Separability of n-particle mixed states: necessary and sufficient   conditions in terms of linear maps
  We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of $n-1$ particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition. 
Michal Horodecki, Pawel Horodecki and Ryszard Horodecki