Controlling Waiting Time Statistics in Monitored Collective Spins: Mitigating Detector's Resolution Barrier in Measurement-Induced Phase Transitions

Abstract

In collective dissipative spin systems, the postselection barrier can be partially mitigated; however, a further obstacle may be posed by the finite temporal resolution of detectors. In this work, we investigate how initial-state inhomogeneities can control waiting-time statistics between quantum jumps, thereby mitigating the detector-resolution problem. We consider a collectively monitored spin model with a boundary time-crystalline phase, introducing inhomogeneity by partitioning the ensemble into two subsystems rotated by an angle $θ$. We find that the measurement-induced phase transition survives under inhomogeneities, with different entanglement scaling regimes. The waiting time increases with $θ$, scaling as $1/N$ but with a prefactor strongly enhanced by orders of magnitude, and in the anti-aligned limit $θ= π$ it remains finite, fully resolving the resolution barrier. This mitigation, however, comes at a cost: the entanglement saturation time becomes significantly longer, partially reintroducing the postselection barrier. Our results highlight a trade-off between detector resolution and postselection overhead, with direct implications for the experimental observation of measurement-induced phenomena.