Majorization ladder in bosonic Gaussian 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 [Formula: see text] energy eigenstate (Fock state) majorizes the channel output resulting from the [Formula: see text] 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.

On a Recent Conjecture by Z. Van Herstraeten and N. J. Cerf for the Quantum Wigner Entropy

“…where z = (x, p) ∈ R 2d , z 0 ∈ R 2d is some fixed constant and M is a real, symmetric, positive-definite 2d × 2d matrix. Then (26) is the Wigner function associated with some pure state f ∈ L 2 (R d ) if and only if M is a symplectic matrix: M ∈ Sp(2d; R).…”

“…( 5)) to the set of positive Wigner distributions only. They called this set the Wigner-positive states [25,26]. They then went on and argued that it is nevertheless a proper measure of quantum uncertainty in phase space and they advocate that it can be an important physical quantity in the context of quantum optics, because it is invariant under affine symplectic transformations (displacements, rotations and squeezing), and it permits one to establish a Wigner entropy-power inequality.…”

On a recent conjecture by Z. Van Herstraeten and N.J. Cerf for the quantum Wigner entropy

“…( 5)) to the set of positive Wigner distributions only. They called this set the Wignerpositive states [12,13]. They then went on and argued that it is nevertheless a proper measure of quantum uncertainty in phase space and they advocate that it can be an important physical quantity in the context of quantum optics, because it is invariant under affine symplectic transformations (displacements, rotations and squeezing), and it permits one to establish a Wigner entropy-power inequality.…”