  We derive lower bounds on the variance of estimators in quantum metrology by
choosing test observables that define constraints on the unbiasedness of the
estimator. The quantum bounds are obtained by analytical optimization over all
possible quantum measurements and estimators that satisfy the given
constraints. We obtain hierarchies of increasingly tight bounds that include
the quantum Cram\'er-Rao bound at the lowest order. In the opposite limit, the
quantum Barankin bound is the variance of the locally best unbiased estimator
in quantum metrology. Our results reveal generalizations of the quantum Fisher
information that are able to avoid regularity conditions and identify threshold
behavior in quantum measurements with mixed states, caused by finite data.

Quantum Physics