Bi-forms Approach to Potential Functions in Information Geometry
Florio M. Ciaglia, Giuseppe Marmo, Marco Pacelli, Luca Schiavone, Alessandro Zampini
TL;DR
The paper extends information-geometric potentials to settings with torsion by introducing contrast bi-forms, a $(1,1)$-bi-form framework on $M\times M$ that simultaneously encodes the metric and both (potentially torsionful) affine connections. Using the diagonal pullback and left/right differential operators $d^L$ and $d^R$, a contrast bi-form $\varpi$ generates a metric $g^\varpi$ and a pair of dual connections $\nabla^\varpi$, with torsion characterized by $\iota^* d^L\varpi$ and $\iota^* d^R\varpi$, thereby unifying contrast, pre-contrast, and super-contrast ideas under a single formalism. The authors develop the theory to address the inverse problem—whether a given Lauritzen manifold can be derived from a potential—within this bi-form framework and illustrate it through teleparallel manifolds and the manifold of faithful quantum states, where quantum monotone metrics $g^f$ and the von Neumann–Umegaki relative entropy $vNU$ arise in natural ways. This provides a coherent, geometry-rich method to study potentials in classical and quantum information geometry, with explicit constructions for quantum-state spaces and practical connections to known quantum divergences. The framework promises a unified approach to potentials and may yield new tools for quantum information geometry and estimation theory.
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.