Geometric phases for mixed states and decoherence
  The gauge invariance of geometric phases for mixed states is analyzed by using the hidden local gauge symmetry which arises from the arbitrariness of the choice of the basis set defining the coordinates in the functional space. This approach gives a reformulation of the past results of adiabatic, non-adiabatic and mixed state geometric phases. The geometric phases are identified uniquely as the holonomy associated with the hidden local gauge symmetry which is an exact symmetry of the Schr\"{o}dinger equation. The purification and its inverse in the description of de-coherent mixed states are consistent with the hidden local gauge symmetry. A salient feature of the present formulation is that the total phase and visibility in the mixed state, which are directly observable in the interference experiment, are manifestly gauge invariant. 
Kazuo Fujikawa