Quantum metrology in a lossless Mach-Zehnder interferometer using entangled photon inputs for a sequence of non-adaptive and adaptive measurements

Using multi-photon entangled input states, we estimate the phase uncertainty in a noiseless Mach-Zehnder interferometer using photon-counting detection. We assume a flat prior uncertainty and use Bayesian inference to construct a posterior uncertainty. By minimizing the posterior variance to get the optimal input states, we first devise an estimation and measurement strategy that yields the lowest phase uncertainty for a single measurement. N00N and Gaussian states are determined to be optimal in certain regimes. We then generalize to a sequence of repeated measurements, using non-adaptive and fully adaptive measurements. N00N and Gaussian input states are close to optimal in these cases as well, and optimal analytical formulae are developed. Using these formulae as inputs, a general scaling formula is obtained, which shows how many shots it would take on average to reduce phase uncertainty to a target level. Finally, these theoretical results are compared with a Monte Carlo simulation using frequentist inference. In both methods of inference, the local non-adaptive method is shown to be the most effective practical method to reduce phase uncertainty.

Automated summary

By using multi-photon entangled input states and photon-counting detection, this study estimates the phase uncertainty in a noiseless Mach-Zehnder interferometer. Bayesian inference is used to construct posterior uncertainty, and the optimal input states are obtained by minimizing the posterior variance. The study expands to a sequence of repeated measurements and develops optimal analytical formulae and scaling formula, which show the average number of shots required to reduce phase uncertainty to a target level. A comparison with a Monte Carlo simulation using frequentist inference suggests that the local non-adaptive method is the most effective practical method to reduce phase uncertainty.