  We revisit the problem of estimating an unknown parameter of a pure quantum
state, and investigate `null-measurement' strategies in which the experimenter
aims to measure in a basis that contains a vector close to the true system
state. Such strategies are known to approach the quantum Fisher information for
models where the quantum Cram\'{e}r-Rao bound is achievable but a detailed
adaptive strategy for achieving the bound in the multi-copy setting has been
lacking. We first show that the following naive null-measurement implementation
fails to attain even the standard estimation scaling: estimate the parameter on
a small sub-sample, and apply the null-measurement corresponding to the
estimated value on the rest of the systems. This is due to non-identifiability
issues specific to null-measurements, which arise when the true and reference
parameters are close to each other. To avoid this, we propose the alternative
displaced-null measurement strategy in which the reference parameter is altered
by a small amount which is sufficient to ensure parameter identifiability. We
use this strategy to devise asymptotically optimal measurements for models
where the quantum Cram\'{e}r-Rao bound is achievable. More generally, we extend
the method to arbitrary multi-parameter models and prove the asymptotic
achievability of the the Holevo bound. An important tool in our analysis is the
theory of quantum local asymptotic normality which provides a clear intuition
about the design of the proposed estimators, and shows that they have
asymptotically normal distributions.
