Volume 5, 2022
|Number of page(s)||11|
|Section||Physics - Applied Physics|
|Published online||21 March 2022|
Logical entropy and negative probabilities in quantum mechanics
Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, 67000 Strasbourg, France
* Corresponding author: firstname.lastname@example.org
Accepted: 10 February 2022
The concept of Logical Entropy, SL = 1-Σni=1 pi2, where the pi are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of SL as a measure of the distinctions of a partition on a given set. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. The quadratic form of the logical entropy lends itself to a generalization of the probabilities that include negative values, an idea that goes back to Feynman and Wigner. Here, we analyze and reinterpret negative probabilities in the light of the concept of logical entropy. Several intriguing quantum-like properties of the logical entropy are derived and discussed in finite dimensional spaces. For infinite-dimensional spaces (continuum), we show that, under the sole hypothesis that the logical entropy and the total probability are preserved in time, one obtains an evolution equation for the probability density that is basically identical to the quantum evolution of the Wigner function in phase space, at least when one considers only the momentum variable. This result suggest that the logical entropy plays a profound role in establishing the peculiar rules of quantum physics.
Key words: Entropy / Negative Probabilities / Quantum Physics / Wigner Functions
© G. Manfredi, Published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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