Generalising quantum fisher information for quantum sensing with error detection

We explore how error characterization techniques can be used in quantum sensing. Entangled states used in sensing are highly sensitive to external parameters—but also fragile to noise and decoherence, limiting scalability. We show that an in-situ error detection can mitigate these effects and extend the scalability of quantum advantage in sensing beyond the current state-of-the-art [1,2]. That is, real-time information about specific noise events can be used to appropriately post-process the data to mitigate their effect. We formalize post-selection by developing generalizations of the Quantum Fisher Information for cases with error detection. We show that the Cramer-Rao inequality extends to all these cases appropriately.   We develop protocols for real-time (in-situ) error detection. While the no-cloning theorem prevents direct observation of quantum errors during evolution, systems of identical bosons provide a natural alternative. Due to their symmetric wavefunctions, the effect of errors is also symmetrized, enabling detection through partial measurements. Focusing on multi-mode atomic ensembles (e.g., Bose-Einstein condensates with more than two spin components), we propose protocols that split the ensemble into subsets and detect errors in one part to infer and mitigate errors in the other, which is used for sensing. This approach improves the robustness and scalability of quantum sensing, providing a practical path toward error-aware quantum metrology.