Geometrical structures for classical and quantum probability spaces

Abstract

On the affine space containing the space [Formula: see text] 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 the relevant geometrical properties of [Formula: see text]. 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 analyze their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.