Witnessing Genuine Multipartite Entanglement with Positive Maps

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.

Figure 1

Here we illustrate the detection power of two different maps for the state, Eq. (18). The regions are labeled according to which criterion detects the state for these values of $p$ and $q$ to be genuinely multipartite entangled (PPT refers to partial transposition and Choi to Choi’s map).