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We study the sensitivity of phase estimation in a lossy Mach-Zehnderinterferometer (MZI) using two general, and practical, resources generated by alaser and a nonlinear optical medium with passive optimal elements, which arereadily available in the laboratory: One is a two-mode separable coherent andsqueezed vacuum state at a beam splitter and the other is a two-mode squeezedvacuum state. In view of the ultimate precision given by quantum Fisherinformation, we show that the two-mode squeezed vacuum state can achieve alower bound of estimation error than the coherent and squeezed vacuum stateunder a photon-loss channel. We further consider practical measurement schemes,homodyne detection and photon number resolving detection (PNRD), tocharacterize the accuracy of phase estimation in reality and find that thecoherent and squeezed vacuum state largely achieves a lower bound than thetwo-mode squeezed vacuum in the lossy MZI while maintaining quantum enhancementover the shot-noise limit. By comparing homodyne detection and PNRD, wedemonstrate that quadrature measurement with homodyne detection is more robustagainst photon loss than parity measurement with PNRD. We also show that doublehomodyne detection can provide a better tool for phase estimation than singlehomodyne detection against photon loss.

Practical resources and measurements for lossy optical quantum metrology