Geometry of positive operators and uhlmann's approach to the geometric phase

Abstract

In Uhlmann's description of the differential geometry of the space Ω of density operators, a relevant role is played by the parallel condition ω*ω = ω*ω, where ω is a lifting of a curve γ in Ω, i.e. ω( t)ω( t)* = γ( t) for all t. In this paper we get a principal bundle with a natural connection over the space G + of all positive invertible elements of a C*-algebra such that the parallel transport is ruled by Uhlmann's parallel equation.