  In a measurement-induced continuous-time quantum walk, we address the problem
of detecting a particle in a subspace, instead of a fixed position. In this
configuration, we develop an approach of bright and dark states based on the
unit and vanishing detection probability respectively for a particle-detection
in the subspace. Specifically, by employing the rank-nullity theorem, we
determine several properties of dark and bright states in terms of energy
spectrum of the Hamiltonian used for a quantum walk and the projectors applied
to detect the subspace. We provide certain conditions on the position and the
rank of the subspace to be detected, resulting in the unit total detection
probability, which has broad implications for quantum computing. Further, we
illustrate the forms of dark as well as bright states and the dependence of
detection probability on the number of dark states by considering a cyclic
graph with nearest-neighbor and next nearest-neighbor hopping. Moreover, we
observe that the divergence in the average number of measurements for detecting
a particle successfully in a subspace can be reduced by performing high rank
projectors.
