Coloquio Junior de matemáticas
The category of non-commutative probabilities in Information Geometry
Ponente: Laura González Bravo (Universidad Carlos III de Madrid)Fecha: martes 25 de noviembre de 2025 - 17:00 (café: 18:00)Lugar: Aula Naranja, ICMAT
Resumen:
The foundational theorems of Čencov and Petz provide a characterization of invariant geometries for finite-dimensional classical and quantum systems, respectively. However, extending these insights into a unified framework that seamlessly incorporates infinite-dimensional and continuous systems remains a significant challenge. This work introduces the category of non-commutative probabilities, \(\mathsf{NCP}\), as a new foundation for information geometry. Built upon the language of operator algebras, \(\mathsf{NCP}\) provides a common environment for both classical and quantum theories, applicable in both finite and infinite dimensions. Within this setting, we reformulate the problem of characterizing invariant geometries. Instead of classifying metric tensors on manifolds of states, we propose the classification of ``fields of covariances," defined as functors from \(\mathsf{NCP}\) to the category of Hilbert spaces. This approach recasts the crucial monotonicity property as a natural condition of functoriality. Furthermore, we demonstrate how this framework accommodates the description of statistical models as dedicated subcategories.
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