  The purpose of this work is to extend the result of previous papers
quant-ph/9611023, quant-ph/9703013 to quantum channels with additive
constraints onto the input signal, by showing that the capacity of such channel
is equal to the supremum of the entropy bound with respect to all apriori
distributions satisfying the constraint. We also make an extension to channels
with continuous alphabet. As an application we prove the formula for the
capacity of the quantum Gaussian channel with constrained energy of the signal,
establishing the asymptotic equivalence of this channel to the semiclassical
photon channel. We also study the lower bounds for the reliability function of
the pure-state Gaussian channel.
