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arXiv:quant-ph/0005112 · Physical Review A

Characterization of separable states and entanglement witnesses

Maciej Lewenstein, Barbara Kraus, Paweł Horodecki, J. I. Cirac

Published Mar 20, 2001• 181 citations
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Abstract

We provide a canonical form of mixed states in bipartite quantum systems in terms of a convex combination of a separable state and a so-called edge state. We construct entanglement witnesses for all edge states, which allows us to introduce a canonical form of nondecomposable entanglement witnesses and the corresponding positive maps. We also present a nontrivial necessary condition for entanglement witnesses and positive maps to be extremal.

DOI
10.1103/physreva.63.044304

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We provide a canonical form of mixed states in bipartite quantum systems in terms of a convex combination of a separable state and a so-called edge state. We construct entanglement witnesses for all edge states, which allows us to introduce a canonical form of nondecomposable entanglement witnesses and the corresponding positive maps. We also present a nontrivial necessary condition for entanglement witnesses and positive maps to be extremal.
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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.
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