We investigate the optimal quantum state for an atomic gyroscope based on a three-site Bose-Hubbard model. In previous studies, various states such as the uncorrelated state, the BAT state and the NOON state are employed as the probe states to estimate the phase uncertainty. In this article, we present a Hermitian operator $mathcal{H}$ and an equivalent unitary parametrization transformation to calculate the quantum Fisher information for any initial states. Exploiting this equivalent unitary parametrization transformation, we can seek the optimal state which gives the maximal quantum Fisher information on both lossless and lossy conditions. As a result, we find that the entangled even squeezed state (EESS) can significantly enhance the precision for moderate loss rates.
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