Measurement-Induced Entanglement Phase Transitions in Constrained Hilbert Spaces

We explore the influence of Hilbert space constraints on the critical properties of measurement-induced phase transitions. Our investigation centers on the low-energy Hilbert space of the 1D PXP model, characterized by the constraint that adjacent states cannot be simultaneously excited. Notably, this constraint leads to a Hilbert space size growth as ~Φ^L instead of the typical ~2^L where Φ is the golden ratio and L is the number of qubits, rendering computational simulations less resource intensive. We study monitored random quantum circuits containing single-qubit Haar random gates (which are entangling by virtue of constraints) and projective measurements. We find compelling numerical evidence that the system undergoes a novel phase transition from area-law to volume-law entanglement as a function of the rate of projective measurements. Remarkably, we discover that the critical behavior of the entanglement transition belongs to a novel universality class, distinct from the behavior observed in Haar random circuits without the constraint. This underscores the profound influence of Hilbert space constraints on the system's dynamics. Hence, we comment on the implications of our results for measurement-induced phase transitions in lattice gauge theories and possibilities for experimental realization in Rydberg atom quantum simulators.