  We establish several upper bounds on the energy-constrained quantum and
private capacities of all single-mode phase-insensitive bosonic Gaussian
channels. The first upper bound, which we call the "data-processing bound," is
the simplest and is obtained by decomposing a phase-insensitive channel as a
pure-loss channel followed by a quantum-limited amplifier channel. We prove
that the data-processing bound can be at most 1.45 bits larger than a known
lower bound on these capacities of the phase-insensitive Gaussian channel. We
discuss another data-processing upper bound as well. Two other upper bounds,
which we call the "$\varepsilon$-degradable bound" and the
"$\varepsilon$-close-degradable bound," are established using the notion of
approximate degradability along with energy constraints. We find a strong
limitation on any potential superadditivity of the coherent information of any
phase-insensitive Gaussian channel in the low-noise regime, as the
data-processing bound is very near to a known lower bound in such cases. We
also find improved achievable rates of private communication through bosonic
thermal channels, by employing coding schemes that make use of displaced
thermal states. We end by proving that an optimal Gaussian input state for the
energy-constrained, generalized channel divergence of two particular Gaussian
channels is the two-mode squeezed vacuum state that saturates the energy
constraint. What remains open for several interesting channel divergences, such
as the diamond norm or the Renyi channel divergence, is to determine whether,
among all input states, a Gaussian state is optimal.
