Characterization of separable states and entanglement witnesses

We provide a canonical form of mixed states in bipartite quantum systems in terms of a convex combination of a separable state and a so-called edge state. We construct entanglement witnesses for all edge states, which allows us to introduce a canonical form of nondecomposable entanglement witnesses and the corresponding positive maps. We also present a nontrivial necessary condition for entanglement witnesses and positive maps to be extremal.