In Wigner phase space, convolution explains why the vacuum majorizes mixtures of Fock states

Luc Vanbever

Published 2021-04-30Version 1

I show that the Wigner function that represents any mixture of the first 300 Fock states is majorized by the Wigner function of the vacuum state. As a consequence, the integration of any concave function over the Wigner phase space has a lower value for the vacuum state than for a mixture of Fock states. The Shannon differential entropy is an example of such concave function of significant physical importance. I demonstrate that the very cause of the majorization lies in the fact that a Wigner function is the result of a convolution. My proof is based on a new majorization result dedicated to the convolution of the negative exponential distribution with a precisely constrained function. I present a geometrical interpretation of the new majorization property in a discrete setting and extend this relation to a continuous setting. The final part of my proof is numerical and consolidates my previous findings: remarkably, I find that the first 300 Fock states match the entry criteria of my new majorization result - with strong indications that it applies to any Fock state.