Facial structures for various notions of positivity and applications to   the theory of entanglement
  In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses. 
Seung-Hyeok Kye
quant-ph