An entanglement transition on a quantum tree and its relation to the measurement transition in all-to-all circuits

The measurement phase transition is a dynamical transition resulting from the competition between entangling gates and non-unitary operations. Its universal properties are known exactly in a type of classical limit obtained by considering the Hartley function or by taking the local Hilbert space dimension to infinity. It can also be understood using numerics on large systems for specific fine tuned models, such as Clifford or automaton circuits. In all these cases the universality of the transition is some version of percolation (or close to it). However, we do not have such a comprehensive understanding of its nature for generic quantum evolution, beyond small system numerics. I will describe our work on all-to-all circuits, where we argue that the transition is controlled by an entanglement transition on a tree tensor network. In this case rigorous results can be obtained for generic entangling gates. We show that this transition occurs at a finite distance from the classical percolation transition and also has a different universality class.