From the Jordan Product to Riemannian Geometries on Classical and Quantum States

The Jordan product on the self-adjoint part of a finite-dimensional [Formula: see text]-algebra [Formula: see text] is shown to give rise to Riemannian metric tensors on suitable manifolds of states on [Formula: see text] , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra [Formula: see text] of linear operators on a finite-dimensional Hilbert space [Formula: see text] , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on [Formula: see text]. Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on [Formula: see text] , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.