Post-selected Criticality in Measurement-induced Phase Transitions

Information-theoretic phase transitions, such as the measurement-induced phase transition (MIPT), characterize the robustness of quantum dynamics to local monitoring and are naturally formulated in terms of trajectories conditioned on typical measurement outcomes, which are naively accessible only through post-selection. Here we implement forced measurements to investigate how explicit post-selection alters the nature of the transition. We find that post-selection fundamentally alters the universality class by reweighting trajectories that are otherwise rare. In particular, we obtain a correlation-length exponent $\nu\approx 2.1$ larger than that of the standard MIPT and a negative effective central charge $c_\mathrm{eff}\approx -0.4$. We also compare the post-selected MIPT to the entanglement transition of Random Tensor Networks (RTN), and demonstrate that their universality class is the same. This setup further allows time-periodic, translationally-invariant circuits with post-selected weak measurements. In both models, we find that an onsite dimension of at least 3 (qutrits but not qubits) is necessary to induce a transition.