  In the present paper we study the entropy gain $H(\Phi [\rho])-H(\rho)$ for
infinite-dimensional channels $\Phi$. We show that unlike finite-dimensional
case where the minimal entropy gain is always nonpositive \cite{al}, there is a
plenty of 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.
