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

TL;DR

This paper introduces a method to construct new positive maps on matrix algebras by merging existing maps, providing conditions for their positivity and revealing new classes of exposed positive maps with applications to entangled PPT states.

Contribution

It presents a novel merging construction for positive maps, establishing conditions for various positivity properties and identifying new classes of exposed positive maps.

Findings

Nondecomposable merging of 2-positive and 2-copositive maps

Canonical merging of extremal maps yields exposed positive maps

Application to constructing entangled PPT states

Abstract

For two positive maps $\phi_i:B(\mathcal{K}_i)\to B(\mathcal{H}_i)$, $i=1,2$, we construct a new linear map $\phi:B(\mathcal{H})\to B(\mathcal{K})$, 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 $\phi_1$ and $\phi_2$. We discuss properties of this construction. In particular, we provide conditions for positivity of $\phi$, 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.