  Higher-order quantum theory is an extension of quantum theory where one
introduces transformations whose input and output are transformations, thus
generalizing the notion of channels and quantum operations. The generalization
then goes recursively, with the construction of a full hierarchy of maps of
increasingly higher order. The analysis of special cases already showed that
higher-order quantum functions exhibit features that cannot be tracked down to
the usual circuits, such as indefinite causal structures, providing provable
advantages over circuital maps. The present treatment provides a general
framework where this kind of analysis can be carried out in full generality.
The hierarchy of higher-order quantum maps is introduced axiomatically with a
formulation based on the language of types of transformations. Complete
positivity of higher-order maps is derived from the general admissibility
conditions instead of being postulated as in previous approaches. The recursive
characterization of convex sets of maps of a given type is used to prove
equivalence relations between different types. The axioms of the framework do
not refer to the specific mathematical structure of quantum theory, and can
therefore be exported in the context of any operational probabilistic theory.
