Bayesian minimum mean square error for transmissivity sensing

We address the problem of estimating the transmissivity of the pure-loss single-mode bosonic channel from the Bayesian point of view, i.e., when a prior probability distribution function (PDF) on the transmissivity is available. We compute the quantum limit of the Bayesian minimum mean square error of estimating transmissivity. Specifically, we consider two prior PDFs: the two-point distribution (relevant for reading from an optical drive) and the beta distribution (relevant for imaging a reflective pixelated scene). If the probe's mean photon number is an integer, for the two-point PDF we prove that the optimal probe is a Fock state and the optimal measurement is photon counting, while for the beta PDF our numerical results provide evidence on the optimality of the Fock state probe and photon counting. When the probe's mean photon number is any (nonnegative) real number, we conjecture the form of the optimal probe state and we study the performance of photon counting, which is a suboptimal yet practical measurement. Our methods can be applied for any prior PDF. We discuss how different precision metrics and priors on the parameter influence the quantum-optimal probe and measurement. To our knowledge, these results on transmissivity estimation with quantum-limited precision represent the first Bayesian analysis of the problem.