Separability of Mixed States: Necessary and Sufficient Conditions
  We provide necessary and sufficient conditions for separability of mixed states. As a result we obtain a simple criterion of separability for $2\times2$ and $2\times3$ systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, it is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered. 
Michal Horodecki, Pawel Horodecki and Ryszard Horodecki