Abstract
A (faithful) density matrix can be represented by a Hilbert-Schmidt operator up to composition with a unitary operator. This ambiguity can be described in terms of a principal fibre bundle which is a modification of a construction by Uhlmann (1986). The HS norm provides a real metric which determines a natural connection via the Kaluza-Klein mechanism. The authors compute the curvature and show that at the boundary its holonomy reproduces the Berry phase for (pure) non-degenerate states and the non-Abelian holonomy of Wilczek and Zee (1984) for k-fold degenerate states.