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 $0\leq q\leq 1$. 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. The detector efficiency behaves instead as a compatible mixed-state estimation parameter, as signaled by the vanishing mean Uhlmann curvature. These results show that incomplete trajectory information destroys the metrological enhancement associated with monitored criticality through a mechanism that differs from ordinary thermal smearing.