Quantum metrology in a lossless Mach-Zehnder interferometer using entangled photon inputs

TL;DR

This paper investigates optimal entangled photon states for phase estimation in a lossless Mach-Zehnder interferometer, developing strategies that minimize phase uncertainty through Bayesian inference and adaptive measurements.

Contribution

It introduces a Bayesian framework for optimizing input states and measurement strategies, identifying N00N and Gaussian states as near-optimal in various measurement regimes.

Findings

N00N and Gaussian states are optimal in certain regimes.

The local non-adaptive measurement method is most effective practically.

Derived analytical formulas for phase uncertainty scaling.

Abstract

Using multi-photon entangled input states, we estimate the phase uncertainty in a noiseless Mach-Zehnder interferometer (MZI) 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.