Bayesian quantum phase estimation with fixed photon states

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.