Practical Sensitivity Bound for Multiple Phase Estimation with Multi-Mode N00N States

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 (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) 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.