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.
Quantum metrology in a lossless Mach–Zehnder interferometer using entangled photon inputs for a sequence of non-adaptive and adaptive measurements
Shreyas Sadugol, Lev Kaplan; Quantum metrology in a lossless Mach–Zehnder interferometer using entangled photon inputs for a sequence of non-adaptive and adaptive measurements. AVS Quantum Sci. 1 March 2023; 5 (1): 014407. https://doi.org/10.1116/5.0137125
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