.
Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. 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 tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.
A state acting on Hilbert space {\\cal H}_1\\otimes{\\cal H}_2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2\\times2 and 2\\times3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability.
The reduction criterion is a well-known necessary condition for separable states, and states violating this condition are entangled and also 1-distillable. In this paper we introduce a set of necessary conditions for separability of multipartite states, obtained from a set of positive but not completely positive maps. These conditions can be thought of as generalizations of the reduction criterion to multipartite systems. We use tripartite Werner states as an example to investigate the entanglement detecting powers of some of these conditions, and we also look at what these conditions mean in terms of distillation. Finally, we show that these maps can be used to give a partial solution to the subsystem problem, as described by Butterley et al.
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.
The recent proposed realignment separability criterion for mixed is analyzed. We identify the essential part of this criterion is a swap operator followed by a partial transposition. Then we analyze the separability criterion of permutation indices of the density matrix for multipartite state which is a generalization of the realignment criterion. We give a method to solve the equivalence problem of the separability criterion of permutation indices of the density matrix. For tripartite state, we show the non-trivial separability criteria are either partial transposition criterion or realignment criterion.