Towards a categorical approach to Information Geometry

Classical and quantum information geometry are shaped by two landmark classification results. Čencov’s theorem singles out the Fisher–Rao metric as the unique invariant geometry of finite classical probability spaces, while the Morozova–Čencov–Petz theory shows that quantum monotonicity allows an entire family of  geometries, parametrized by operator monotone functions. Although these results are deeply connected, they are usually formulated in different mathematical languages. This raises a natural question: can they be understood as two instances of a single structural principle?

In this talk, I will present a categorical approach to this question using the category of non-commutative probability spaces, NCP. This framework provides a common language for classical and quantum systems by treating probability distributions and quantum states as states on operator algebras. The central objects will be fields of covariances: functorial assignments of inner products to the Gelfand–Naimark–Segal GNS Hilbert spaces associated with states. In this setting, invariance and monotonicity are not imposed as external conditions, but are encoded directly by functoriality.

I will explain how this viewpoint leads to a classification problem for covariances, in the spirit of Čencov’s original problem of invariant geometries. In finite dimensions, the tracial case recovers the classical statistical covariance underlying the Fisher–Rao metric, while the faithful non-tracial case recovers the Morozova–Čencov–Petz classification of quantum monotone metrics in contravariant form. Finally, I will discuss how the same framework naturally extends the classification to non-faithful states, going beyond the usual faithful setting.