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On a characterization of positive maps. (English) Zbl 0987.46047
Summary: Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras $ℬ\left(ℋ\right)$ and $ℬ\left(𝒦\right)$, where $ℋ$ and $𝒦$ are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on $ℬ\left(ℋ\otimes 𝒦\right)$. Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone $ℬ{\left(ℋ\right)}^{+}\otimes ℬ{\left(𝒦\right)}^{+}$ is given.
##### MSC:
 46L60 Applications of selfadjoint operator algebras to physics 47L90 Applications of operator algebras to physics 15A30 Algebraic systems of matrices 81P15 Quantum measurement theory 46L30 States of ${C}^{*}$-algebras