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arXiv:1703.07586

Mapping cones and separable states

Erling Størmer

Published Mar 22, 2017• 0 citations
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Abstract

We study mapping cones and their dual cones of positive maps of the n by n matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular the composition of a map from the cone with a map in the dual cone is super-positive, and so the natural state it defines is separable.

DOI
10.48550/arxiv.1703.07586

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We study mapping cones and their dual cones of positive maps of the n by n matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular the composition of a map from the cone with a map in the dual cone is super-positive, and so the natural state it defines is separable.
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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( ) and ℬ( ), where and 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( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.
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Let and let be the n-dimensional second-order cone, or Lorentz cone. A linear map M from to is called positive if M[Lm]⊂Ln . The set of positive maps forms a convex cone, the positive cone. Its dual cone, the separable cone, is the convex hull of tensor products x⊗y, where x∈Ln , y∈Lm . The structure of the positive and the separable cones is the subject of the present contribution. We study the action of the automorphism group of the positive cone and provide a complete classification of its orbits by constructing explicitly canonical forms. This enables us to identify and to classify the extreme rays of the positive cone, as well as their dual counterparts, the largest faces of the separable cone. We also characterize positive maps which are scalable to doubly stochastic ones.