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We prove that Gaussian thermal input states minimize the output von Neumann entropy of the one-mode Gaussian quantum-limited attenuator for fixed input entropy. This result opens the way to its multimode generalization, that is crucial to prove the converse theorems for both the triple trade-off region of the Gaussian quantum-limited attenuator and the classical capacity region of the Gaussian degraded quantum broadcast channel. The first step of our proof is a result relevant by its own. We prove that passive states are the optimal inputs of one-mode Gaussian quantum channels, in the sense that the output generated by a passive state majorizes the output generated by any other state with the same spectrum. The passive states of a quantum system minimize the average energy for a given spectrum. The proof of the main result is therefore restricted to input states diagonal in the Fock basis.