Positive maps, states, entanglement and all that; some old and new problems, e-print quant-ph/0411043

We outline a new approach to the characterization as well as to the classification of positive maps. This approach is based on the facial structures of the set of states and of the cone of positive maps. In particular, the equivalence between Schroedinger’s and Heisenberg’s pictures is reviewed in this more general setting. Furthermore, we discuss in detail the structure of positive maps for two and three dimensional systems. In particular, the explicit form of decomposition of a positive map and the uniqueness of this decomposition for extremal positive maps for 2 dimensional case are described. The difference of the structure of positive maps between 2 dimensional and 3 dimensional cases is clarified. The resulting characterization of positive maps is applied to the study of quantum correlations and entanglement. Mathematical Subject Classification: Primary: 46L53, 46L60: Secondary: 46L45, 46L30