Measurement induced faster symmetry restoration in quantum trajectories

Continuous measurement of quantum systems provides a standard route to quantum trajectories through the successive acquisition of information which further results in measurement back-action. In this work, we harness this back-action as a resource for global $U(1)$ symmetry restoration where continuous measurement is combined with a $U(1)$-preserving unitary evolution. Starting from a $U(1)$ symmetry-broken initial state, we simulate quantum trajectories generated by continuous measurements of both global and local observables. We show that under global monitoring, states containing superpositions of distant charge sectors restore symmetry faster than those involving nearby sectors. We establish the universality of this behavior across different measurement protocols. Finally, we demonstrate that local monitoring can further accelerate symmetry restoration for certain states that relax slowly under global monitoring.