  Measurements in quantum mechanics cannot perfectly distinguish all states and
necessarily disturb the measured system. We present and analyse a proposal to
demonstrate fundamental limits on quantum control of a single qubit arising
from these properties of quantum measurements. We consider a qubit prepared in
one of two non-orthogonal states and subsequently subjected to dephasing noise.
The task is to use measurement and feedback control to attempt to correct the
state of the qubit. We demonstrate that projective measurements are not optimal
for this task, and that there exists a non-projective measurement with an
optimum measurement strength which achieves the best trade-off between gaining
information about the system and disturbing it through measurement back-action.
We study the performance of a quantum control scheme that makes use of this
weak measurement followed by feedback control, and demonstrate that it realises
the optimal recovery from noise for this system. We contrast this approach with
various classically inspired control schemes.
