  In quantum metrology, quantum probe states are capable of estimating unknown
physical parameters to precisions beyond classical limits. What qualities do
such states possess? Here we relate the performance of a probe state at
estimating a parameter $\phi$ -- as quantified by the quantum Fisher
information -- to the amount of purity it loses when $\phi$ undergoes
statistical fluctuation. This indicates that the better a state is for
estimating $\phi$, the more it decoheres when $\phi$ is subject to noise.
