  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 analytically tractable all-to-all Brownian hybrid
circuit model composed of qubits. The system is initially entangled with an
equal sized reference, and the subsequent hybrid system dynamics either
partially preserves or totally destroys this entanglement depending on the
measurement rate. Our approach can access a variety of entropic observables
which are distinguished by the averaging procedure, and for concreteness we
focus on a particular purity quantity for which the averaging is particularly
simple. We represent the purity as a path integral coupling four replicas with
twisted boundary conditions. Saddle-point analysis reveals a second-order phase
transition corresponding to replica permutation symmetry breaking below a
critical measurement rate. The transition is mean-field-like and we
characterize the critical properties near the transition in terms of a simple
Ising field theory in 0+1 dimensions. In addition to studying the purity of the
entire system, we study subsystem purities and relate these results to
manifestations of quantum error correction in the model. We also comment on the
experimental feasibility for simulating this averaged purity, and corroborate
our results with exact diagonalization for modest system sizes.
