On the Riemannian metric on the space of density matrices

Abstract

The concept of purification of mixed states of quantum systems leads naturally to an interesting Riemannian metric on the space of normalized density matrices D n . More precisely, this Riemannian metric introduced by Uhlmann is defined on submanifolds (of all hermitean matrices) D nk 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 D nn = n 2 − 1, which contains ∪ k = 1 n D nk as a topological subspace such that D nk , 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 D nn diverges if ρ tends to a state of rank less than n − 1. Roughly speaking, the metrics on the manifolds D nk cannot be glued together, because D nn contains geodesically complete submanifolds which look like conical singularities in the neighbourhood of ∂D nn .