Program:
1. Basic notions: positivty, k-(co)positivity, complete (co)positivity, decomposability
2. Basic properties: relations between k- and l- positivity for different k,l; case when domain or predomain is commutative; Stinespring form of completely positive maps
3. Stormer-Kye duality: convex structures; duality between positive maps and states on tensor product
4. Characterization of dual sets to positive, decomposable and completely positive maps as separable, PPT and all states; concept of an entanglement witness
5. Choi matrix method: Choi's theorem on characterization of completely positive maps on matrix algebras, Kraus form; Choi matrix vs duality
6. Problem of classification of positive maps: low dimensional case (Stormer, Woronowicz); examples in higher dimensional algebras; extremal and exposed maps, Straszecicz theorem; optimality
All lectures are elemetary and instroductive!
