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

Abstract

The Jordan product on the self-adjoint part of a finite-dimensional C*-algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A, 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 B(H) of linear operators on a finite-dimensional Hilbert space H, and that the Bures-Helstrom metric tensors is recovered when we consider faithful states on B(H). 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 B(H), this alternative geometrical description clarifies the analogy between the Fubini-Study and the Bures-Helstrom metric tensor.