Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria |
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Authors: | Micha? Horodecki Pawe? Horodecki Ryszard Horodecki |
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Affiliation: | (1) Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland;(2) Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, Poland |
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Abstract: | 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.
An early version of the present paper has appeared in e-print archive as quant-ph/0206008. The premutation criterion has been
then further developed in 23, 24, 25], where the problem of classification of inequivalent permutation criteria has been
investigated. |
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