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A formulation of the generalized minimal output entropy conjecture forGaussian channels is presented. It asserts that, for states with fixed inputentropy, the minimal value of the output entropy of the channel (i.e. theminimal output entropy increment for fixed input entropy) is achieved byGaussian states. In the case of centered channels (i.e. channels which do notadd squeezing to the input state) this implies that the minimum is obtained bythermal (Gibbs) inputs. The conjecture is proved to be valid in some specialcases.

Generalized minimal output entropy conjecture for one-mode Gaussian channels: definitions and some exact results