Sensing applications utilizing the quantum Bayesian minimum square error

We consider the genuine quantum sensing Bayesian approach, i.e., the unknown parameter is a random value that follows a prior probability distribution function (pPDF). We compute the minimum mean square error (MMSE) which is always attainable by a measurement. We consider sensing of loss rate and optical phase.

For sensing loss rate, we first consider the two-point pPDF and we prove analytically the optimality of the Fock state and photon-counting measurement. For the beta pPDF our numerical investigation provides evidence on the optimality of the Fock state and photon-counting.

For our optical phase sensing probe, we consider the generic fixed-photon state. For the NOON state, which is a special case of our probe state and for a flat pPDF we find that the Heisenberg scaling is lost and the attainable MMSE is found to be proportional to -n^-2, which is a manifestation of the fundamental difference between the Fisherian and Bayesian approaches. Then, we consider an adaptive technique: We start with a flat pPDF, and for each next step we use as pPDF the posterior probability of the previous step, while for each step we update the optimal state and optimal measurement. We compute the Bayesian MMSE per step. We draw conclusions on the performance of optimal probe states.