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.

arXiv comment20 pages

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• Quantum entanglement
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Cited by in corpus (12)

• An Efficient Algorithm for Optimizing Adaptive Quantum Metrology Processes
• Positive finite rank elementary operators and characterizing entanglement of states
• A characterization of optimal entanglement witnesses
• Operational Entanglement Detection Based on $\Lambda$-Moments
• A geometric characterization of invertible quantum measurement maps
• Measurement-induced nonlocality over two-sided projective measurements
• Entanglement detection beyond the CCNR criterion for infinite-dimensions
• Constructing separable states in infinite-dimensional systems by operator matrices
• Detection of entanglement via moments of positive maps
• Entanglement certification from moments of positive maps
• When quantum channel preserves product states
• The CHSH-type inequalities for infinite-dimensional quantum systems