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We utilize the concept of a measurement-induced entanglement transition toanalyze the interplay and competition of processes that generate and destroyentanglement in a one-dimensional quantum spin chain evolving under a locallynoisy and disordered Hamiltonian. We employ continuous measurements of variablestrength to induce a transition from volume to area-law scaling of thesteady-state entanglement entropy. While static background disordersystematically reduces the critical measurement strength, this critical valuedepends non-monotonically on the strength of non-static noise. According to theextracted fine-size scaling exponents, the universality class of the transitionis independent of the noise and disorder strength. We interpret the results interms of the effect of static and non-static disorder on the intricate dynamicsof the entanglement generation rate due to the Hamiltonian in the absence ofmeasurement, which is fully reflected in the behavior of the criticalmeasurement strength. Our results establish a firm connection between thisentanglement growth and the steady-state behavior of the measurement-controlledsystems, which therefore can serve as a tool to quantify and investigatefeatures of transient entanglement dynamics in complex many-body systems via asteady-state phase transition.

Diagnosing entanglement dynamics in noisy and disordered spin chains via the measurement-induced steady-state entanglement transition