  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 bi-separable 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.
