  Continuity properties of the output entropy of positive linear maps between
Banach spaces of trace class operators are investigated with the special
attention to the classes of quantum channels and operations. It is shown that
finiteness of the output entropy of a positive map on the whole input state
space implies its continuity. Sufficient conditions for this property expressed
in terms of the Kraus representation of quantum channels and operations are
presented. The characterization of a positive map "preserving continuity of the
entropy" (in the sense that continuity of the entropy on a set of input states
implies continuity of its output entropy on this set) is obtained and its
applications to the class of quantum operations are considered. The special
relation between continuity properties of the output entropies of complementary
quantum operations is established and its corollaries are discussed. Continuity
conditions for the output entropy of positive maps considered as a function of
a pair (map, input state) are also obtained. Several applications to the
quantum information theory are presented in the last part of the paper.
