  Weak-value amplification employs postselection to enhance the measurement of
small parameters of interest. The amplification comes at the expense of reduced
success probability, hindering the utility of this technique as a tool for
practical metrology. Following other quantum technologies that display a
quantum advantage, we formalize a quantum advantage in the success probability
and present a scheme based on non-linear collective Hamiltonians that shows a
super-extensive growth in success probability while simultaneously displaying
an extensive growth in the weak value. We propose an experimental
implementation of our scheme.
