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We establish several upper bounds on the energy-constrained quantum andprivate capacities of all single-mode phase-insensitive bosonic Gaussianchannels. The first upper bound, which we call the "data-processing bound," isthe simplest and is obtained by decomposing a phase-insensitive channel as apure-loss channel followed by a quantum-limited amplifier channel. We provethat the data-processing bound can be at most 1.45 bits larger than a knownlower bound on these capacities of the phase-insensitive Gaussian channel. Wediscuss 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 ofapproximate degradability along with energy constraints. We find a stronglimitation on any potential superadditivity of the coherent information of anyphase-insensitive Gaussian channel in the low-noise regime, as thedata-processing bound is very near to a known lower bound in such cases. Wealso find improved achievable rates of private communication through bosonicthermal channels, by employing coding schemes that make use of displacedthermal states. We end by proving that an optimal Gaussian input state for theenergy-constrained, generalized channel divergence of two particular Gaussianchannels is the two-mode squeezed vacuum state that saturates the energyconstraint. What remains open for several interesting channel divergences, suchas the diamond norm or the Renyi channel divergence, is to determine whether,among all input states, a Gaussian state is optimal.

Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels