Uncertainty and symmetry bounds for the quantum total detection probability

Abstract

We investigate a generic discrete quantum system prepared in state |ψ_{in}〉 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 P_{det} is unity. Due to destructive interference, one may find initial states |ψ_{in}〉 with P_{det}<1. We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between P_{det} and energy fluctuations: ΔPVar[H[over ̂]]_{d}≥|〈d|[H[over ̂],D[over ̂]]|ψ_{in}〉|^{2}, where ΔP=P_{det}−|〈ψ_{in}|d〉|^{2} and D[over ̂]=|d〉〈d| is the measurement projector. Secondly, exploiting symmetry we show that P_{det}≤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.