Metrology and multipartite entanglement in measurement-induced phase transition

Measurement-induced phase transition arises from the competition between a deterministic quantum evolution and a repeated measurement process. We explore the measurement-induced phase transition in the no-click limit of a one dimensional Ising chain through the quantum Fisher information (QFI) in two different metrological scenarios. In the first one, where we use an interferometric approach, we demonstrate through the scaling behavior of the QFI the transition of the multi-partite entanglement across the phases. In the second one, we consider the QFI with respect to the measurement strength. In this case, we show that the QFI signals the measurement induced criticality through its non-analytic behavior, in analogy to the fidelity susceptibility, in standard continuous phase transitions. Our results offer novel insights into the features of a quantum system undergoing measurement-induced phase transition, and indicate potential avenues for further exploration in the field of quantum physics.