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 (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 (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 (ea) is not unusual for constrained c-q Gaussian channel.