  We develop an asymptotic theory of estimation of a shift parameter in a pure
quantum state to study the relation between entangled and unentangled covariant
estimates in the analytically most transparent way. After recollecting basics
of estimation of shift parameter in Sec. 2, we study the structure of the
optimal covariant estimate in Sec. 3, showing how entanglement comes into play
for several independent trials. In Secs. 4,5 we give the asymptotics of the
performance of the optimal covariant estimate comparing it with the
``semiclassical'' unentangled covariant estimation in the regular case of
finite variance of the generator of the shift group. Sec. 6 is devoted to
covariant estimation in the case where the regularity assumption is violated.
