The interplay between quantum measurement and localization significantly alters the spreading of quantum information. In this work, we investigate monitored free fermion chains with localized potentials. With the aid of the quantum trajectory method and finite-size analysis, we numerically reveal that the Berezinskii-Kosterlitz-Thouless transition, which emerges in monitored free fermion chains, is robust in the presence of localization. To understand this, we construct a phase diagram that diverges at a boundary described by entanglement propagation. We find that the monitored system with Stark-localized decays quickly under small measurement strength, whereas the Anderson-localized system decays more slowly. By incorporating localization effects, this work advances our understanding of how measurement competes with the coherent spreading of quantum information. Our findings can potentially be realized in cold atom systems, trapped ions, and quantum dot arrays.

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• Entanglement entropy
• Localization
• Quantum measurements
• Lattices
• Schroedinger equation