  We study quantum circuits consisting of unitary gates, projective
measurements, and control operations that steer the system towards a pure
absorbing state. Two types of phase transition occur as the rate of these
control operations is increased: a measurement-induced entanglement transition,
and a directed percolation transition into the absorbing state (taken here to
be a product state). In this work we show analytically that these transitions
are generically distinct, with the quantum trajectories becoming disentangled
before the absorbing state transition is reached, and we analyze their critical
properties. We introduce a simple class of models where the measurements in
each quantum trajectory define an Effective Tensor Network (ETN) -- a subgraph
of the initial spacetime graph where nontrivial time evolution takes place. By
analyzing the entanglement properties of the ETN, we show that the entanglement
and absorbing-state transitions coincide only in the limit of infinite local
Hilbert-space dimension. Focusing on a Clifford model which allows numerical
simulations for large system sizes, we verify our predictions and study the
finite-size crossover between the two transitions at large local Hilbert space
dimension. We give evidence that the entanglement transition is governed by the
same fixed point as in hybrid circuits without feedback.
