Completely positive transformations of quantum operations

Quantum supermaps are higher-order maps transforming quantum operations into quantum operations and satisfying suitable requirements of normality and complete positivity. Here we present the extension of the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations on generic von Neumann algebras. In this setting, we provide two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. A structure theorem for probability measures with values in the set of quantum supermaps is also illustrated. Finally, some applications are given, and in particular it is shown that all the supermaps defined in this paper can be implemented by connecting quantum devices in quantum circuits.