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The problem of geometric phase for an open quantum system is reinvestigatedin a unifying approach. Two of existing methods to define geometric phase, oneby Uhlmann's approach and the other by kinematic approach, which have beenconsidered to be distinct, are shown to be related in this framework. Themethod is based upon purification of a density matrix by its uniformdecomposition and a generalization of the parallel transport condition obtainedfrom this decomposition. It is shown that the generalized parallel transportcondition can be satisfied when Uhlmann's condition holds. However, it does notmean that all solutions of the generalized parallel transport condition arecompatible with those of Uhlmann's one. It is also shown how to recover theearlier known definitions of geometric phase as well as how to generalize themwhen degeneracy exists and varies in time.Comment: 4 pages, extended result

General setting for a geometric phase of mixed states under an arbitrary nonunitary evolution