Generalization of geometric phase to completely positive maps
  We generalize the notion of relative phase to completely positive maps with known unitary representation, based on interferometry. Parallel transport conditions that define the geometric phase for such maps are introduced. The interference effect is embodied in a set of interference patterns defined by flipping the environment state in one of the two paths. We show for the qubit that this structure gives rise to interesting additional information about the geometry of the evolution defined by the CP map. 
Marie Ericsson, Erik Sj\"oqvist, Johan Br\"annlund, Daniel K.L. Oi,   and Arun K. Pati