On the Necessary and Sufficient Conditions for Separability of Quantum Mixed States

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.