  We theoretically study the quantum Fisher information (QFI) of the SU(1,1)
interferometer with phase shifts in two arms taking account of realistic noise
effects. A generalized phase transform including the phase diffusion effect is
presented by the purification process. Based on this transform, the analytical
QFI and the bound to the quantum precision are derived when considering the
effects of phase diffusion and photon losses simultaneously. To beat the
standard quantum limit with the reduced precision of phase estimation due to
noisy, the upper bounds of decoherence coefficients as a function of total mean
photon number are given.
