  In conventional measurement, to reach the greatest accuracy of parameter
estimation, all samples must be measured since each independent sample contains
the same quantum Fisher information. In postselected metrology, postselection
can concentrate the quantum Fisher information of the initial samples into a
tiny post-selected sub-ensemble. It has been proven that this quantum advantage
can not be realized in any classically commuting theory. In this work, we
present that the advantage of postselection in weak value amplification (WVA)
can not be achieved without quantum coherence. The quantum coherence of the
initial system is closely related to the preparation costs and measurement
costs in parameter estimation. With the increase of initial quantum coherence,
the joint values of preparation costs and measurement costs can be optimized to
smaller. Moreover, we derive an analytical tradeoff relation between the
preparation, measurement and the quantum coherence. We further experimentally
test the tradeoff relation in a linear optical setup. The experimental and
theoretical results are in good agreement and show that the quantum coherence
plays a key role in bounding the resource costs in the postselected metrology
process.
