The Theory Of Binary-outcome Quantum Phase Measurement

The combination of quantum mechanics and statistical informatics has led to the theory of quantum phase estimation.And it provides a good theoretical foundation in the fields of atomic clock,high-precision quantum measurement and gravitational wave detection and other fields.In fact,quantum phase measurement is a process of reasonable data processing on the results of random measurements to obtain phase estimates.In the evaluation of the quantum phase,the key point is to find a suitable quantum state,optimize the related mechanical quantities and data processing,so as to achieve the purpose that the fluctuation of the estimator is lower than the standard quantum limit.In the quantum measurement,artificially categorizing and processing the output results can realize the binary-outcome measurement and then obtain the estimator.According to even/odd and zero/nonzero number of photons detected at one output port of the Mach-Zehnder interferometer,the so-called parity detection and zero/nonzero photon counting measurement are realized successfully in experiments.Both of the above examples can be attributed to the dual output problem.Recently,the binary-outcome measurements have also begun to be adopted in quantum gain microscopes.The main innovative research results of this paper are as follows:1.For a binary-outcome measurements,the data processing based on inverting the output signal always saturate the Cramer-Rao lower bound(CRLB).Considering the imperfect of the experiment,one can obtain the analytical expressions of the output signal,the optimal phase working point and the phase sensitivity.In addition,numerically simulate the phase estimator using MLE method and inversion method,respectively,which prove that the results are unbiased.2.Due to the imperfect experiments,the microscope image is affected by phase sensitivity in the microscopy that uses twin-Fock states of light for illumination.Two solutions are proposed in this paper to completely eliminate this singularity:(i)locking the phase shift sensed by the beam at the optimal operating point through a spatially-related offset phase;(ii)combining two d binary-outcome photon count measurements,one with a fixed offset phase and the other without any offset phase.This result is valid for any kind of binary-outcome measurement and may provide a method for a quantum gain microscope with an N photon state.3.For binary-outcome detection with enough repetitions,the inversion method can be regarded as the MLE method.