The entropy gain of quantum channels

In this paper we study the entropy gain H(Φ[ρ]) - H(ρ) for infinite-dimensional quantum channels Φ. We show that unlike finite-dimensional case where the minimal entropy gain is always nonpositive, there are channels with positive minimal entropy gain. We obtain the new lower bound and compute the minimal entropy gain for a broad class of Bosonic Gaussian channels by proving that the infimum is attained on the Gaussian states.