On geometric phases for quantum trajectories

A sequence of completely positive maps can be decomposed into quantum trajectories. The geometric phase or holonomy of such a trajectory is delineated. For nonpure initial states, it is shown that well-defined holonomies can be assigned by using Uhlmann's concept of parallel transport along the individual trajectories. We put forward an experimental realization of the geometric phase for a quantum trajectory in interferometry. We argue that the average over the phase factors for all quantum trajectories that build up a given open system evolution, fails to reflect the geometry of the open system evolution itself.