Usefulness of Quantum Entanglement for Enhancing Precision in Frequency Estimation

We investigate strategies for reaching the ultimate limit on the precision of frequency estimation when the number of probes used in each run of the experiment is fixed. That limit is set by the quantum Cram\'er-Rao bound (QCRB), which predicts that the use of maximally entangled probes enhances the estimation precision, when compared with the use of independent probes. However, the bound is only achievable if the statistical model used in the estimation remains identifiable throughout the procedure. This in turn sets different limits on the maximal sensing time used in each run of the estimation procedure, when entangled and independent probes are used. When those constraints are taken into account, one can show that, when the total number of probes and the total duration of the estimation process are counted as fixed resources, the use of entangled probes is, in fact, disadvantageous when compared with the use of independent probes. In order to counteract the limitations imposed on the sensing time by the requirement of identifiability of the statistical model, we propose a time-adaptive strategy, in which the sensing time is adequately increased at each step of the estimation process, calculate an attainable error bound for the strategy and discuss how to optimally choose its parameters in order to minimize that bound. We show that the proposed strategy leads to much better scaling of the estimation uncertainty with the total number of probes and the total sensing time than the traditional fixed-sensing-time strategy. We also show that, when the total number of probes and the total sensing time are counted as resources, independent probes and maximally entangled ones have now the same performance, in contrast to the non-adaptive strategy, where the use of independent is more advantageous than the use of maximally entangled ones.