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

• URL: http://arxiv.org/abs/2104.14996v2
• Date: Mon, 23 Aug 2021 07:49:15 GMT
• Title: In Wigner phase space, convolution explains why the vacuum majorizes mixtures of Fock states
• Authors: Luc Vanbever
• Abstract summary: I show that a nonnegative Wigner function that represents a mixture of Fock states is majorized by the Wigner function of the vacuum state. Findings presented in this article might be expanded upon to explain why the Wigner function of the vacuum majorizes.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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