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In this work, we investigate the localization properties of a paradigmaticmodel, coupled to a monitoring environment and possessing a many-body localized(MBL) phase. We focus on the post-selected no-click limit with quench randomrates, i.e., random gains and losses. In this limit, the system is modeled byadding an imaginary random potential, rendering non-Hermiticity in the system.Numerically, we provide an evidence that the system is localized for any finiteamount of disorder. To analytically understand our results, we extend thequantum random energy model (QREM) to the non-Hermitian scenario. The HermitianQREM has been used previously as a benchmark model for MBL. The QREM exhibits asize-dependent MBL transition, where the critical value scales as $W_c\sim\sqrt{L} \ln{L}$ with system size and presenting many-body mobility edges. Wereveal that the non-Hermitian QREM with random gain-loss offers a significantlystronger form of localization, evident in the nature of the many-body mobilityedges and the value for the transition, which scales as $W_c\sim \ln^{1/2}{L}$with the system size.

Stable many-body localization under random continuous measurements in the no-click limit