Bayesian quantum phase estimation with fixed photon states

TL;DR

This paper develops a Bayesian approach to quantum phase estimation using fixed photon states, optimizing input states and measurements to minimize mean square error, and explores adaptive strategies with truncated priors.

Contribution

It introduces a method to optimize fixed photon number states for Bayesian quantum phase estimation, including adaptive strategies with prior updates.

Findings

Optimal Fock coefficients for minimal MSE identified

Adaptive measurement strategies improve phase estimation accuracy

Truncated priors enhance estimation when prior variance is large

Abstract

We consider a two-mode bosonic state with fixed photon number $n \in \mathbb{N}$, whose upper and lower modes pick up a phase $\phi$ and $-\phi$ respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating $\phi$ is minimized while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider $\phi$ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique we consider $n=1$ and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step and at the same time we update the optimal state and optimal measurement.