  A quantum system may be purified, i.e., projected into a pure state, faster
if one applies feedback operations during the measurement process. However
existing results suggest that such an enhancement is only possible when the
measurement efficiency exceeds 0.5, which is difficult to achieve
experimentally. We address the task of finding the global optimal feedback
control for purifying a single qubit in the presence of measurement
inefficiency. We use the Bloch vector length, a more physical and practical
quantity than purity, to assess the quality of the state, and employ a backward
iteration algorithm to find the globally optimal strategy. Our results show
that a speedup is available for quantum efficiencies well below 0.5, which
opens the possibility of experimental implementation in existing systems.
