  Ideal projective quantum measurement makes the system state collapse in one
of the observable operator eigenstates $|\phi_\alpha\rangle$, making it a
powerful tool for preparing the system in the desired pure state. Nevertheless,
experimental realisations of projective measurement are not ideal. During the
measurement time needed to overcome the classical noise of the apparatus, the
system state is often (slightly) perturbed, which compromises the fidelity of
initialisation. In this paper, we propose an analytical model to analyse the
initialisation fidelity of the system performed by the single-shot readout. We
derive a method to optimise parameters for the three most used cases of photon
counting based readouts for NV colour centre in diamond, charge state, nuclear
spin and low temperature electron spin readout. Our work is of relevance for
the accurate description of initialisation fidelity of the quantum bit when the
single-shot readout is used for initialisation via post-selection or real-time
control.
