  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.
