Results and Discussions A strategy is proposed to eliminate the phase ambiguity of the twin-Fock state. Numerical simulation results verify that the strategy can eliminate phase ambiguity, and the phase sensitivity of the fluctuation of the estimator is better than the standard quantum limit in the whole phase interval. The main work is as follows: firstly, based on the related literature, the numerical simulation of the coincidence count measurement of the twin-Fock state light source and the single-photon light source in the Mach-Zehnder interferometer is carried out. Then the measurement strategy proposed in this paper is simulated, and the effectiveness of the simulation results is analyzed. Finally, the reduction sensitivity of the measurement strategy is given, and the whole phase interval is sampled to obtain statistical data. Furthermore, the unbiasedness of the estimator is verified numerically. The standard deviation of the measurement is compared with the theoretical expectation, the standard quantum limit, and the Heisenberg limit. The maximum likelihood estimation of the multi-output method for the twin-Fock state without bias phase was previously available, and it can be found that its likelihood function has two maximum points in a phase period. In this paper, we propose that when using multi-output measurement and maximum likelihood estimation, there is only one maximum point in a phase period if we combine the measurement results with the bias phase and without the bias phase to obtain the joint likelihood function. This paper gives an analysis of this phenomenon based on the approximation of linear combination estimators of maximum likelihood estimators. By generalizing the joint likelihood function measurement method of binary-outcome measurements to the multiple-output case, we find that it is possible to reduce the phase estimator from 4 to 2 in using only the twin-Fock state. Through the relationship between the maximum likelihood estimator and the inverse function estimator in the case of multiple outputs [Eqs. (9)-(14)], we have semi-analytically explained the reason why this method works. We then use the single-photon state for joint measurement, which ultimately eliminates phase ambiguity. Using the Monte Carlo method, we simulate the measurement process, and the results have validated our analysis results (Fig. 3). Finally, we calculate the Fisher information and the Cramer-Rao lower bound (CRB) of our measurement scheme as the analytical analysis of phase sensitivity. The numerical simulation results (Fig. 4) show that our estimator can beat the standard quantum limit.

Metrics play a central role in science and engineering. It is concerned with the final reachable accuracy of parameters or phase estimation and the construction of measurement schemes to achieve this accuracy. By combining quantum mechanics and basic theories of statistics, quantum metrics find that the final lower limit of estimation accuracy is related to input state preparation, phase accumulation modes, and measurement schemes, and the main goal is to break through the standard quantum limit and reach the Heisenberg limit of measurement accuracy. In recent years, due to the progress of experimental conditions, quantum metrics have been widely used in the frontier fields such as gravitational wave detection and atomic clocks. A major research direction of quantum metrics is phase estimation in optical interferometers, which was first proposed in research on the input coherent light and compressed light in Mach-Zende interferometers by Caves et al., and its theoretical phase sensitivity can reach the physical limit (Heisenberg limit). In recent years, other kinds of non-classical light sources have also been studied, such as the NOON state and twin-Fock state. The NOON state is a numerical light source that can theoretically reach the Heisenberg limit, while the twin-Fock state has theoretical phase sensitivity up to the Heisenberg scale and is more robust to photon loss than the NOON state. However, coincidence count detection for the twin-Fock state results in a multi-peak structure of the phase distribution (i.e., the likelihood function), which is the so-called phase ambiguity. Aiming at this problem, we propose a simple scheme to eliminate phase ambiguity and analyze its performance.

A binary-outcome photon counting and joint likelihood function measurement are employed in this work, where the detection event with an equal number of photons is a measurement outcome. All the other detection events are treated as another outcome. We generalize it to a multi-output scenario and use single-photon states for joint measurement. According to the relationship between the maximum likelihood estimator and the inverse function estimator in the case of multiple outputs, we have semi-analytically explained the reason why this method works. Using the Monte Carlo method, we simulate the measurement probabilities of the six-photon twin-Fock state and the single-photon state and get a numerical simulation of the measurement scheme, where the experimental imperfection is added artificially.

We propose a simple scheme to eliminate phase ambiguity of coincidence count detection for the twin-Fock state. Our scheme relies on a sequence of the N-photons Fock states and the single-photon state that are injected into the interferometer to realize a single-peak structure of the total phase distribution, which determines the maximum likelihood estimator. Phase uncertainty of the estimator can beat the standard quantum limit over the entire phase interval.