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

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 $(\Phi \otimes I)\rho \ge 0$ 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.

20 pages

References in corpus (3)

• Experimental quantum teleportation
• Constructing entanglement witnesses for infinite-dimensional systems
• A general scheme for construction of scalar separability criteria from positive maps

Cited by in corpus (7)

• Positive finite rank elementary operators and characterizing entanglement of states
• Operational Entanglement Detection Based on $\Lambda$-Moments
• A geometric characterization of invertible quantum measurement maps
• Entanglement detection beyond the CCNR criterion for infinite-dimensions
• Detection of entanglement via moments of positive maps
• When quantum channel preserves product states
• Entanglement certification from moments of positive maps