arXiv:quant-ph/0607035

Constructions of indecomposable positive maps based on a new criterion for indecomposability

Hall, William

Published Jul 5, 2006• 3 citations

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: 10.48550/arxiv.quant-ph/0607035

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arXiv:quant-ph/0607035

Constructions of indecomposable positive maps based on a new criterion for indecomposability

Hall, William

Published Jul 5, 2006• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0607035

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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

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.

{M}_{d}(\mathbb{C})

into a difference of two completely positive maps ${\ensuremath{\Lambda}}_{1}$, ${\ensuremath{\Lambda}}_{2}$, i.e., $\ensuremath{\Lambda}={\ensuremath{\Lambda}}_{1}\ensuremath{-}{\ensuremath{\Lambda}}_{2}$. The scheme may also be treated as a generalization of the known entropic inequalities, which are obtained from the reduction map. Analyses performed on a few classes of states show that the scalar criteria are stronger than the entropic inequalities and when derived from indecomposable maps allow for detection of bound entanglement.