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arXiv:0903.2320

Witness for an entangled state with positive partial transpose

Nirman Ganguly, Binayak Samadder Choudhury

Published Mar 13, 2009• 0 citations
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Abstract

In this letter we have analyzed an entangled state in $C^{3} \bigotimes C^{3}$ having a positive partial transposition and have shown that it is an edge state. Further we have constructed explicitly a witness operator $W$ which detects the entanglement.

DOI
10.48550/arxiv.0903.2320

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In this letter we have analyzed an entangled state in C3⨂C3C^{3} \bigotimes C^{3}C3⨂C3 having a positive partial transposition and have shown that it is an edge state. Further we have constructed explicitly a witness operator WWW which detects the entanglement.
Edge states lying at the edge of positive-partial-transpose- (PPT-) entangled states have a very intriguing existence and their detection is equally interesting. We present here a witness for detection of edge states. We then compare between our proposed witness operator and the witness operator proposed by Lewenstein et al. [Phys. Rev. A 62, 052310 (2000)] in terms of the efficiency in the detection of PPT entangled states. In this regard we show that this operator is finer than the operator of Lewenstein et al. in some restrictions. We also discuss about its experimental realization via Gell-Mann matrices.
An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.
Recently, Qi and Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated with indecomposable positive linear maps, but whether partial transposes of these witnesses are also optimal has not been evaluated. We show that partial transposes of these entanglement witnesses have the spanning property and that these witnesses are indeed optimal witnesses detecting positive-partial- transposed entangled states (nondecomposable optimal entanglement witnesses).
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 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.