Asymptotic estimation of a shift parameter of a quantum state

The asymptotics of the optimal covariant estimate of a shift parameter of a pure quantum state is studied. In the regular case, where the variance of the generator of the shift group is finite, this problem of quantum statistics turns out to be connected with the information-theoretic approach to the classical central limit theorem proposed by Linnik in 1959. We also consider the quantum analogue of estimation of the shift parameter of the uniform distribution in which the regularity assumption is violated. The advantage of using entanglement is demonstrated in the class of covariant estimates.