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Multi-path (or multi-mode) entanglement has been proved to be a useful resource for sub-shot-noise sensitivity of phase estimation, which has aroused much research interest inquantum metrology recently. Various schemes of multi-path interferometers based on optical systems have been put forward. Here, we study a multi-state interferometer with multi-level atoms by projective measurements. Specifically, we investigate its ultimate sensitivity described by quantum Fisher information theory and find that the Cramerâ€"Rao bound can be achieved. In particular, we investigate a specific scheme to improve the sensitivityof magnetometery with a three-state interferometry delivered by a single nitrogen-vacancy (NV) center of diamond with tailor pulses. The impacts of imperfections of the atomic beam-splitter, described by the three-level quantum Fourier transform, on the sensitivity of phase estimation is also discussed. [ABSTRACT FROM AUTHOR]