Bayesian estimation of one-parameter qubit gates

We address estimation of one-parameter unitary gates for qubit systems and seek optimal probes and measurements. Single- and two-qubit probes are analysed in detail focusing on precision and stability of the estimation procedure. Bayesian inference is employed and compared with the ultimate quantum limits to precision, taking into account the biased nature of the Bayes estimator in the non-asymptotic regime. In addition, through the evaluation of the asymptotic a posteriori distribution for the gate parameter and a comparison with the results of Monte Carlo simulated experiments, we show that asymptotic optimality of the Bayes estimator is actually achieved after a limited number of runs. The robustness of the estimation procedure against fluctuations of the measurement settings is investigated and the use of entanglement to improve the overall stability of the estimation scheme is also analysed in some detail.