  The application of postselection to a weak quantum measurement leads to the
phenomenon of weak values. Expressed in units of the measurement strength, the
displacement of a quantum coherent measuring device is ordinarily bounded by
the eigenspectrum of the measured observable. Postselection can enable an
interference effect that moves the average displacement far outside this range,
bringing practical benefits in certain situations. Employing the Fisher
information metric, we argue that the amplified displacement offers no
fundamental metrological advantage, due to the necessarily reduced probability
of success. Our understanding of metrological advantage is the possibility of a
lower uncertainty in the estimate of an unknown parameter with a large number
of trials. We analyze a situation in which the detector is pixelated with a
finite resolution, and in which the detector is afflicted by random
displacements: imperfections which degrade the fundamental limits of parameter
estimation. Surprisingly, weak-value amplification is no more robust to them
than a technique making no use of the amplification effect brought about by a
final, postselected measurement.
