Cones of positive maps and their duality relations

Details Cones of positive maps and their duality relations Cones of positive maps and their duality relations

2009-02-27

arxiv.org/abs/0902.4877v1

J. Math. Phys. 50, 062106 (2009)

Physics

Quantum Physics

22 pages, 3 figures

Scientific paper

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamiolkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.

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