Measurement induced phase transition in a solvable all-to-all Brownian circuit model

Competition between unitary dynamics that scrambles quantum information non-locally and local measurements that probe and collapse the quantum state can result in a measurement induced entanglement phase transition. Here we study this phenomenon in an all-to-all Brownian hybrid circuit model of qubits that is analytically tractable. A part of the system is initially entangled with a reference which remains mixed at low measurement rates but is purified at high measurement rates. After circuit averaging, purity of the reference can be represented as a path integral coupling four replicas with twisted boundary conditions. Saddle point analysis reveals a second-order bulk phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. This model also allows us to characterize the transition analytically and derive a simple mean field type theory for this transition.