Logical Entropy and Negative Probabilities in Quantum Mechanics

TL;DR

This paper explores the concept of Logical Entropy in quantum mechanics, linking it to negative probabilities and deriving quantum-like properties, suggesting a fundamental role in quantum theory's rules.

Contribution

It introduces a reinterpretation of negative probabilities through Logical Entropy and connects it to quantum evolution, offering new insights into quantum foundations.

Findings

Logical entropy relates to quantum state purity.

Negative probabilities can be understood via logical entropy.

Quantum evolution equations resemble those derived from logical entropy considerations.

Abstract

The concept of Logical Entropy, $S_L = 1- \sum_{i=1}^n p_i^2$, where the $p_i$ 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 $S_L$ 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.