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

DOI10.1016/J.LAA.2017.04.026arXiv1605.02219MaRDI QIDQ6273262

Adam J. Rutkowski, Marcin Marciniak

Publication date: 7 May 2016

Abstract: For two positive maps $ph{i}_i:B(mathcal{K}_i)oB(mathcal{H}_i)$, $i=1,2$, we construct a new linear map $phi:B(mathcalH)oB(mathcalK)$, where $mathcalK=mathcal{K}_{1}oplusmathcal{K}_{2}oplusmathbbC$, $mathcalH=mathcal{H}_{1}oplusmathcal{H}_{2}oplusmathbbC$, by means of some additional ingredients such as operators and functionals. We call it a merging of maps $ph{i}_1$ and $ph{i}_2$. 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.

Mathematics Subject Classification ID

Positive matrices and their generalizations; cones of matrices (15B48) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) General theory of (C^*)-algebras (46L05) Linear operators on ordered spaces (47B60) Quantum coherence, entanglement, quantum correlations (81P40)

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