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.