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arXiv:1005.5530 · Physical Review A

Constructing entanglement witnesses for infinite-dimensional systems

Jinchuan Hou, Xiaofei Qi

Published Jun 30, 2010• 35 citations
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Abstract

It is shown that every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $\ensuremath{\alpha}I+T$ with $\ensuremath{\alpha}$ a non-negative number and $T$ a finite rank self-adjoint operator. We also provide two methods of constructing an entanglement witness and apply them to obtain some entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion.

DOI
10.1103/physreva.81.062351

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It is shown that every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form \ensuremathαI+T\ensuremath{\alpha}I+T\ensuremathαI+T with \ensuremathα\ensuremath{\alpha}\ensuremathα a non-negative number and TTT a finite rank self-adjoint operator. We also provide two methods of constructing an entanglement witness and apply them to obtain some entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion.
We study the structure of k-Schmidt witnesses in infinite dimensions. We show that for any states with Schmidt number greater than , there exists a -Schmidt witnesses of the form to detect it, where and is a finite rank self-adjoint operator. We provide a method to construct -Schmidt witnesses with these forms. An elementary operator criterion to determine the lower bound of the Schmidt number of a state is obtained. We also construct a class of 2-Schmidt witnesses, from which we can detect the entanglement of states by entries of their matrix representations.
Entanglement detection is one of the important problems in quantum information theory. To deal with this problem, many entanglement detection criteria have been proposed. Among the proposed criteria, the detection of entanglement through witness operator (also known as linear entanglement witness (LEW) operator) may be considered as the most practical. Although the witness operator approach to detect entanglement is experimentally friendly, the construction of these operators is not a very simple task. Even if we are able to construct a LEW operator, our problem is not solved as it may either detect a few entangled states or not a single entangled state from a given family of entangled states. Thus, we need a constructive approach in order to tackle this type of problem. In this work, we provide a few constructions of the non-linear entanglement witnesses (NLEW) for d1⊗d2d_1\otimes d_2d1​⊗d2​ dimensional system from any linear entanglement witness (LEW) operator. The advantage of these constructions is that, if a LEW is unable to detect any particular entangled state described by the density operator ρentρ^{ent}ρent then our construction of NLEW may detect the same entangled state ρentρ^{ent}ρent. Further, we have constructed NLEW operator that may detect not only a class of bipartite negative partial transpose entangled state (NPTES), but also positive partial transpose entangled state (PPTES). Moreover, we have shown that the constructed NLEW operators may be decomposed in terms of the tensor product of local observables and hence may be realizable in an experiment.
We present a multi-parameter family of positive maps between spaces of arbitrary but finite dimensions. This framework facilitates the construction of Entanglement Witnesses (EWs) specifically designed for systems living in [Formula: see text] dimensions. A sufficient condition for positivity is presented. Interestingly, it is shown that all EWs constructed this way are equivalent to a single realignment-like criterion which for [Formula: see text] and [Formula: see text] are different is in general stronger than the original realignment criterion. We illustrate effectiveness of this criterion considering examples of Positive Partial Transpose entangled states in [Formula: see text] and [Formula: see text] dimensions.
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.