  The Loschmidt amplitude of the purified states of mixed-state density
matrices is shown to have zeros when the system undergoes a quasistatic,
quench, or Uhlmann process. While the Loschmidt-amplitude zero of a quench
process corresponds to a dynamical quantum phase transition (DQPT) accompanied
by the diverging dynamical free energy, the Loschmidt-amplitude zero of the
Uhlmann process corresponds to a topological phase transition (TQPT)
accompanied by a jump of the Uhlmann phase. Although the density matrix remains
intact in a quasistatic process, the Loschmidt amplitude can have zeros not
associated with a phase transition. We present examples of two-level and
three-level systems exhibiting finite- or infinite- temperature DQPTs and
finite-temperature TQPTs associated with the Loschmidt-amplitude zeros.
Moreover, the dynamical phase or geometrical phase of mixed states can be
extracted from the Loschmidt amplitude. Those phases may become quantized or
exhibit discontinuity at the Loschmidt-amplitude zeros. A spinor representation
of the purified states of a general two-level system is presented to offer more
insights into the change of purification in different processes. The
quasistatic process, for example, is shown to cause a rotation of the spinor.
