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arXiv · 1605.02219

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

Marcin Marciniak·Adam Rutkowski
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Abstract

For two positive maps $ϕ_i:B(\mathcal{K}_i)\to B(\mathcal{H}_i)$, $i=1,2$, we construct a new linear map $ϕ:B(\mathcal{H})\to B(\mathcal{K})$, 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 $ϕ_1$ and $ϕ_2$. We discuss properties of this construction. In particular, we provide conditions for positivity of $ϕ$, 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.

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BibTeXRIS

Marcin Marciniak, Adam Rutkowski. 2016-09-20. Merging of positive maps: a construction of various classes of positive maps on matrix algebras. https://doi.org/10.1016/j.laa.2017.04.026

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