Geometrical Structures for Classical and Quantum Probability Spaces

TL;DR

This paper explores the geometric structures of classical and quantum probability spaces using tensor fields, drawing parallels between quantum states and classical probability distributions, with detailed analysis on a three-level system.

Contribution

It introduces a unified geometric framework for classical and quantum probability spaces using contravariant tensor fields inspired by Dirac's analogy.

Findings

Defined Hamiltonian and gradient vector fields on probability spaces

Analyzed geometrical properties of these tensor fields

Applied framework to a three-level quantum system

Abstract

On the affine space containing the space $\mathcal{S}$ of quantum states of finite-dimensional systems there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding relevant geometrical properties of $\mathcal{S}$. Guided by Dirac's analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyse their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.