Complementary channels and the additivity problem

We explore the complementarity between output and environment of a quantum channel (or, more generally, completely positive (CP) map), making an observation that the output purity characteristics for complementary CP maps coincide. Hence, the 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 a new class of CP maps (channels) for which the multiplicativity/additivity holds. Covariant and Gaussian channels are discussed briefly in this context.