  Quantum supermaps are higher-order maps transforming quantum operations into
quantum operations. Here we extend the theory of quantum supermaps, originally
formulated in the finite dimensional setting, to the case of higher-order maps
transforming quantum operations with input in a separable von Neumann algebra
and output in the algebra of the bounded operators on a given separable Hilbert
space. In this setting we prove two dilation theorems for quantum supermaps
that are the analogues of the Stinespring and Radon-Nikodym theorems for
quantum operations. Finally, we consider the case of quantum superinstruments,
namely measures with values in the set of quantum supermaps, and derive a
dilation theorem for them that is analogue to Ozawa's theorem for quantum
instruments. The three dilation theorems presented here show that all the
supermaps defined in this paper can be implemented by connecting devices in
quantum circuits.
