  We study the problem of transmitting classical information using quantum
Gaussian states on a family of phase-noise channels with a finite decoherence
time, such that the phase-reference is lost after $m$ consecutive uses of the
transmission line. This problem is relevant for long-distance communication in
free space and optical fiber, where phase noise is typically considered as a
limiting factor. The Holevo capacity of these channels is always attained with
photon-number encodings, challenging with current technology. Hence for
coherent-state encodings the optimal rate depends only on the total-energy
distribution and we provide upper and lower bounds for all $m$, the latter
attainable at low energies with on/off modulation and photodetection. We
generalize this lower bound to squeezed-coherent encodings, exhibiting for the
first time to our knowledge an unconditional advantage with respect to any
coherent encoding for $m=1$ and a considerable advantage with respect to its
direct coherent counterpart for $m>1$. This advantage is robust with respect to
moderate attenuation, and persists in a regime where Fock encodings with up to
two-photon states are also suboptimal. Finally, we show that the use of part of
the energy to establish a reference frame is sub-optimal even at large
energies. Our results represent a key departure from the case of
phase-covariant Gaussian channels and constitute a proof-of-principle of the
advantages of using non-classical, squeezed light in a motivated communication
setting.
