  We study the dynamics of a charge qubit, consisting of a single electron in a
double well potential, coupled to a point-contact (PC) electrometer using the
quantum trajectories formalism. In contrast with previous work, our analysis is
valid for arbitrary source-drain bias across the PC, but is restricted to the
sub-Zeno limit. We find that the dynamics is strongly affected by inelastic
tunnelling processes in the PC. These processes reduce the efficiency of the PC
as a qubit readout device, and induce relaxation even when the source-drain
bias is zero. We show that the sub-Zeno dynamics are divided into two regimes:
low- and high-bias in which the PC current and current power spectra show
markedly different behaviour. To further illustrate the division between the
regimes and the inefficiency of the detector, we present simulated quantum
trajectories of the conditional qubit and detector dynamics. We also describe
how single shot measurements in an arbitrary basis may be achieved in the
sub-Zeno regime.
