Kinematic approach to off-diagonal geometric phases of nondegenerate and   degenerate mixed states
  Off-diagonal geometric phases have been developed in order to provide information of the geometry of paths that connect noninterfering quantal states. We propose a kinematic approach to off-diagonal geometric phases for pure and mixed states. We further extend the mixed state concept proposed in [Phys. Rev. Lett. {\bf 90}, 050403 (2003)] to degenerate density operators. The first and second order off-diagonal geometric phases are analyzed for unitarily evolving pairs of pseudopure states. 
D.M. Tong, Erik Sj\"oqvist, Stefan Filipp, L.C. Kwek, C.H. Oh