A characterization of positive linear maps and criteria of entanglement for quantum states

TL;DR

This paper characterizes positive linear maps and provides criteria for entanglement detection in quantum states, including representations of quantum channels and examples of entangled states beyond standard criteria.

Contribution

It offers a new characterization of positive linear maps, including elementary operators, and introduces a necessary and sufficient separability criterion for quantum states.

Findings

Characterization of positive completely bounded normal linear maps.

Representation of quantum channels for infinite-dimensional systems.

Identification of entangled states not detectable by PPT or realignment criteria.

Abstract

Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ 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 $\rho$ on $H\otimes K$ is separable if and only if $(\Phi\otimes I)\rho\geq 0$ for all positive finite rank elementary operators $\Phi$. 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.