Cones of positive maps and their duality relations

TL;DR

This paper explores the structure and duality of cones of positive and k-positive maps in quantum information theory, providing characterizations and generalizations relevant to entanglement detection.

Contribution

It introduces new characterizations of k-positive and k-superpositive maps, and extends duality relations to mapping cones beyond the Choi isomorphism.

Findings

Characterization of k-positive and k-superpositive maps.

Duality relations between positive maps and their dual cones.

Identification of extreme entanglement witnesses as optimal tools.

Abstract

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamiolkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.