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We study one-mode Gaussian quantum channels in continuous-variable systems byperforming a black-box characterization using complete positivity and tracepreserving conditions, and report the existence of two subsets that do not havea functional Gaussian form. Our study covers as particular limit the case ofsingular channels, thus connecting our results with their known classificationscheme based on canonical forms. Our full characterization of Gaussian channelswithout Gaussian functional form is completed by showing how Gaussian statesare transformed under these operations, and by deriving the conditions for theexistence of master equations for the non-singular cases. We show that althoughevery functional form can be found in the vicinity of the identity, one of themdoes not parametrize unitary channels.

Position representation of single-mode Gaussian channels beyond the Gaussian functional form