Bi-forms Approach to Potential Functions in Information Geometry

TL;DR

This paper introduces contrast bi-forms, a new mathematical framework that generalizes contrast functions to encode geometric structures with torsion, advancing the understanding of information geometry, especially in quantum contexts.

Contribution

It presents contrast bi-forms as a systematic way to encode metric and connection data, including torsion, providing a unified framework for statistical potentials in information geometry.

Findings

Bi-forms accommodate torsion in geometric structures.

Application to teleparallel manifolds demonstrates natural fit.

Offers new insights into the inverse problem in information geometry.

Abstract

Contrast functions play a fundamental role in information geometry, providing a means for generating the geometric structures of a statistical manifold: a pseudo-Riemannian metric and a pair of torsion-free conjugate affine connections. Conventional contrast-based approaches become indeed insufficient within settings where torsion is naturally present, such as quantum information geometry. This paper introduces contrast bi-forms, a generalisation of contrast functions that systematically encode metric and connection data, allowing for arbitrary affine connections regardless of torsion. It will be shown that they provide a unified framework for statistical potentials, offering new insights into the inverse problem in information geometry. As an example, we consider teleparallel manifolds, where torsion is intrinsic to the geometry, and show how bi-forms naturally accommodate these structures.