On the Riemannian metric on the space of density matrices

The concept of purification of mixed states of quantum systems leads naturally to an interesting Riemannian metric on the space of normalized density matrices Dn. More precisely, this Riemannian metric introduced by Uhlmann is defined on submanifolds (of all hermitean matrices) Dnk of density matrices of fixed rank k, but not for vectors transversal to this submanifolds (at least using definitions known in the literature). A natural question is, whether there exists a manifold M of dimension dim Dnn = n2 − 1, which contains ∪k = 1n Dnk as a topological subspace such that Dnk, k = 1,…,n, is isometrically embedded. For n = 2 this is true, as Uhlmann observed. We show that for higher n such a manifold does not exist, because the sectional curvature at ρ ϵ Dnn diverges if ρ tends to a state of rank less than n − 1. Roughly speaking, the metrics on the manifolds Dnk cannot be glued together, because Dnn contains geodesically complete submanifolds which look like conical singularities in the neighbourhood of ∂Dnn.