  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 $\rho_a$ 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 $\rho_a$ 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 ($S_z s_z$-type in spin language)
coupling, the critical values $M_c$ for the measurement-induced entanglement
transition depends sensitively on $\rho_a$. Moreover, our results indicate that
for a sufficiently small $\rho_a$ 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 ($S_z s_x$-type) coupling. Specifically, the dynamics is much
less sensitive to the concentration $\rho_a$ of detectors than in the first
model. Furthermore, the dependence of entanglement on the coupling strength $M$
is strongly non-monotonic, indicating re-entrance of the entangling phase at
large $M$.
