  The quantum Cram\'er-Rao bound (QCRB) provides an ultimate precision limit
allowed by quantum mechanics in parameter estimation. Given any quantum state
dependent on a single parameter, there is always a positive-operator valued
measurement (POVM) saturating the QCRB. However, the QCRB-saturating POVM
cannot always be implemented efficiently, especially in multipartite systems.
In this paper, we show that the POVM based on local operations and classical
communication (LOCC) is QCRB-saturating for arbitrary pure states or rank-two
mixed states with varying probability distributions over fixed eigenbasis.
Local measurements without classical communication, however, is not
QCRB-saturating in general.
