  We study the additivity problems for the classical capacity of quantum
channels, the minimal output entropy and its convex closure. We show for each
of them that additivity for arbitrary pairs of channels holds iff it holds for
arbitrary equal pairs, which in turn can be taken to be unital. In a similar
sense, weak additivity is shown to imply strong additivity for any convex
entanglement monotone. The implications are obtained by considering direct sums
of channels (or states) for which we show how to obtain several information
theoretic quantities from their values on the summands. This provides a simple
and general tool for lifting additivity results.
