Volume-law supression in (2 + 1)D subsystem-symmetric monitored quantum circuits

The competing effects of unitary evolution and measurements, and the measurement-induced phase transition they precipitate, have brought a new perspective to the study many-body entanglement. Key to this phenomenon is a stable phase of volume-law entanglement, which persists even with growing system size. This is despite measurements having the capacity to reduce entanglement anywhere they're performed, whereas local unitary dynamics affect entanglement only when operating across subsystem boundaries. We analyze a collection of monitored (2+1)D circuits subject to 2D cluster state stabilizer and computational basis measurements and unitary dynamics respecting varying levels of symmetry -- both global and subsystem -- inspired by measurement-based quantum computation. Our investigation shows how the imposition of symmetry leads to the emergence of a new universality class of the associated phase transitions, accompanied by the suppression or annihilation of the volume-law entanglement phase. These results on the volume-law phase's stability and change in transition class may have more general implications for the persistence of entanglement in monitored circuits with conserved charges.