Reflections upon separability and distillability

An abstract formulation is presented of the so-called Innsbruck-Hannover programme that investigates quantum correlations and entanglement in terms of convex sets. A unified description is given of optimal decompositions of quantum states and the optimization of witness operators that detect whether a given state belongs to a given convex set. The abstract formulation is illustrated with several examples, and relations between optimal entanglement witnesses and n -copy non-distillable states with non-positive partial transpose are discussed.