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

For two positive maps ${\varphi }_i:B({\mathcal{K}}_i)\to B({\mathcal{H}}_i)$, $i=1,2$, we construct a new linear map $\varphi :B(\mathcal{H})\to B(\mathcal{K})$, where $\mathcal{K}={\mathcal{K}}_1\oplus {\mathcal{K}}_2\oplus \mathbb{C}$, $\mathcal{H}={\mathcal{H}}_1\oplus {\mathcal{H}}_2\oplus \mathbb{C}$, by means of some additional ingredients such as operators and functionals. We call it a merging of maps ${\varphi }_1$ and ${\varphi }_2$. We discuss properties of this construction. In particular, we provide conditions for positivity of $\varphi$, as well as for $2$-positivity, complete positivity and nondecomposability. In particular, we show that for a pair composed of $2$-positive and $2$-copositive maps, there is a nondecomposable 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.