Abstract: Studying the effect of local projective measurements on the scaling of entanglement entropy is an intense topic of research in the context of measurement-induced phase transitions. While it is traditionally studied in discrete circuit models, a close continuous-time analogy can be drawn with monitored open quantum dynamics, where a record of the registered quantum-jump clicks allows one to reconstruct the pure-state stochastic trajectories. I will first show how monitoring a cavity array with two competing protocols, one generating coherence and the other performing a weak number measurement, induces an entanglement scaling transition for the trajectory states across a critical point. While for this problem the associated master equation of the open system is not invariant, I will illustrate next that solely changing the monitoring protocol of the same dissipative processes can also have an impact on the stochastic trajectory dynamics. In this setup, the quantum jumps of spontaneous emission are monitored after passing through a linear interferometer, affecting both the scaling of trajectory entanglement entropy and the hardness of sampling the click outcomes. We show that this problem is equivalent to the famous problem of Fock-state boson sampling.
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