  We utilize the concept of a measurement-induced entanglement transition to
analyze the interplay and competition of processes that generate and destroy
entanglement in a one-dimensional quantum spin chain evolving under a locally
noisy and disordered Hamiltonian. We employ continuous measurements of variable
strength to induce a transition from volume to area-law scaling of the
steady-state entanglement entropy. While static background disorder
systematically reduces the critical measurement strength, this critical value
depends non-monotonically on the strength of non-static noise. According to the
extracted fine-size scaling exponents, the universality class of the transition
is independent of the noise and disorder strength. We interpret the results in
terms of the effect of static and non-static disorder on the intricate dynamics
of the entanglement generation rate due to the Hamiltonian in the absence of
measurement, which is fully reflected in the behavior of the critical
measurement strength. Our results establish a firm connection between this
entanglement growth and the steady-state behavior of the measurement-controlled
systems, which therefore can serve as a tool to quantify and investigate
features of transient entanglement dynamics in complex many-body systems via a
steady-state phase transition.
