Bayesian minimum mean square error for transmissivity sensing

We address the problem of estimating the transmissivity of the pure-loss channel from the Bayesian point of view, i.e., we consider that some prior probability distribution function (PDF) on the unknown variable is available and we employ methods to compute the Bayesian minimum mean square error (MMSE). Specifically, we consider two prior PDFs: the two-point and the beta distributions. By fixing the input mean photon number to an integer, for the two-point PDF we prove analytically that the optimal state is the Fock state and the optimal measurement is photon-counting, while for the beta PDF our numerical investigation provides evidence on the optimality of the Fock state and photon-counting. Moreover, we investigate the situation where the input mean photon number is any (non-negative) real number. For said case, we conjecture the form of the optimal input states and we study the performance of photon-counting, which is a sub-optimal yet practical measurement. Our methods can be applied for any prior PDF. We emphasize that we compute the MMSE instead of Bayesian lower bounds on the mean square error based on the Fisherian approach.