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I show that a nonnegative Wigner function that represents a mixture of Fockstates is majorized by the Wigner function of the vacuum state. As aconsequence, the integration of any concave function over the Wigner phasespace 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 ofsignificant physical importance. I demonstrate that the very cause of themajorization lies in the fact that a Wigner function is the result of aconvolution. My proof is based on a new majorization result dedicated to theconvolution of the negative exponential distribution with a preciselyconstrained function. I present a geometrical interpretation of the newmajorization property in a discrete setting and extend this relation to acontinuous setting. Findings presented in this article might be expanded uponto explain why the Wigner function of the vacuum majorizes - beyond mixtures ofFock states - many other physical states represented by a nonnegative Wignerfunction.

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