Majorization preservation of Gaussian bosonic channels

It is shown that phase-insensitive Gaussian bosonic channels are majorization-preserving over the set of passive states of the harmonic oscillator. This means that comparable passive states under majorization are transformed into equally comparable passive states. The proof relies on a new preorder relation called Fock-majorization, which coincides with regular majorization for passive states but also induces a mean photon number order, thereby connecting the concepts of energy and disorder of a quantum state. As an application, the consequences of majorization preservation are investigated in the context of the broadcast communication capacity of bosonic Gaussian channels. Most of our results being independent of the bosonic nature of the system under investigation, they could be generalized to other quantum systems and Hamiltonians, providing a general tool that could prove useful in quantum information theory and quantum thermodynamics.