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The Jordan product on the self-adjoint part of a finite-dimensional$C^{*}$-algebra $\mathscr{A}$ is shown to give rise to Riemannian metrictensors on suitable manifolds of states on $\mathscr{A}$, and the covariantderivative, the geodesics, the Riemann tensor, and the sectional curvature ofall these metric tensors are explicitly computed. In particular, it is provedthat the Fisher--Rao metric tensor is recovered in the Abelian case, that theFubini--Study metric tensor is recovered when we consider pure states on thealgebra $\mathcal{B}(\mathcal{H})$ of linear operators on a finite-dimensionalHilbert space $\mathcal{H}$, and that the Bures--Helstrom metric tensors isrecovered when we consider faithful states on $\mathcal{B}(\mathcal{H})$.Moreover, an alternative derivation of these Riemannian metric tensors in termsof the GNS construction associated to a state is presented. In the case of pureand faithful states on $\mathcal{B}(\mathcal{H})$, this alternative geometricaldescription clarifies the analogy between the Fubini--Study and theBures--Helstrom metric tensor.

From the Jordan product to Riemannian geometries on classical and quantum states