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Journal of Physics A Mathematical and General

A new criterion for indecomposability of positive maps

William Hall

Published Oct 24, 2006• 67 citations
Publisher

Abstract

We give a criterion for a positive mapping on the space of operators on a Hilbert space to be indecomposable. We show that this criterion can be applied to two families of positive maps. These families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states.

DOI
10.1088/0305-4470/39/45/020

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We give a criterion for a positive mapping on the space of operators on a Hilbert space to be indecomposable. We show that this criterion can be applied to two families of positive maps. These families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states.
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Let HHH and KKK be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from B(H){\mathcal B}(H)B(H) into B(K){\mathcal B}(K)B(K) is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state ρ\rhoρ on H⊗KH\otimes KH⊗K is separable if and only if (Φ⊗I)ρ≥0(\Phi\otimes I)\rho\geq 0(Φ⊗I)ρ≥0 for all positive finite rank elementary operators Φ\PhiΦ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.
We present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres‐Horodecki separability criterion. Their inseparability is demonstrated using Choi‐type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.