Maps

A crucial aspect in the formulation of quantum theory is the description of state transformations and correspondingly, in a dual way, of observable transformations. We address in detail their structure and interpretation, building on the notion of map. Relevant transformations include time evolutions due to the free dynamics of the system, the coupling with other degrees of freedom acting as an environment, as well as the interaction with a measurement apparatus. The general characterization of these maps puts into evidence the notion of complete positivity, related to non-commutativity and entanglement, reflecting at the level of a single system the quantum features arising in the description of composite systems.