  Based on a kinematic approach in defining a geometric phase for a density
matrix, we define the generalized Loschmidt overlap amplitude (GLOA) for an
open system for arbitrary quantum evolution. The GLOA reduces to the Loschmidt
overlap amplitude (LOA) with a modified dynamic phase for unitary evolution of
a pure state, with the argument of the GLOA well-defined by the geometric
phase, thus possessing similar physical interpretation to that of the LOA. The
rate function for the GLOA exhibits non-analyticity at a critical time, which
corresponds to the dynamical quantum phase transition. We observe that the
dynamical quantum phase transition related to GLOA is not destroyed under a
finite temperature and weak enough dissipation. In particular, we find that a
new type of dynamical quantum phase transition emerges in a dissipation system.
The proposed GLOA provides a powerful tool in the investigation of a dynamical
quantum phase transition in an arbitrary quantum system, which not only can
characterize the robustness of the dynamical quantum phase transition but also
can be used to search for new transitions.
