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arXiv:quant-ph/0412220 · Physical Review Letters

Entanglement Detection by Local Orthogonal Observables

Sixia Yu, Nai-Le Liu

Published Oct 7, 2005• 103 citations
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Abstract

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.

DOI
10.1103/physrevlett.95.150504

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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.
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.
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.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {\it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of ddd maximally entangled states in a d⊗dd\otimes dd⊗d system are systematically constructed for all d≥4d\ge 4d≥4 as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of kkk locally indistinguishable maximally entangled states with the ratio k/dk/dk/d approaching 3/4. In particular, in a d⊗dd\otimes dd⊗d system with even d≥6d\ge 6d≥6, there always exist d−1d-1d−1 maximally entangled states that are locally indistinguishable by separable measurements.