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We prove multiplicativity of maximal output $p$ norm of classical noisechannels and thermal noise channels of arbitrary modes for all $p>1$ under theassumption that the input signal states are Gaussian states. As a directconsequence, we also show the additivity of the minimal output entropy and thatof the energy-constrained Holevo capacity for those Gaussian channels underGaussian inputs. To the best of our knowledge, newly discovered majorizationrelation on symplectic eigenvalues, which is also of independent interest,plays a central role in the proof.Comment: 9 pages, no figures. Published Versio

Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs