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We consider a quantum many-body lattice system that is coupled to ancillary degrees of freedom (``detectors''), which are periodically measured by means of strong projective measurements. The concentration ₀ of ancillae and their coupling M to the main system are considered as parameters. We explore the dynamics of density and of entanglement entropy in the chain, for various values of ₀ and M for two models of the detector-chain interaction that couple the local density in the chain to a detector degree of freedom. It is found that, for the density-density (Sₙsₙ-type in spin language) coupling, the critical values M₂ for the measurement-induced entanglement transition depends sensitively on ₀. Moreover, our results indicate that for a sufficiently small ₀ the transition in this model disappears; i. e. , a finite density of detectors is needed to reach a disentangling phase. The behavior is qualitatively different for the second model, with density-hopping (Sₙsₗ-type) coupling. Specifically, the dynamics is much less sensitive to the concentration ₀ of detectors than in the first model. Furthermore, the dependence of entanglement on the coupling strength M is strongly nonmonotonic, indicating reentrance of the entangling phase at large M.
Doggen et al. (Tue,) studied this question.