Aspects of geodesical motion with Fisher-Rao metric: classical and quantum

TL;DR

This paper explores the geometric structure of classical and quantum statistical manifolds using Fisher-Rao metric, revealing how quantum properties influence geodesic endpoints and relate to the uncertainty principle.

Contribution

It demonstrates the difference in geodesic endpoints between classical and quantum cases, linking quantum effects to the geometry of statistical manifolds.

Findings

Classical geodesic endpoints minimize Shannon's Entropy.

Quantum geodesic endpoints depend on initial conditions.

Quantum effects relate to the uncertainty principle through geometry.

Abstract

The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon's Entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.