  Quantum parameter estimation holds the promise of quantum technologies, in
which physical parameters can be measured with much greater precision than what
is achieved with classical technologies. However, how to obtain a best
precision when the optimal measurement is not accessible is still an open
problem. In this work, we present a theoretical framework to explore the
parameter estimation with limited access of measurements by analyzing the
effect of non-optimal measurement on the estimation precision. We define a
quantity to characterize the effect and illustrate how to optimize observables
to attain a bound with limited accessibility of observables. On the other side,
we introduce the minimum Euclidean distance to quantify the difference between
an observable and the optimal ones in terms of Frobenius norm and find that the
measurement with a shorter distance to the optimal ones benefits the
estimation. Two examples are presented to show our theory. In the first, we
analyze the effect of non-optimal measurement on the estimation precision of
the transition frequency for a driven qubit. While in the second example, we
consider a bipartite system, in which one of them is measurement inaccessible.
To be specific, we take a toy model, the NV-center in diamond as the bipartite
system, where the NV-center electronic spin interacts with a single nucleus via
the dipole-dipole interaction. We achieve a precise estimation for the nuclear
Larmor frequency by optimizing only the observables of the electronic spin. In
these two examples, the minimum Euclidean distance between an observable and
the optimal ones is analyzed and the results show that the observable closed to
the optimal ones better the estimation precision.
