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Jordan algebras arise naturally in (quantum) information geometry, and wewant to understand their role and their structure within that framework.Inspired by Kirillov's discussion of the symplectic structure on coadjointorbits, 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 thedual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on theleaves of the distribution. In particular, these leaves are the orbits of a Liegroup, which is the structure group of ${\mathcal J}$, in clear analogy withwhat happens for coadjoint orbits. However, this time in contrast with theLie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on aleaf of the canonical Jordan distribution. When the leaves are contained in thecone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannianstructure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, itcoincides with the Fisher-Rao metric on non-normalized probabilitydistributions on a finite sample space, or with the Bures-Helstrom metric fornon-normalized, faithful quantum states of a finite-level quantum system, thusshowing a direct link between the mathematics of Jordan algebras and bothclassical and quantum information geometry.

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