Post-Selected Criticality in Measurement-Induced Phase Transitions in Locally Constrained Hilbert Spaces

We investigate the post-selected critical properties of measurement-induced phase transitions (MIPT) in a locally constrained Hilbert space. Specifically, we focus on the low-energy Hilbert space of the 1D PXP model, where adjacent qubits cannot be simultaneously excited.

Our study examines monitored random quantum circuits consisting of single-qubit Haar random gates, which are entangling by virtue of the constraint, and post-selected projective measurements, where the qubits are consistently projected onto the ground state. Post-selected measurements profoundly impact the dynamics of the 1D PXP model, as projecting a qubit onto an excited state forces its adjacent qubits into the ground state due to the local constraint, whereas projecting a qubit onto the ground state leaves its neighbors unaffected. As a result, we anticipate that consistently projecting qubits onto the ground state may give rise to new criticality in the system. Our numerical findings provide strong evidence for a novel phase transition from area-law to volume-law entanglement driven by the rate of post-selected projective measurements. Interestingly, we also discover that the universality of post-selected measurements differs from that of non-selective measurements in this constrained Hilbert space, offering deeper insights on the influence of local constraints and post-selection in our system.