  Utilizing nonlinear elements, SU(1,1) interferometers demonstrate superior
phase sensitivity compared to passive interferometers. However, the precision
is significantly impacted by photon losses, particularly internal losses. We
propose a theoretical scheme to improve the precision of phase measurement
using homodyne detection by implementing number-conserving operations
(PA-then-PS and PS-then-PA) within the SU(1,1) interferometer, with the
coherent state and the vacuum state as the input states. We analyze the effects
of number-conserving operations on the phase sensitivity, the quantum Fisher
information, and the quantum Cramer-Rao bound under both ideal and photon
losses scenarios. Our findings reveal that the internal non-Gaussian operations
can enhance the phase sensitivity and the quantum Fisher information, and
effectively improve the robustness of the SU(1,1) interferometer against
internal photon losses. Notably, the PS-then-PA scheme exhibits superior
improvement in both ideal and photon losses cases in terms of phase
sensitivity. Moreover, in the ideal case, PA-then-PS scheme slightly
outperforms PS-then-PA scheme in terms of the quantum Fisher information and
the Quantum Cramer-Rao. However, in the presence of photon losses, PS-then-PA
scheme demonstrates a greater advantage.
