Evolution of any finite open quantum system always admits a Kraus-type representation, although it is not always completely positive -: art. no. 054102

We argue that a full account of the set of quantum states prevents one from applying the rigorous definition of complete positivity. However, we give three equivalent proofs that any quantum evolution of a finite system always admits a Kraus-type decomposition, i.e., a Kraus decomposition but with Kraus matrices dependent on the initial state upon which they apply.