Relaxation Critical Dynamics in Measurement-induced Phase Transitions

Abstract: 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. This framework offers significant advantages for experimental MIPT detection. Our novel scheme, leveraging relaxation dynamical scaling, drastically reduces post-selection overhead, and can eliminate it completely with trackable classical simulation.