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

Competition between unitary scrambling dynamics that protects 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 at large N. 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 permutation symmetry breaking below a critical measurement rate. We demonstrate that this bulk transition is directly responsible for the purity transition and check our analytical results against exact diagonalization numerics. This model allows us to explicitly observe and characterize the transition without relying on numerics or the limit of large local Hilbert space dimension which were crucial in previous analytical studies.