Uncertainty and symmetry bound for the total detection probability of quantum walks

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