2025/09/12 by Florio M. Ciaglia, Ciaglia, Florio M., Fabio Di Cosmo +3 · 1 voice · 1 citation
Computer Science · Physics and Astronomy · #Cognitive Computing and Networks #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Rough Sets and Fuzzy Logic #Topological and Geometric Data Analysis #math-ph #quant-ph
paper · pdf · doi:10.48550/arxiv.2509.10262
openalex publication_date 2025/09/12 · arxiv published 2025/09/12 · arxiv updated 2025/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the category NCP, whose objects are pairs of W^∗-algebras and normal states and whose morphisms are state-preserving unital completely positive (CPU) maps, as a common stage for classical and quantum information geometry, and we formulate two results that will appear in forthcoming works. First, we recast the problem of classifying admissible Riemannian geometries on classical and quantum statistical models in terms of functors \mathfrakC:NCP\toHilb.These functors provide a generalization of classical statistical covariance, and we call them fields of covariances. A prominent example being the so-called GNS functor arising from the Gelfand-Naimark-Segal (GNS) construction. The classification of fields of covariances on NCP entails both Čencov's uniqueness of the Fisher-Rao metric tensor and Petz's classification of monotone quantum metric tensors as particular cases. Then, we show how classical and quantum statistical models can be realized as subcategories of NCP in a way that takes into account symmetries. In this setting, the fields of covariances determine Riemannian metric tensors on the model that reduce to the Fisher-Rao, Fubini-Study, and Bures-Helstrom metric tensor in particular cases.