The relation between positive maps on Mn and subsets of self-adjoint elements from Mn ⊗ Mn is investigated. It has been shown that there exists the isomorphism between the set of all positive normalized maps on Mn and the subset of separable states on Mn ⊗ Mn.
Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras ℬ( ) and ℬ( ), where and are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on ℬ( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.
Using the natural duality between linear functionals on tensor products of C∗-algebras with the trace class operators on a Hilbert space H and linear maps of the C∗-algebra into B(H), we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF C∗-algebras.