Quantum superchannels on the space of quantum channels

Quantum channels, defined as completely-positive and trace-preserving maps on matrix algebras, are an important object in quantum information theory. In this thesis we are concerned with the space of these channels. This is motivated by the study of quantum superchannels, which are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps, as has been done in the past, we motivate and define superchannels by considering them as transformations on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.