A fundamental characteristic of non-Hermitian quantum systems is the nonunitary time evolution under a Euclidean inner product. Until now, a consensus on the mathematical formulation for the dynamics of non-Hermitian quantum systems has been lacking. Here, we focus on the quantum metric framework, in which the state normalization is preserved through a fundamental redefinition of the inner product of the Hilbert space. We generalize this framework to the $\mathcal{PT}$-broken region and provide a method to construct the metric operator of a general two-band non-Hermitian model. Then we redefine the Loschmidt echo and investigate the nonadiabatic dynamics in non-Hermitian quantum systems. Taking the $\mathcal{PT}$-symmetric staggered tight-binding chain model as an example, we demonstrate that the long-time average of the rate function can be used to characterize the $\mathcal{PT}$ phase transitions. In the non-Hermitian SSH model, we investigate the dynamical quantum phase transitions (DQPTs) and show that this framework can remove the accidental DQPTs and thus build a bridge between DQPTs and equilibrium topological phase transitions.

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• Dynamical phase transitions
• Quantum quench
• Non-Hermitian systems