  A formulation of the generalized minimal output entropy conjecture for
Gaussian channels is presented. It asserts that, for states with fixed input
entropy, the minimal value of the output entropy of the channel (i.e. the
minimal output entropy increment for fixed input entropy) is achieved by
Gaussian states. In the case of centered channels (i.e. channels which do not
add squeezing to the input state) this implies that the minimum is obtained by
thermal (Gibbs) inputs. The conjecture is proved to be valid in some special
cases.
