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arXiv:2308.07019 · Scientific Reports

A class of entanglement witnesses and a realignment-like criterion

Vahid Jannesary, Vahid Karimipour, Dariusz Chruściński

Published Feb 17, 2025• 2 citations
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Abstract

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.

DOI
10.1038/s41598-025-90528-9

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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.
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.
Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.
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.