Merging of positive maps: A construction of various classes of positive maps on matrix algebras

Abstract

For two positive maps φ i : B(K i ) → B(H i ), i = 1, 2, we construct a new linear map φ : B(H) → B(K), where K = K 1 ⊕ K 2 ⊕ C, H = H 1 ⊕ H 2 ⊕ C, by means of some additional ingredients such as operators and functionals. We call it a merging of maps φ 1 and φ 2 . The properties of this construction are discussed. In particular, conditions for positivity of φ, as well as for 2-positivity, complete positivity, optimality and nondecomposability, are provided. In particular, we show that for a pair composed of 2-positive and 2-copositive maps, there is a nondecomposable merging of them. One of our main results asserts, that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps, their canonical merging is an exposed positive map. This result provides a wide class of new examples of exposed positive maps. As an application, new examples of entangled PPT states are described.2010 Mathematics Subject Classification. Primary: 46L60, 15B48, 81P40; Secondary: 81Q10, 46L05.

On the origin of non-decomposable maps

“…The important point to note here is that the form of C * -extreme completely positive map provides a motivation for such studies, see Section 2. For a deeper discussion on this topic we refer the reader to [37], see also [38].…”

Optimality of generalized Choi maps in M 3

“…Moreover, it serves as a powerful tool for characterizing quantum entanglement and hence plays a key role in various aspects of quantum information theory. Although there exist several results regarding the classification of positive maps [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], the complete analysis of positive maps still remains an open question. In this direction, examples of positive linear maps given by Choi [19][20][21] play a prominent role in investigating the structure of the positive cones of positive linear maps.…”

A class of optimal positive maps in $M_n$

“…Equivalently, Φ is positive if Φ(xx † ) ≥ 0 for any x ∈ C n . Positive maps from M n to M m form a convex cone P n,m and the structure of P n,m in spite of the considerable effort is still rather poorly understood (for some recent works see [9][10][11][12][13][14][15][16]). Positive maps play an important role both in physics and mathematics providing generalization of * -homomorphisms, Jordan homomorphisms and conditional expectations.…”