Mapping cones and separable states

Erling Størmer

Erling Størmer

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.

Kil-Chan Ha, Seung-Hyeok Kye

We exhibit examples of separable states which are on the boundary of the convex cone generated by all separable states but in the interior of the convex cone generated by all PPT states. We also analyze the geometric structures of the smallest face generated by those examples. As a byproduct, we obtain a large class of entangled states with positive partial transposes.

Kil-Chan Ha, Seung-Hyeok Kye

Erling Størmer

Erling Størmer

This is a revised form of the previous paper in which we study cones of positive maps of B(H) into itself. We add the result that the dual cone of a symmetric mapping cone is itself a symmetric mapping cone. As applications we obtain characterizations of linear functionals with strong positivity conditions with respect to a class of mapping cones called symmetric mapping cones. Applications are given to separable and PPT-states.