Transitions of the Lyapunov spectrum and entanglement entropy in monitored quantum dynamics with homogeneous unitary gates

Abstract We explore the Lyapunov spectrum and entanglement entropy in quantum trajectories evolved by quantum measurements and spatially homogeneous unitary gates. In models with temporally random and Floquet unitary gates, we find that the Lyapunov exponents typically converge to values independent of measurement outcomes and that spectral transitions of the Lyapunov spectrum and entanglement transitions of ground states occur at the same measurement thresholds. Our results indicate that (i) randomness due to quantum measurements is enough, or equivalently, randomness of unitary gates is not essential, for the typical convergence of the Lyapunov spectrum and that (ii) the correspondence between the spectral transition and entanglement transition is ubiquitous in a wide range of systems. We also discuss why the typical convergence occurs, in light of the uniqueness and positive definiteness of stationary states in the completely-positive trace-preserving (CPTP) dynamics averaged over random measurement outcomes. Our discussions indicate that properties of the averaged CPTP dynamics play a key role in characterizing measurement-induced transitions in quantum trajectories.