Beyond the Quantum Cramér-Rao Bound

The quantum Cramér-Rao bound (QCRB) stands as a cornerstone of quantum metrology. Yet, akin to its classical counterpart, it provides only local information and overlooks higher-order details. We leverage the theory of higher-order asymptotics to circumvent these issues, providing corrections to the performance of estimators beyond the QCRB. While the QCRB often yields a whole family of optimal states and several optimal measurements, our approach allows us to identify optimal states and measurements within the family that are otherwise equivalent according to the QCRB alone; these are requisite for optimal metrology before reaching the asymptotic limit. These results are especially pertinent when dealing with unitary processes.