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

TL;DR

This paper proves that the vacuum state's Wigner function majorizes mixtures of Fock states due to convolution properties, impacting the understanding of quantum state representations and entropy measures.

Contribution

It introduces a new majorization result for convolutions of the negative exponential distribution, explaining why the vacuum state dominates in Wigner phase space.

Findings

Vacuum Wigner function majorizes mixtures of Fock states.

Concave functions like Shannon entropy are minimized by the vacuum state.

Convolution structure underpins the majorization relationship.

Abstract

I show that a nonnegative Wigner function that represents a mixture of 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. Findings presented in this article might be expanded upon to explain why the Wigner function of the vacuum majorizes - beyond mixtures of Fock states - many other physical states represented by a nonnegative Wigner function.