Comparison Between the Cramer-Rao and the Mini-max Approaches in Quantum Channel Estimation

In a unified viewpoint in quantum channel estimation, we compare the Cramér-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramér-Rao bound in the phase estimation case while both bounds coincide when the minimum mean square error decreases with the order O(\frac1n){O(\frac{1}{n})} . We also derive a sufficient condition so that the minimum mean square error decreases with the order O(\frac1n){O(\frac{1}{n})} .