## A characterization of positive linear maps and criteria of entanglement for quantum states.(English)Zbl 1198.81055

Summary: 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 to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion for separability is given which shows that a state $$\rho$$ on $$H \otimes K$$ is separable if and only if ($$\Phi \otimes I)\rho \geqslant 0$$ 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.

### MSC:

 81P40 Quantum coherence, entanglement, quantum correlations 81P15 Quantum measurement theory, state operations, state preparations
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### References:

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