Parametric models and information geometry on W*-algebras

Information Geometry - Tập 7 - Trang 329-354 - 2023
F. M. Ciaglia1, F. Di Nocera2, J. Jost, L. Schwachhöfer3
1Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain
2Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
3Fakultät für Mathematik, Technische Universität Dortmund, Dortmund, Germany

Tóm tắt

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional $$W^{\star }$$ -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 $$W^{\star }$$ -algebra and the models are made.

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