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 w*w =w*w, where w is a lifting of acurve y in Ω, i.e. w(t)o(t)* = y(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.