  A class of problems in quantum information theory, having an elementary
formulation but still resisting solution, concerns the additivity properties of
various quantities characterizing quantum channels, notably the "classical
capacity", and the "maximal output purity". All known results, including
extensive numerical work, are consistent with the conjecture that these
quantities are indeed additive (resp. multiplicative) with respect to tensor
products of channels. A proof of this conjecture would have important
consequences in quantum information theory. In particular, according to this
conjecture, the classical capacity or the maximal purity of outputs cannot be
increased by using entangled inputs of the channel. In this paper we state the
additivity/multiplicativity problems, give some relations between them, and
prove some new partial results, which also support the conjecture.
