  We address the problem of estimating pure qubit states with non-ideal (noisy)
measurements in the multiple-copy scenario, where the data consists of a number
N of identically prepared qubits. We show that the average fidelity of the
estimates can increase significantly if the estimation protocol allows for
inconclusive answers, or abstentions. We present the optimal such protocol and
compute its fidelity for a given probability of abstention. The improvement
over standard estimation, without abstention, can be viewed as an effective
noise reduction. These and other results are exemplified for small values of N.
For asymptotically large N, we derive analytical expressions of the fidelity
and the probability of abstention, and show that for a fixed fidelity gain the
latter decreases with N at an exponential rate given by a Kulback-Leibler
(relative) entropy. As a byproduct, we obtain an asymptotic expression in terms
of this very entropy of the probability that a system of N qubits, all prepared
in the same state, has a given total angular momentum. We also discuss an
extreme situation where noise increases with N and where estimation with
abstention provides a most significant improvement as compared to the standard
approach.
