Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra $\mathcal{J}$, we exploit the generalized distribution determined by the Jordan product on the dual $\mathcal{J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of $\mathcal{J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in $\mathcal{J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on $\mathcal{J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of $\mathcal{J}$, it coincides with the Fisher–Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures–Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.