  The reduction criterion is a well known necessary condition for separable
states, and states violating this condition are entangled and also
1-distillable. In this paper we introduce a new set of necessary conditions for
separability of multipartite states, obtained from a set of positive but not
completely positive maps. These conditions can be thought of as generalisations
of the reduction criterion to multipartite systems. We use tripartite Werner
states as an example to investigate the entanglement detecting powers of some
of these new conditions, and we also look at what these conditions mean in
terms of distillation. Finally, we show that these maps can be used to give a
partial solution to the subsystem problem, as described in Ref. [14].
