Single parameter phase estimation for a sequence of measurements

In this work, we investigate estimation of phase uncertainty in a noiseless Mach–Zehnder interferometer using multi-photon entangled input states and photon-counting detection. Our approach leverages Bayesian inference to address phase uncertainty estimation. By assuming a flat prior uncertainty and applying Bayesian inference, we obtain a posterior uncertainty. Our main goal is to minimize the posterior variance to determine the optimal input states, consequently formulating an innovative estimation and measurement strategy for achieving minimal phase uncertainty in a single measurement [1].

To achieve this objective, we explore the suitability of “N00N” and “Gaussian” states as optimal input states within specific operational regimes. For single measurements, both N00N and Gaussian states offer excellent performance. Moving beyond single measurements, we extend our investigation to sequences of repeated measurements, encompassing both non-adaptive and fully adaptive measurement scenarios. Remarkably, N00N and Gaussian input states retain their proximity to optimality across these scenarios, enabling us to derive analytical formulae for optimal performance.

Building upon these analytical formulae, we formulate a comprehensive scaling formula that predicts the average number of measurements required to reduce phase uncertainty to a pre-defined target level. This scaling formula provides a valuable insight into the relationship between measurement repetition and uncertainty reduction.

To validate our theoretical findings, we employ Monte Carlo simulations along with frequentist inference. We conclude that the local non-adaptive approach emerges as the superior strategy for reducing phase uncertainty. Our study thus offers a comprehensive framework for phase uncertainty estimation in noiseless Mach–Zehnder interferometers, exploiting multi-photon entangled states, Bayesian inference, and optimal input state selection for both single and multiple measurements.