Channel estimation theory of low-noise multiple parameters: Attainability problem of the Cramer-Rao bounds

We consider the achievability problem of the Cramer-Rao bound for multiparameter estimations. In general, it is not achievable due to the noncommutativity of optimal measurements for the corresponding parameters. However, we show that, under certain conditions, it can always be attained up to the leading order in the parameters as long as D <= N-1, where D and N denote the number of parameters and the dimension of the system, respectively. After proving that, we discuss the achievability in the context of channel estimation for a general channel called a low-noise channel, which is very useful for investigating parameter estimation in the leading order. This allows us to find an ancilla-assisted enhancement effect: if entangled input states with an ancilla system are utilized for the channel estimation together with collective measurements on those output states, the bound becomes achievable for D <= N-2-1.