Parametric models and information geometry on W*-algebras

TL;DR

This paper develops a generalized framework for parametric models on W*-algebras using information geometry, unifying classical and quantum metrics like Fisher-Rao and Bures-Helstrom.

Contribution

It introduces smooth parametric models on W*-algebras and defines a Riemannian metric tensor that generalizes key information metrics in a unified setting.

Findings

Defines a general notion of parametric models on W*-algebras.

Constructs a Riemannian metric tensor using the Jordan product.

Recovers classical and quantum information metrics as special cases.

Abstract

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional W*-algebras generalizing the notions of parametric models used in classical and quantum information geometry. We then use the Jordan product naturally available in this context in order to define a Riemannian metric tensor on parametric models satsfying suitable regularity conditions. This Riemannian metric tensor reduces to the Fisher-Rao metric tensor, or to the Fubini-Study metric tensor, or to the Bures-Helstrom metric tensor when suitable choices for the W*-algebra and the models are made.