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.

中文翻译：

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不同物理过程中混合态的Loschmidt振幅零点的普遍性及其含义

当系统进行准静态，猝灭或Uhlmann过程时，混合态密度矩阵的纯态的Loschmidt振幅显示为零。淬灭过程的Loschmidt振幅零对应于伴随着发散的动态自由能的动态量子相变（DQPT），而Uhlmann过程的Loschmidt振幅零对应于伴随着跃迁的拓扑相变（TQPT）乌尔曼相。尽管密度矩阵在准静态过程中保持不变，但Loschmidt振幅可以具有不与相变相关的零。我们提供了两级和三级系统的示例，这些系统展示了与Loschmidt振幅零点相关的有限温度或无限温度DQPT和有限温度TQPT。此外，混合态的动态相位或几何相位可以从Loschmidt振幅中提取出来。这些相位可能会在Loschmidt振幅零点处量化或显示不连续性。提出了一般两级系统纯化状态的Spinor表示，以提供有关不同过程中纯化变化的更多见解。例如，准静态过程显示为导致旋转子旋转。