Geometrical Structures for Classical and Quantum Probability Spaces

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.

16 pages

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Cited by in corpus (3)

• From the Jordan product to Riemannian geometries on classical and quantum states
• Stratified Manifold of Quantum States, actions of the complex special linear group
• The quantum geometric tensor from generating functions