Stabilizing Criticality and Long-Ranged Entanglement in Measurement-Only Dynamics

Quantum circuit dynamics with local, projective measurements provide a way of realizing a rich spectrum of entangled states of quantum matter. We study a (2+1)-dimensional quantum circuit involving only repeated, local projective measurements, and demonstrate that this circuit stabilizes a phase with (i) a logarithmic violation of area-law entanglement which coexists with (ii) true long-ranged entanglement, as quantified by the conditional mutual information. This phase provides a non-equilibrium generalization of a gapless spin liquid state which can arise as a stable, zero-temperature phase of quantum matter in two spatial dimensions. We relate the entanglement properties of this highly-entangled, critical phase to a classical loop ensemble in three spatial dimensions. A phase transition out of this long-range-entangled phase can be driven by tuning the rate of different local projective measurements, and which we study in numerical simulations of these dynamics.