Fields of covariances on non-commutative probability spaces in finite dimensions

Abstract

We introduce the notion of a field of covariances, a contravariant functor from non-commutative probability spaces to Hilbert spaces, as a categorical analogue of statistical covariance. In the finite-dimensional setting, we obtain a complete characterization of the fields of covariances satisfying an additional continuity property in terms of a strictly positive constant $\alpha$ and a continuous function $F\colon[0,+\infty)\rightarrow(0,+\infty)$ that is operator monotone on $(0,+\infty)$. In the commutative case, our result recovers a contravariant formulation of \v{C}encov's characterization of the Fisher--Rao metric tensor, and extends the corresponding uniqueness statement to tracial states on possibly noncommutative algebras. On faithful states, a subfamily satisfying a symmetry constraint recovers the regular Morozova--\v{C}encov--Petz monotone metric tensors. More generally, the covariance fields induce positive contravariant symmetric tensors on the homogeneous manifolds of states generated by the action of the invertible elements of an arbitrary finite-dimensional $C^*$-algebra. The geometry internal to the support is determined by the positive spectrum of the modular operator, whereas support-changing directions are governed by $F(0)$. For $F(t)=(1+t)/2$, and with a suitable normalization convention, the resulting tensors recover the inverse of the Riemannian metrics induced by the Jordan product. In particular, on pure states, every $F$ induces a multiple of the inverse Fubini--Study tensor, with coefficient $2F(0)$ in our normalization, in agreement after inversion with the Petz--Sud\'ar radial extension.