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arXiv:quant-ph/0306041 · Physical Review A

Test for entanglement using physically observable witness operators and positive maps

Kai Chen, Ling-An Wu

Published Feb 20, 2004• 50 citations
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Abstract

Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but noncompletely positive maps of arbitrary high dimensions, which provide a much better test than the witness operators themselves. Moreover, we find that there are two types of positive maps that can detect $2\ifmmode\times\else\texttimes\fi{}N$ and $4\ifmmode\times\else\texttimes\fi{}N$ bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our construction could be of great significance for future experiments.

DOI
10.1103/physreva.69.022312

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Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but noncompletely positive maps of arbitrary high dimensions, which provide a much better test than the witness operators themselves. Moreover, we find that there are two types of positive maps that can detect 2\ifmmode×\else\texttimes\fiN2\ifmmode\times\else\texttimes\fi{}N2\ifmmode×\else\texttimes\fiN and 4\ifmmode×\else\texttimes\fiN4\ifmmode\times\else\texttimes\fi{}N4\ifmmode×\else\texttimes\fiN bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our construction could be of great significance for future experiments.
We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of a 3x3 bound entangled state [P. Horodecki, Phys. Lett. A 232, 333 (1997)] is explicitly constructed and a family of dxd bound entangled states is introduced, whose entanglement can be detected by permuting local orthogonal observables. The proposed criterion of separability can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables.
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We develop a strong and computationally simple entanglement criterion. The criterion is based on an elementary positive map Phi which operates on state spaces with even dimension N > or = 4. It is shown that Phi detects many entangled states with a positive partial transposition (PPT) and that it leads to a class of optimal entanglement witnesses. This implies that there are no other witnesses which can detect more entangled PPT states. The map Phi yields a systematic method for the explicit construction of high-dimensional manifolds of bound entangled states.
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We propose a family of entanglement witnesses. These witnesses are constructed by using local orthogonal observables, and therefore can be easily measured by means of local measurements and classical communications. From the Jamiokowski isomorphism we obtain the corresponding positive maps that are not completely positive. These maps lead to a new separability criterion——the O-reduction criterion, which includes the reduction criterion and the realignment criterion as special cases. In addition, we find that the O-reduction criterion can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables. As applications, we construct an explicit entanglement witness for the first bound entangled state found by Horodecki in 1997,and as well introduce a family of dd bound entangled states, whose entanglement can be detected by permuting local orthogonal observables.