On the Riemannian Geometry of Finite Dimensional Mixed States

We consider the Riemannian geometry of the space of nonsingular density matrices D 1 equipped with the Bures metric g B .This space is of certain physical relevance on the background of a generalization of the Berry phase to mixed states.We determine the covariant derivative and the curvature tensor field related to the Levi-Cevita connection of (D 1 , g B ), which allow us to calculate other curvature quantities .It turns out that D 1 is not a space of constant curvature and not even a locally symmetric space, in contrast to what the case of two-dimensional density matrices might suggest.Moreover, we give a local description of D 1 and explicit formulae for the Bures metric in terms of natural matrix operations containing and d only.