On the LOCC Classification of Bipartite Density Matrices
  We provide a unifying framework for exact, probabilistic, and approximate conversions by local operations and classical communication (LOCC) between bipartite states. This framework allows us to formulate necessary and sufficient conditions for LOCC conversions from pure states to mixed states and it provides necessary conditions for LOCC conversions between mixed states. The central idea is the introduction of convex sets for exact, probabilistic, and approximate conversions, which are closed under LOCC operations and which are largely characterized by simple properties of pure states. 
Patrick Hayden, Barbara M. Terhal and Armin Uhlmann