Cones of positive maps and their duality relations

Abstract

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 Jamio lkowski-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.

Interpolation by completely positive maps

“…. , F r in the last assertion of the theorem, we see that each F j has rank at most = min{m, n} so that the corresponding completely positive map Φ is a super -positive map [18].…”

How often is a random quantum statek-entangled?

“…This turns out to be equivalent to the existence of a Kraus representation (ρ) = i A † i ρA i , of such that all the operators A i are of rank smaller than or equal to k [21]. We denote The cross section of the cones with the hyperplane corresponding to the trace-preserving condition yields a sequence of nested convex bodies, P TP k , the volumes of which we aim to estimate.…”

“….g., [21]). Thus, we can identify the cone P k (M d ) via the Jamiołkowski-Choi isomorphism → C with the cone of k-block positive operators on…”

“…Furthermore, the (larger) set of k-positive maps is the dual of the set of k-superpositive maps (see section 3 for details and fine points). These maps, also called k-entanglement breaking channels, correspond by the Choi-Jamiołkowski isomorphism to k-entangled states of a bipartite system [21,25]. Thus, investigating relations between sets of maps of different degree of positivity, one can establish the properties of the subsets of M d 2 characterized by different classes of quantum entanglement [21,26], and vice versa.…”

“…In general, the characterization of a set of k-positive maps is not easy [20,21], and the geometry of the set of maps acting on M d is nontrivial even in the simplest case of d = 2 [2,22].…”