The entropy gain of infinite-dimensional quantum channels

In the present paper we study the entropy gain $H(\mathrm{\Phi }[\rho ])-H(\rho )$ for infinite-dimensional channels $\mathrm{\Phi }$. 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.