Measurement-Induced Spectral Transition

We show that noisy quantum dynamics exposed to generalized measurements exhibit a spectral transition between gapless and gapped phases. To this end, we employ the Lyapunov spectrum obtained through singular values of a nonunitary matrix describing the dynamics. We discover that the gapless and gapped phases, respectively, correspond to the volume-law and area-law phases of the entanglement entropy for the dominant Lyapunov vector. This correspondence between the spectral gap and the scaling of entanglement offers an intriguing common structure with ground-state phase transitions. We also discuss some crucial differences from ground-state transitions, such as the extensive entanglement and the exponentially small gaps. Furthermore, we show that the spectral transition leads to the transition of the timescale for the memory loss and purification of initial states.