Quantum enhanced multiple phase estimation is essential for various applications in quantum sensors and imaging. For multiple phase estimation, the sensitivity enhancement is dependent on both quantum probe states and measurement. It is known that multi-mode $N00N$ states can outperform other probe states for estimating multiple phases. However, it is generally not feasible in practice to implement an optimal measurement to achieve the quantum Cramer–Rao bound (QCRB) under a practical measurement scheme using a multi-mode beam splitter in interferometric phase estimation. Here, a strategy to achieve the best practical sensitivity by optimizing both mode-amplitudes of multi-mode $N00N$ states and a split ratio of a multi-mode beam splitter is investigated. Then, it is experimentally demonstrated that the best sensitivity is achieved when an amplitude-balanced multi-mode $N00N$ state and a multi-mode beam splitter with an unbalanced ratio are used in three-mode interferometric phase estimation. The results show that the lower QCRB cannot guarantee better sensitivity under a practical measurement scheme, thus it is more desirable to enhance the practical sensitivity rather than the QCRB. It is believed that this strategy can provide a powerful tool for practical applications in multiple phase estimation.

中文翻译：

---

多模式 N00N$N00N$ 状态下多相位估计的实用灵敏度界限

量子增强多相位估计对于量子传感器和成像中的各种应用至关重要。对于多相位估计，灵敏度增强取决于量子探针状态和测量。众所周知，多模$ñ00ñ$在估计多个阶段时，状态可以优于其他探测状态。然而，在实际测量方案下，在干涉相位估计中使用多模分束器的实际测量方案下，实现最佳测量以实现量子克拉默-饶界 (QCRB) 通常是不可行的。在这里，通过优化多模式的两个模式幅度来实现最佳实际灵敏度的策略$ñ00ñ$研究了多模分束器的状态和分光比。然后，实验证明，当幅度平衡的多模$ñ00ñ$状态和具有不平衡比的多模分束器用于三模干涉相位估计。结果表明，在实际测量方案下，较低的QCRB不能保证更好的灵敏度，因此提高实际灵敏度比提高QCRB更为可取。相信该策略可以为多相位估计的实际应用提供强大的工具。