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arXiv:1405.3415 · Russian Journal of Mathematical Physics

On positive maps in quantum information

W. A. Majewski

Published Jul 1, 2014• 8 citations
arXivHTMLPDF

Abstract

Using the Grothendieck approach to the tensor product of locally convex spaces, we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.
DOI
10.1134/s106192081403008x

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We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send kkk-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to kkk. They also send kkk-positive maps onto kkk-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between kkk-positivity and Schmidt numbers ≤k\le k≤k. They also retain the duality between kkk-superpositivity and kkk-block-positivity.
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A natural and intrinsic characterization of the structure of the set C\mathfrak{C}C of positive unital maps is given, i.e. it is shown that C\mathfrak{C}C is isometrically isomorphic to the subset $\gD$ of bp-positive density matrices endowed with the geometry given by the norm α\alphaα dual to the Grothendieck projective norm π\piπ, 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 α\alphaα is crucial.
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