Merging of positive maps: A construction of various classes of positive maps on matrix algebras

For two positive maps ϕi:B(Ki)→B(Hi), i=1,2, we construct a new linear map ϕ:B(H)→B(K), where K=K1⊕K2⊕C, H=H1⊕H2⊕C, by means of some additional ingredients such as operators and functionals. We call it a merging of maps ϕ1 and ϕ2. The properties of this construction are discussed. In particular, conditions for positivity of ϕ, as well as for 2-positivity, complete positivity, optimality and indecomposability, are provided. In particular, we show that for a pair composed of 2-positive and 2-copositive maps, there is an indecomposable merging of them. One of our main results asserts, that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps, their canonical merging is an exposed positive map. This result provides a wide class of new examples of exposed positive maps. As an application, new examples of entangled PPT states are described.