Passive states optimize the output of bosonic Gaussian quantum channels

An ordering between the quantum states emerging from a single-mode gauge-covariant bosonic Gaussian channel is proved. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e., diagonal with decreasing eigenvalues) produces an output, which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of Mari et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others.