  Many-body unitary dynamics interspersed with repeated measurements display a
rich phenomenology hallmarked by measurement-induced phase transitions.
Employing feedback-control operations that steer the dynamics toward an
absorbing state, we study the entanglement entropy behavior at the absorbing
state phase transition. For short-range control operations, we observe a
transition between phases with distinct sub-extensive scalings of entanglement
entropy. In contrast, the system undergoes a transition between volume-law and
area-law phases for long-range feedback operations. The fluctuations of
entanglement entropy and of the order parameter of the absorbing state
transition are fully coupled for sufficiently strongly entangling feedback
operations. In that case, entanglement entropy inherits the universal dynamics
of the absorbing state transition. This is, however, not the case for arbitrary
control operations, and the two transitions are generally distinct. We
quantitatively support our results by introducing a framework based on
stabilizer circuits with classical flag labels. Our results shed new light on
the problem of observability of measurement-induced phase transitions.
