  This paper strengthens the interpretation and understanding of the classical
capacity of the pure-loss bosonic channel, first established in [Giovannetti et
al., Physical Review Letters 92, 027902 (2004), arXiv:quant-ph/0308012]. In
particular, we first prove that there exists a trade-off between communication
rate and error probability if one imposes only a mean-photon number constraint
on the channel inputs. That is, if we demand that the mean number of photons at
the channel input cannot be any larger than some positive number N_S, then it
is possible to respect this constraint with a code that operates at a rate
g(\eta N_S / (1-p)) where p is the code's error probability, \eta\ is the
channel transmissivity, and g(x) is the entropy of a bosonic thermal state with
mean photon number x. We then prove that a strong converse theorem holds for
the classical capacity of this channel (that such a rate-error trade-off cannot
occur) if one instead demands for a maximum photon number constraint, in such a
way that mostly all of the "shadow" of the average density operator for a given
code is required to be on a subspace with photon number no larger than n N_S,
so that the shadow outside this subspace vanishes as the number n of channel
uses becomes large. Finally, we prove that a small modification of the
well-known coherent-state coding scheme meets this more demanding constraint.
