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Majorization ladder in bosonic Gaussian channels

par Van Herstraeten, Zacharie ;Jabbour, Michael ;Cerf, Nicolas
Référence AVS Quantum Science, 5, 1, 011401
Publication Publié, 2023-03-01

Article révisé par les pairs

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Résumé :

We show the existence of a majorization ladder in bosonic Gaussian channels, that is, we prove that the channel output resulting from the n th energy eigenstate (Fock state) majorizes the channel output resulting from the (n + 1) 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 non-trivial 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 [Jabbour and Cerf, Phys. Rev. Res. 3, 043065 (2021)]. Possible generalizations and implications of these results are then discussed.