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The implementation of weak-value amplification requires the pre- andpost-selection of states of a quantum system, followed by the observation ofthe response of the meter, which interacts weakly with the system. Dataacquisition from the meter is conditioned to successful post-selection events.Here we derive an optimal post-selection procedure for estimating the couplingconstant between system and meter, and show that it leads both to weak-valueamplification and to the saturation of the quantum Fisher information, underconditions fulfilled by all previously reported experiments on theamplification of weak signals. For most of the pre-selected states, fullinformation on the coupling constant can be extracted from the meter data setalone, while for a small fraction of the space of pre-selected states, it mustbe obtained from the post-selection statistics.

Weak-value amplification as an optimal metrological protocol