  Improving interferometric phase sensitivity is crucial for high-precision
measurements in rapidly developing quantum technologies. The Mach-Zehnder
interferometer (MZI) is a versatile tool for analyzing this phenomenon. By
splitting and recombining a light beam using beam splitters, MZIs allow for
precise phase sensitivity analysis using tools like the quantum Cram\'er-Rao
bound (QCRB) and the quantum Fisher information (QFI). This paper analyzes the
phase sensitivity of a MZI in various scenarios using different detection
schemes and input states. We compare the single- and two-parameter quantum
estimation and their associated QCRB for three phase-shift situations: in both
arms, only in the upper arm (asymmetric), and in both arms symmetrically. We
then investigate the phase sensitivity under three detection schemes:
difference intensity, single-mode intensity, and balanced homodyne.
Additionally, we explore the use of Perelomov and Barut-Girardello coherent
states, two types of SU(1,1) coherent states, in all scenarios. Notably, we
demonstrate that under optimal conditions, all detection schemes can achieve
the QCRB by utilizing entangled SU(1,1) coherent states as input states.
