Capacities of Gaussian classical-quantum channels

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