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MATHEMATICA SCANDINAVICA

Mapping cones of positive maps

Erling Størmer

Published Jun 1, 2011• 13 citations
Publisher

Abstract

We study mapping cones of positive maps of $B(H)$ into itself, i.e., cones which are closed under composition with completely positive maps. As applications we obtain characterizations of linear functionals with strong positivity properties with respect to so-called symmetric mapping cones, with special emphasis on separable and PPT states.

DOI
10.7146/math.scand.a-15168

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STØRMERWe show that each positive map from B(K ) to B(H) is a scalar multiple of a map of the form Trψ with ψ completely positive.This is used to give necessary and sufficient conditions for maps to be Ꮿ-positive for a large class of mapping cones; in particular, we apply the results to k-positive maps.P(B(K ), Ꮿ) = {a ∈ B(K ⊗ H ) : ι ⊗ α(a) ≥ 0 for all α ∈ Ꮿ}, MSC2010: 46L60, 46L99.

arXiv:1009.5809•Pacific Journal of Mathematics•Oct 29, 2011•6 citations
We study mapping cones of positive maps of B(H)B(H)B(H) into itself, i.e., cones which are closed under composition with completely positive maps. As applications we obtain characterizations of linear functionals with strong positivity properties with respect to so-called symmetric mapping cones, with special emphasis on separable and PPT states.
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 Jamio̷lkowski-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.
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-entangled operators due to the Jamiołkowski–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.