  We explore complementarity between output and environment of a quantum
channel (or, more generally, CP map), making an observation that the output
purity characteristics for complementary CP maps coincide. Hence, validity of
the mutiplicativity/additivity conjecture for a class of CP maps implies its
validity for complementary maps. The class of CP maps complementary to
entanglement-breaking ones is described and is shown to contain diagonal CP
maps as a proper subclass, resulting in new class of CP maps (channels) for
which the multiplicativity/additivity holds. Covariant and Gaussian channels
are discussed briefly in this context.
