Majorization ladder in bosonic Gaussian channels

TL;DR

This paper proves a majorization relation for outputs of bosonic Gaussian channels, linking input energy states to output disorder, extending previous special cases to general single-mode phase-covariant channels.

Contribution

It generalizes the majorization ladder result to all single-mode phase-covariant bosonic Gaussian channels using a new constructive proof.

Findings

Majorization relation holds for all single-mode phase-covariant Gaussian channels.

Explicit construction of a column-stochastic matrix relates outputs for consecutive Fock states.

Results extend previous special-case findings to a broader class of channels.

Abstract

We show the existence of a majorization ladder in bosonic Gaussian channels, that is, we prove that the channel output resulting from the $n\text{th}$ energy eigenstate (Fock state) majorizes the channel output resulting from the $(n\!+\!1)\text{th}$ energy eigenstate (Fock state). This reflects a remarkable link between the energy at the input of the channel and a disorder relation at its output as captured by majorization theory. This result was previously known in the special cases of a pure-loss channel and quantum-limited amplifier, and we achieve here its nontrivial generalization to any single-mode phase-covariant (or -contravariant) bosonic Gaussian channel. The key to our proof is the explicit construction of a column-stochastic matrix that relates the outputs of the channel for any two subsequent Fock states at its input. This is made possible by exploiting a recently found recurrence relation on multiphoton transition probabilities for Gaussian unitaries [M. G. Jabbour and N. J. Cerf, Phys. Rev. Research 3, 043065 (2021)]. Possible generalizations and implications of these results are then discussed.