  We investigate a generic discrete quantum system prepared in state
$|\psi_\text{in}\rangle$, under repeated detection attempts aimed to find the
particle in state $|d\rangle$, for example a quantum walker on a finite graph
searching for a node. For the corresponding classical random walk, the total
detection probability $P_\text{det}$ is unity. Due to destructive interference,
one may find initial states $|\psi_\text{in}\rangle$ with $P_\text{det}<1$. We
first obtain an uncertainty relation which yields insight on this deviation
from classical behavior, showing the relation between $P_\text{det}$ and energy
fluctuations: $ \Delta P \,\mathrm{Var}[\hat{H}]_d \ge | \langle d| [\hat{H},
\hat{D}] | \psi_\text{in} \rangle |^2$ where $\Delta P = P_\text{det} -
|\langle\psi_\text{in}|d\rangle |^2$, and $\hat{D} = |d\rangle\langle d|$ is
the measurement projector. Secondly, exploiting symmetry we show that
$P_\text{det}\le 1/\nu$ where the integer $\nu$ is the number of states
equivalent to the initial state. These bounds are compared with the exact
solution for small systems, obtained from an analysis of the dark and bright
subspaces, showing the usefulness of the approach. The upper bounds works well
even in large systems, and we show how to tighten the lower bound in this case.
