  We theoretically studied the quantum Cram\'{e}r-Rao bound of an actively
correlated Mach-Zehnder interferometer (ACMZI), where the quantum Fisher
information obtained by the phase-averaging method can give the proper
phase-sensing limit without any external phase reference. We numerically
calculate the phase sensitivities with the method of homodyne detection and
intensity detection in the presence of losses. Under lossless and very low loss
conditions, the ACMZI is operated in a balanced case to beat the standard
quantum limit (SQL). As the loss increases, the reduction in sensitivity
increases. However within a certain range, we can adjust the gain parameters of
the beam recombination process to reduce the reduction in sensitivity and
realize the sensitivity can continue to beat the SQL in an unbalanced
situation. Our scheme provides an optimization method of phase estimation in
the presence of losses.
