The von Neumann entropy and quantum channels

The improper properties of entropic characteristics of infinite dimensional quantum systems (among which the von Neumann entropy of a quantum state is the main one) represent a real obstacle for study of the informational capabilities of these systems. In this talk, we give a survey of the analytical properties of the von Neumann entropy and the results permitting to overcome problems associated with the entropy discontinuity and the non-compactness of the set of quantum states.
The main part of the talk is devoted to the properties of the output entropy of a quantum channel (non-commutative Markov map), through which the most information characteristics of a quantum channel, for example, its classical and quantum capacities are expressed. We, in particular, specify the necessary and sufficient conditions for continuity of the output entropy on the whole set of input states and on some its special subsets, and analyze relations between the local continuity properties of the output entropies of mutually complementary channels. Some concrete examples are also considered.