  We consider the quantum measurement properties of a driven cavity with a
Kerr-type nonlinearity which is used to amplify a dispersively coupled input
signal. Focusing on an operating regime which is near a bifurcation point, we
derive simple asymptotic expressions describing the cavity's noise and
response. We show that the cavity's backaction and imprecision noise allow for
quantum limited linear amplification and position detection only if one is able
to utilize the sizeable correlations between these quantities. This is possible
when one amplifies a non-resonant signal, but is not possible in QND qubit
detection. We also consider the possibility of using the nonlinear cavity's
backaction for cooling a mechanical mode.
