  We investigate measurement-induced phase transitions in the Quantum Ising
chain coupled to a monitoring environment. We compare two different limits of
the measurement problem, the stochastic quantum-state diffusion protocol
corresponding to infinite small jumps per unit of time and the no-click limit,
corresponding to post-selection and described by a non-Hermitian Hamiltonian.
In both cases we find a remarkably similar phenomenology as the measurement
strength $\gamma$ is increased, namely a sharp transition from a critical phase
with logarithmic scaling of the entanglement to an area-law phase, which occurs
at the same value of the measurement rate in the two protocols. An effective
central charge, extracted from the logarithmic scaling of the entanglement,
vanishes continuously at the common transition point, although with different
critical behavior possibly suggesting different universality classes for the
two protocols. We interpret the central charge mismatch near the transition in
terms of noise-induced disentanglement, as suggested by the entanglement
statistics which displays emergent bimodality upon approaching the critical
point. The non-Hermitian Hamiltonian and its associated subradiance spectral
transition provide a natural framework to understand both the extended critical
phase, emerging here for a model which lacks any continuous symmetry, and the
entanglement transition into the area law.
