Measurement-induced phase transitions (MIPTs), characterizing abrupt changes in entanglement properties in quantum many-body systems subjected to unitary evolution with interspersed projective measurements, have garnered increasing interest. Here, we investigate the driven critical dynamics of MIPTs by linearly tuning the measurement probability $p$ across the critical point $p_c$ at velocity $R$. While the Kibble-Zurek mechanism (KZM) has been extensively validated for driven dynamics across conventional phase transitions, we demonstrate that it exhibits discrepancies when applied to MIPTs, depending on the initial states. Specifically, we show that although the KZM remains valid for quenches starting from the area-law phase, it breaks down for quenches initiated from the volume-law phase. Surprisingly, a unified finite-time scaling (FTS) form can be established to describe these distinct scaling behaviors. A mechanism in which the FTS is protected by a driving-induced quasi-steady state is then proposed. In addition, from the volume-law phase, we uncover a new scaling relation of the entanglement entropy $S$, namely, $S\propto R^{1/r}$ in which $r=z+1/\nu$ with $z$ and $\nu$ being the dynamic and correlation length exponents, respectively. By establishing a dynamic scaling theory for this entanglement transition, we not only bring a perspective on MIPTs that is amenable to experimental detection in rapidly advancing quantum computing platforms, but also fundamentally extend the framework of nonequilibrium critical theory beyond the KZM.

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• Dynamical phase transitions
• Nonequilibrium statistical mechanics
• Phase transitions
• Quantum circuits