Charge/Spin Sharpening Transition on Dynamical Quantum Trees

The dynamics of monitored quantum systems can generally be described by either a entangling phase where the initial information is protected by quantum scrambling and a disentangling phase where information is lost due to measurement. The transition between these two phases is so called the measurement induced phase transition. Recent work shows that the dynamics inside the entangling phase can be enriched by symmetries of quantum dynamics. However, although a random walk picture about Abelian sharpening transition has been established, the understanding about phase transitions with non-Abelian symmetry like SU(2) is still limited. In this work, we study the charge/spin sharpening in monitored dynamical quantum trees, which has more analytical trackabilities. We find that when the dynamics has U(1)/SU(2) symmetry, an extra transition can be defined based on whether the measurements can fully determine the final charge/spin or not, besides the usual entanglement transition critical point in previous work.