Information geometry of density matrices and state estimation

Given a pure state vector vertical bar x > and a density matrix (rho) over cap, the function p(x (rho) over cap) = < x vertical bar(rho) over cap vertical bar x > defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher-Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.