  The problem of geometric phase for an open quantum system is reinvestigated
in a unifying approach. Two of existing methods to define geometric phase, one
by Uhlmann's approach and the other by kinematic approach, which have been
considered to be distinct, are shown to be related in this framework. The
method is based upon purification of a density matrix by its uniform
decomposition and a generalization of the parallel transport condition obtained
from this decomposition. It is shown that the generalized parallel transport
condition can be satisfied when Uhlmann's condition holds. However, it does not
mean that all solutions of the generalized parallel transport condition are
compatible with those of Uhlmann's one. It is also shown how to recover the
earlier known definitions of geometric phase as well as how to generalize them
when degeneracy exists and varies in time.
