  We study the sensitivity of phase estimation in a lossy Mach-Zehnder
interferometer (MZI) using two general, and practical, resources generated by a
laser and a nonlinear optical medium with passive optimal elements, which are
readily available in the laboratory: One is a two-mode separable coherent and
squeezed vacuum state at a beam splitter and the other is a two-mode squeezed
vacuum state. In view of the ultimate precision given by quantum Fisher
information, we show that the two-mode squeezed vacuum state can achieve a
lower bound of estimation error than the coherent and squeezed vacuum state
under a photon-loss channel. We further consider practical measurement schemes,
homodyne detection and photon number resolving detection (PNRD), to
characterize the accuracy of phase estimation in reality and find that the
coherent and squeezed vacuum state largely achieves a lower bound than the
two-mode squeezed vacuum in the lossy MZI while maintaining quantum enhancement
over the shot-noise limit. By comparing homodyne detection and PNRD, we
demonstrate that quadrature measurement with homodyne detection is more robust
against photon loss than parity measurement with PNRD. We also show that double
homodyne detection can provide a better tool for phase estimation than single
homodyne detection against photon loss.
