Catalyst-assisted Probabilistic Entanglement Transformation
  We are concerned with catalyst-assisted probabilistic entanglement transformations. A necessary and sufficient condition is presented under which there exist partial catalysts that can increase the maximal transforming probability of a given entanglement transformation. We also design an algorithm which leads to an efficient method for finding the most economical partial catalysts with minimal dimension. The mathematical structure of catalyst-assisted probabilistic transformation is carefully investigated. 
Yuan Feng, Runyao Duan and Mingsheng Ying