Some properties of the Riemannian Bures metric on mixed states

Abstract

We consider the Riemannian geometry of the space of nonsingular density matrices D equipped with the Bures metric g. This space is of certain physical relevance on the background of generalization of the Berry phase to mixed states. The main result is the determination of the covariant derivative and the curvature tensor field related to the Levi-Cività connection of ( D , g). It turns out that D is not a space of constant curvature and even not a locally symmetric space in contrast to the suggestions one gets from the case of two-dimensional density matrices. Moreover, we give a local description of D and explicit formulae for g in terms of natural matrix operations containing ϱ and dϱ only.