  The confluence of unitary dynamics and non-unitary measurements gives rise to
intriguing and relevant phenomena, generally referred to as measurement-induced
phase transitions. These transitions have been observed in quantum systems
composed of trapped ions and superconducting quantum devices. However, their
experimental realization demands substantial resources, primarily owing to the
classical tracking of measurement outcomes, known as post-selection of
trajectories. In this work, we first describe the statistical properties of an
interacting transmon array which is repeatedly measured, and predict the
behavior of relevant quantities in the area-law phase using a combination of
the replica method and non-Hermitian perturbation theory. We show numerically
that a transmon array, modeled by an attractive Bose-Hubbard model, in which
local measurements of the number of bosons are probabilistically interleaved,
exhibits a phase transition in the entanglement entropy properties of the
ensemble of trajectories in the steady state. Furthermore, by using
deterministic feedback operations after the local number measurements, the
distribution of the number of bosons measured at a single site carries
information on the phase in the entanglement of individual trajectories.
Interestingly, we can extract information about the phase and the phase
transition from simple observables without considering an absorbing state in
the feedback pattern. This implies that the feedback measurement approach might
be a viable experimental option to use simple observables to study some aspects
of the entanglement phase transition in individual trajectories.
