  Many works have stated that nonlinear interactions can improve phase
sensitivity beyond the Heisenberg limit scaling of $1/N$ with $N$ being the
mean photon number. This raises some open questions---among them the conclusive
sensitivity limits with respect to single-mode inputs. Namely, when one of two
inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear
Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase
shifts, we make an attempt to provide an answer to this question. Based upon
phase-averaging approach, this puzzle is partially resolved with careful
calculations of the quantum Fisher information regarding three kinds of common
inputs: Gaussian states, squeezed number states, and Schr\"odinger cat states.
The results suggest that shot-noise-style sensitivity limit is no longer
available, and the ideal candidate is squeezed vacuum.
