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

Conclusive nonlinear phase sensitivity limit for a Mach-Zehnder interferometer with single-mode non-vacuum inputs