  Recent theoretical work has shown that the competition between coherent
unitary dynamics and stochastic measurements, performed by the environment,
along wavefunction trajectories can give rise to transitions in the
entanglement scaling. In this work, complementary to these previous studies, we
analyze a situation where the role of Hamiltonian and dissipative dynamics is
reversed. We consider an engineered dissipation, which stabilizes an entangled
phase of a quantum spin$-1/2$ chain, while competing single-particle or
interacting Hamiltonian dynamics induce a disentangled phase. Focusing on the
single-particle unitary dynamics, we find that the system undergoes an
entanglement transition from a logarithmic growth to an area law when the
competition ratio between the unitary evolution and the non-unitary dynamics
increases. We evidence that the transition manifests itself in state-dependent
observables at a finite competition ratio for Hamiltonian and measurement
dynamics. On the other hand, it is absent in trajectory-averaged steady-state
dynamics, governed by a Lindblad master equation: although purely dissipative
dynamics stabilizes an entangled state, for any non-vanishing Hamiltonian
contribution the system ends up irremediably in a disordered phase. In
addition, a single trajectory analysis reveals that the distribution of the
entanglement entropy constitutes an efficient indicator of the transition.
Complementarily, we explore the competition of the dissipation with coherent
dynamics generated by an interacting Hamiltonian, and demonstrate that the
entanglement transition also occurs in this second model. Our results suggest
that this type of transition takes place for a broader class of Hamiltonians,
underlining its robustness in monitored open quantum many-body systems.
