  The intensely studied measurement-induced entanglement phase transition has
become a hallmark of non-unitary quantum many-body dynamics. Usually, such a
transition only shows up at the level of each individual quantum trajectory,
and is absent for the density matrix averaged over measurement outcomes. In
this work, we introduce a class of adaptive random circuit models with feedback
that exhibit transitions in both settings. After each measurement, a unitary
operation is either applied or not depending on the measurement outcome, which
steers the averaged density matrix towards a unique state above a certain
measurement threshold. Interestingly, the transition for the density matrix and
the entanglement transition in the individual quantum trajectory in general
happen at \textit{different} critical measurement rates. We demonstrate that
the former transition belongs to the parity-conserving universality class by an
explicit mapping to a classical branching-annihilating random walk process.
