The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive Maps

Abstract

Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and more generally as witnesses for states having Schmidt number not greater than k. It is known that such witnesses are related to k-positive maps. In this article we propose a new proof for the correspondence between vectors having Schmidt number k and k-positive maps using Jamio lkowski's criterion for positivity of linear maps; to this aim, we also investigate the precise notion of the term "Jamio lkowski isomorphism". As consequences of our proof we get the Jamio lkowski criterion for complete positivity, and we find a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.

Cones of positive maps and their duality relations

“…By pointing out that the extreme points of the set of unital witnesses are optimal, we tried to spill the idea that future efforts could concentrate on witnesses which are not only optimal, but also extreme. 6 Within this paper several results by other authors [4,26,30,38,49,53] appear as special cases of general theorems. Presented in the way we did it, they start to reveal a mathematical structure of a certain degree of generality.…”

“…Proposition 2.2 appears in the early work by Takasaki and Tomiyama, [10] (it was also proved in [53] using different methods). Thus we have found the B (H ⊗ H) counterparts of the sets P k (H).…”

“…This is nothing else as the definition of an operator with the Schmidt number equal to ksee [38,39,53].…”

Hierarchy of non-Markovianity andk-divisibility phase diagram of quantum processes in open systems

“…To ensure that one can always obtain a legitimate quantum state in the presence the such nonclassical correlations, the notion of PP must be generalized to a series of k positivity: A TP map E is said to be k positive if I k ⊗ E : M k ⊗ A → M k ⊗ A is PP and CP if E is k positive for all positive integers k, where I k is the identity map acting on the k × k matrix algebra M k . Although the structure of CP maps have been studied thoroughly [25,26], there is still no efficient criterion for determining whether a map is k positive or not [33][34][35].…”

How often is a random quantum statek-entangled?

“…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.…”