  We introduce the concept of quantum supermap, describing the most general
transformation that maps an input quantum operation into an output quantum
operation. Since quantum operations include as special cases quantum states,
effects, and measurements, quantum supermaps describe all possible
transformations between elementary quantum objects (quantum systems as well as
quantum devices). After giving the axiomatic definition of supermap, we prove a
realization theorem, which shows that any supermap can be physically
implemented as a simple quantum circuit. Applications to quantum programming,
cloning, discrimination, estimation, information-disturbance trade-off, and
tomography of channels are outlined.
