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We study free fermion systems under adaptive quantum dynamics consisting of unitary gates and projective measurements followed by corrective unitary operations. We further introduce a classical flag for each site, allowing for an active or inactive status which determines whether or not the unitary gates are allowed to apply. In this dynamics, the individual quantum trajectories exhibit a measurement-induced entanglement transition from critical to area-law scaling above a critical measurement rate, similar to previously studied models of free fermions under continuous monitoring. Furthermore, we find that the corrective unitary operations can steer the system into a state characterized by charge-density-wave order. Consequently, an additional phase transition occurs, which can be observed at both the level of the quantum trajectory and the quantum channel. We establish that the entanglement transition and the steering transition are fundamentally distinct. The latter transition belongs to the parity-conserving (PC) universality class, arising from the interplay between the inherent fermionic parity and classical labelling. We demonstrate both the entanglement and the steering transitions via efficient numerical simulations of free fermion systems, which confirm the PC universality class of the latter. 

The intensely studied measurement-induced entanglement phase transition has become a hallmark of nonunitary quantum many-body dynamics. Usually, such a transition only appears 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 different critical measurement rates. We demonstrate that the former transition belongs to the parity-conserving universality class by explicitly mapping to a classical branching-annihilating random-walk process.