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O-reduction criterion for the separability of quantum states

Yu Si

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Entanglement detection and estimation are fundamental problems in quantum information science. Compared with discrete-variable states, for which lots of efficient entanglement detection criteria and lower bounds of entanglement measures have been proposed, the continuous-variable entanglement is much less understood. Here we shall present a family of entanglement witnesses based on continuous-variable local orthogonal observables (CVLOOs) to detect and estimate entanglement of Gaussian and non-Gaussian states, especially for bound entangled states. By choosing an optimal set of CVLOOs, our entanglement witness is equivalent to the realignment criterion and can be used to detect bound entanglement of a class of 2+2 mode Gaussian states. Via our entanglement witness, lower bounds of two typical entanglement measures for arbitrary two-mode continuous-variable states are provided.
Recently, a toolkit of highly symmetric techniques employing matrix inequalities has been developed to detect entanglement in various ways. Here we unifiedly explain in detail these methods, and expand them to a new family of positive maps with further detection capabilities. In the simplest case, we generalize the reduction map to detect more generic states using both multiple copies and local filters. Through the Choi-Jamiołkowski isomorphism, this family of maps leads to a construction of multipartite entanglement witnesses. Discussions and examples are provided regarding the detection of states with local positive partial transposition and the use of multiple copies.
Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.
Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analog of permutation separability criteria in the even-partite systems.