  The additivity problem asks if the use of entanglement can boost the
information-carrying capacity of a given channel beyond what is achievable by
coding with simple product states only. This has recently been shown not to be
the case for phase-insensitive one-mode Gaussian channels, but remains
unresolved in general. Here we consider two general classes of bosonic noise
channels, which include phase-insensitive Gaussian channels as special cases:
these are beamsplitters with general, potentially non-Gaussian environment
states and classical noise channels with general probabilistic noise. We show
that additivity violations, if existent, are rather minor for all these
channels: the maximal gain in classical capacity is bounded by a constant
independent of the input energy. Our proof shows that coding by simple
classical modulation of coherent states is close to optimal.
