Quantum Fisher information and imperfect detection in a monitored fermion chain

We study the metrological properties of a continuously monitored Kitaev chain in the presence of imperfect detection. The system is conditioned on a no-click record, while each emitted fermion is registered only with probability 0≤ q≤ 1. Because the conditional dynamics remains Gaussian, the steady state is fully characterized by the fermionic correlation matrix. This allows a direct evaluation of the quantum Fisher information and of the mean Uhlmann curvature. For perfect detection, the monitored steady state retains a singular critical structure and the quantum Fisher information with respect to the chemical potential becomes super-extensive. For any $q<1$, imperfect detection introduces a finite smoothing length that rounds the singularity and restores extensive scaling. Treating the detector efficiency as an additional estimation parameter reveals a generic multiparameter incompatibility for imperfect detection, with compatibility recovered only in the perfect-detection limit, where the mean Uhlmann curvature vanishes. These results show that incomplete trajectory information destroys the metrological enhancement associated with monitored criticality through a mechanism distinct from ordinary thermal smearing.