Equivalence relations for the classical capacity of single-mode Gaussian quantum channels

We prove the equivalence of an arbitrary single-mode Gaussian quantum channel and a newly defined fiducial channel preceded by a phase shift and followed by a Gaussian unitary operation. This equivalence implies that the energy-constrained classical capacity of any single-mode Gaussian channel can be calculated based on this fiducial channel, which is furthermore simply realizable with a beam splitter, two identical single-mode squeezers, and a two-mode squeezer. In a large domain of parameters, we also provide an analytical expression for the Gaussian classical capacity, exploiting its additivity, and prove that the classical capacity cannot exceed it by more than 1/ln2 bits.