  We analyze the information that one can learn about the state of a quantum
two-level system, i.e. a qubit, when probed weakly by a nearby detector. We
consider the general case where the qubit Hamiltonian and the qubit's operator
probed by the detector do not commute. Because the qubit's state keeps evolving
while being probed and the measurement data is mixed with a detector-related
background noise, one might expect the detector to fail in this case. We show,
however, that under suitable conditions and by proper analysis of the
measurement data useful information about the initial state of the qubit can be
extracted. Our approach complements the usual master-equation and
quantum-trajectory approaches, which describe the evolution of the qubit's
quantum state during the measurement process but do not keep track of the
acquired measurement information.
