  A pure-loss bosonic channel is a simple model for communication over
free-space or fiber-optic links. More generally, phase-insensitive bosonic
channels model other kinds of noise, such as thermalizing or amplifying
processes. Recent work has established the classical capacity of all of these
channels, and furthermore, it is now known that a strong converse theorem holds
for the classical capacity of these channels under a particular photon number
constraint. The goal of the present paper is to initiate the study of
second-order coding rates for these channels, by beginning with the simplest
one, the pure-loss bosonic channel. In a second-order analysis of
communication, one fixes the tolerable error probability and seeks to
understand the back-off from capacity for a sufficiently large yet finite
number of channel uses. We find a lower bound on the maximum achievable code
size for the pure-loss bosonic channel, in terms of the known expression for
its capacity and a quantity called channel dispersion. We accomplish this by
proving a general "one-shot" coding theorem for channels with classical inputs
and pure-state quantum outputs which reside in a separable Hilbert space. The
theorem leads to an optimal second-order characterization when the channel
output is finite-dimensional, and it remains an open question to determine
whether the characterization is optimal for the pure-loss bosonic channel.
