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arXiv:1404.7449 · Physical Review Letters

Witnessing Genuine Multipartite Entanglement with Positive Maps

Marcus Huber, Ritabrata Sengupta

Published Sep 2, 2014• 105 citations
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Abstract

We derive a general framework that lifts any set of bipartite to multipartite entanglement witnesses and we show how positive maps can naturally be incorporated into this framework. We show that some previous approaches for multipartite entanglement detection are intimately connected to the witnesses derived from partial transposition and that such criteria can easily be outperformed in higher dimensions by nondecomposable maps. As an exemplary case we present a witness that is capable of detecting genuine multipartite entanglement in bound entangled states.

DOI
10.1103/physrevlett.113.100501

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We derive a general framework that lifts any set of bipartite to multipartite entanglement witnesses and we show how positive maps can naturally be incorporated into this framework. We show that some previous approaches for multipartite entanglement detection are intimately connected to the witnesses derived from partial transposition and that such criteria can easily be outperformed in higher dimensions by nondecomposable maps. As an exemplary case we present a witness that is capable of detecting genuine multipartite entanglement in bound entangled states.
Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of biseparable states but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitly construct examples of multipartite non-positive maps, obtained from positive maps via a lifting procedure, that in this fashion can reveal genuine multipartite entanglement in a robust way.
We derive a general framework to identify genuinely multipartite entangled mixed quantum states in arbitrary-dimensional systems and show in exemplary cases that the constructed criteria are stronger than those previously known. Our criteria are simple functions of the given quantum state and detect genuine multipartite entanglement that had not been identified so far. They are experimentally accessible without quantum state tomography and are easily computable as no optimization or eigenvalue evaluation is needed.
Abstract The problem of entanglement detection is a long standing problem in quantum information theory. One of the primary procedures of detecting entanglement is to find the suitable positive but non-completely positive maps. Here we try to give a generic prescription to construct a positive map that can be useful for such scenarios. We study a class of positive maps arising from Lindblad structures. We show that two famous positive maps viz. transposition, reduction map and Choi map can be obtained as a special case of a class of positive maps having Lindblad structure. Generalizing the transposition map to a one parameter family we have used it to detect genuine multipartite entanglement. Finally being motivated by the negativity of entanglement, we have defined a similar measure for genuine multipartite entanglement.
We introduce a general framework for detecting genuine multipartite entanglement and non-full-separability in multipartite quantum systems of arbitrary dimensions based on correlation tensors. Regarding genuine multipartite entanglement, our conditions are comparable to those of previous approaches in the case of qubits, while they show particular strength in the relatively unexplored case of higher-dimensional systems. In the case of non-full-separability, our conditions prove to be advantageous in situations where more than two-body correlations are relevant, where most previous conditions turned out to be weak. Moreover, they allow for the detection of fully bound entangled states. Finally, we also discuss experimentally friendly ways of implementing our conditions, which are based on directly measurable quantities.
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.
Abstract We present a formalism to detect genuine multipartite entanglement by considering projection map which is a positive but not completely positive map. Projection map has been motivated by the no-pancake theorem which repudiates the existence of a quantum operation that maps the Bloch sphere onto a disk along its equator. The not-complete positivity feature of projection map is explored to investigate genuine multipartite entanglement in arbitrary N-qubit quantum systems. Our proposed framework can detect some important classes of genuinely entangled states in tripartite and quadripartite scenarios. We provide illustrative example to show the efficacy of our formalism to detect a class of tripartite PPT bound entangled states. Finally, we construct a suitable witness operator based on projection map to certify genuine tripartite entanglement, which is likely to be feasible experimentally.