Relaxation critical dynamics in measurement-induced phase transition

The measurement-induced phase transition (MIPT) describes the nonanalytical change of the entanglement entropy resulting from the interplay between measurement and unitary evolution. In this paper, we investigate the relaxation critical dynamics near the MIPT for different initial states in a one-dimensional quantum circuit. Specifically, when the initial state is in the volume-law phase with vanishing measurement probability, we find that the half-chain entanglement entropy S decays as S \propto t^{-1} with the coefficients proportional to the size of the system in the short-time stage; In contrast, when the initial state is the product state, S increases with time as S \propto \ln{t}, consistent with previous studies. Despite these contrasting behaviors, we develop a unified scaling form to describe these scaling behaviors for different initial states where the off-critical-point effects can also be incorporated. We also confirm that this scaling form is also applicable for other initial states. Possible experimental realizations are discussed.