Measurement induced scrambling and emergent symmetries in random circuits

Quantum entanglement is affected by unitary evolution, which spreads the entanglement through the whole system, and also by measurements, which usually tends to disentangle subsystems from the rest. Their competition has been known to result in the measurement-induced phase transition. But more intriguingly, measurement alone has the ability to drive a system into different entanglement phases. In this work, we map the entanglement evolution under unitaries and/or measurements into a classical spin problem. This framework is used to understand a myriad of random circuit models analytically, including measurement-induced and measurement-only transitions. Regarding many-body joint measurements, a lower bound of measurement range that is necessary for a global scrambled phase is derived. Moreover, emergent continuous symmetries (U(1) or SU(2)) are discovered in some random measurement models in the large-$d$ (qudit dimension) limit. The emergent continuous symmetry allows a variety of spin dynamics phenomena to find their counterparts in random circuit models.