On the structure of positive maps: Finite-dimensional case

Abstract

A natural and intrinsic characterization of the structure of the set \documentclass[12pt]{minimal}\begin{document}$\mathfrak {C}$\end{document}C of positive unital maps is given, i.e., it is shown that \documentclass[12pt]{minimal}\begin{document}$\mathfrak {C}$\end{document}C is isometrically isomorphic to the subset \documentclass[12pt]{minimal}\begin{document}${\mathfrak D}$\end{document}D of bp-positive density matrices endowed with the geometry given by the norm α dual to the Grothendieck projective norm π, the structure of \documentclass[12pt]{minimal}\begin{document}${\mathfrak D}$\end{document}D is determined by the set of its exposed points, and finally a characterization of exposed points of \documentclass[12pt]{minimal}\begin{document}${\mathfrak D}$\end{document}D in terms of convex analysis is presented. This seems to be an answer to an old open problem, characterization of the structure of the set of positive maps, studied both in quantum information and operator algebras. Our arguments are based on the concept of exposed points and links between tensor products and mapping spaces. It should be emphasized that the concept of exposed point depends strongly on the geometry; hence, the role of the norm α is crucial.