arXiv:1009.5809 · Pacific Journal of Mathematics

A completely positive map associated with a positive map

Erling Størmer

Published Oct 29, 2011• 6 citations

Abstract

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.

DOI: 10.2140/pjm.2011.252.487

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arXiv:1009.5809 · Pacific Journal of Mathematics

A completely positive map associated with a positive map

Erling Størmer

Published Oct 29, 2011• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.2140/pjm.2011.252.487

Related papers

Mapping cones of positive maps

Erling Størmer

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.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.

MATHEMATICA SCANDINAVICAJun 1, 201113 citations

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Erling Størmer

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Kennett L. Dela Rosa, Iana Angela C. Fajardo, Cedie B. Reyes

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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.

Given

α\in \mathbb R

and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on

α

and

B_1,\ldots,B_q

are given that guarantee

Φ_{α,\mathfrak{B}}^\pm

k

-positive,

k

-copositive, and

k

-PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.