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

Abstract

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$N00N$ 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$N00N$ 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$N00N$ 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.

Optimal multiple-phase estimation with multi-mode NOON states against photon loss

“…with coefficients λ j = l̸ =j (1 − γ l ) 1/2 (see appendix B for the details). The minimum QCRB written in equation ( 10) successfully encapsulates the previous results in [20,[23][24][25] as lossless case. The difference between p 0 and p j (j ̸ = 0) arises because the reference phase is assumed to be known beforehand but the other phases are unknown.…”

“…As a result, we find that our scheme is not only more robust against photon loss than the scheme in [20], but also exhibits a quantum advantage over the SQL even in the presence of loss. These findings are elaborated in more details for the cases of three-and four-mode two-photon NOON states that have been experimentally demonstrated [23,24], followed by the generalization of the scheme with increasing the numbers of modes and photons. The proposed schemes are also investigated with a measurement scheme with general structure consisting of a multi-mode beam-splitter and multiple photon-number-resolving detectors (PNRDs), offering a practical means to achieve the quantum advantage in multiple-phase estimation under a lossy environment.…”

“…For example, when multiple phases are encoded in a multi-mode NOON state |ψ N ⟩ in equation ( 1), this measurement is represented as a positive-operator-valued measure (POVM) including a projector |ψ N ⟩⟨ψ N | [20,35]. The fact that the multi-mode NOON state is generated by nonlinear optics implies the difficulty in experimentally realizing the measurement [23][24][25]. Thus, it is necessary to consider the CRB provided by a measurement experimentally configurable by linear optics.…”

“…N,ϕ |. However, it is known that such the measurement is experimentally challenging to implement in general [24,25]. Thus, for experimental feasibility, we need to verify whether an estimation scheme using a measurement with a linear optical structure can provides the SQL.…”

“…Quantum resources can enhance performance of precisely estimating multiple parameters over classical counterparts. For this reason, most studies have focused on finding optimal quantum resources achieving the ultimate quantum limit in multiple-parameter estimation [1][2][3], including the use of the Greenberger-Horne-Zeilinger states [4,5], single-photon Fock states [6], squeezed states [7][8][9][10][11][12][13], entangled coherent states [14][15][16], Holland-Burnett states [17][18][19], and multi-mode NOON states [20][21][22][23][24][25]. Among them, the multi-mode NOON states are particularly known to achieve the enhanced estimate precision outperforming the other probe states in the mean photon number for multiple-phase estimation, consequently beating the standard quantum limit (SQL) [20].…”