Gaussian classical-quantum channels: gain from entanglement-assistance

In the present paper we introduce and study Bosonic Gaussian classical-quantum (c-q) channels; embedding of the classical input in quantum input is always possible, and therefore the classical entanglement-assisted capacity $C_\mathrm{ea}$ under an appropriate input constraint is well defined. We prove the general property of entropy increase for a weak complementary channel, which implies the equality $C=C_\mathrm{ea}$ (where $C$ is the unassisted capacity) for a certain class of c-q Gaussian channels under an appropriate energy-type constraint. On the other hand, we show by an explicit example that the inequality $C<C_\mathrm{ea}$ is not unusual for constrained c-q Gaussian channel.