Uncertainty and symmetry bounds for the quantum total detection probability

We investigate a generic discrete quantum system prepared in state |ψin) under repeated detection attempts, aimed to find the particle in state |d), for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability Pdet is unity. Due to destructive interference, one may find initial states |ψin) with Pdet<1. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between Pdet and energy fluctuations: ΔPVar[H]d≥|(d|[H,D]|ψin)|2, where ΔP=Pdet-|(ψin|d)|2 and D=|d)(d| is the measurement projector. Secondly, exploiting symmetry we show that Pdet≤1/ν, where the integer ν 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 bound works well even in large systems, and we show how to tighten the lower bound in this case.