Separability of mixed states: necessary and sufficient conditions

Abstract We provide necessary and sufficient conditions for the separability of mixed states. As a result we obtain a simple criterion of the separability for 2 × 2 and 2 × 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, this is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered.

• Publication year

  1996

• Venue

  Physics Letters A

• Publication date

  1996-05-28

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1016/S0375-9601(96)00706-2arXiv quant-ph/9605038
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Separability Criterion for Density Matrices.

  1996influential reference
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