On the structure of positive maps; finite dimensional case

Details On the structure of positive maps; finite dimensional case On the structure of positive maps; finite dimensional case

2010-05-21

arxiv.org/abs/1005.3949v3

J. Math. Phys. 53, 023515(2012)

Physics

Mathematical Physics

I would be grateful for any comment. Copyright 2012 American Institute of Physics. This article may be downloaded for personal

Scientific paper

10.1063/1.3685617

A natural and intrinsic characterization of the structure of the set $\mathfrak{C}$ of positive unital maps is given, i.e. it is shown that $\mathfrak{C}$ is isometrically isomorphic to the subset $\gD$ of bp-positive density matrices endowed with the geometry given by the norm $\alpha$ dual to the Grothendieck projective norm $\pi$, the structure of $\gD$ is determined by the set of its exposed points, and finally a characterization of exposed points of $\gD$ 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 $\alpha$ is crucial.

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