Uncertainty And Symmetry Bounds For The Quantum Total Detection Probability

We investigate a generic discrete quantum system prepared in state vertical bar psi(in)> under repeated detection attempts, aimed to find the particle in state vertical bar d >, for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability Pd et is unity. Due to destructive interference, one may find initial states vertical bar psi(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: Delta P Var [<(H)over cap>](d) >= vertical bar < d vertical bar[(H) over cap, (D) over cap]vertical bar psi(in)>vertical bar(2), where Delta P = P-det - vertical bar vertical bar(2), and (D) over cap= vertical bar d >< d vertical bar is the measurement projector. Secondly, exploiting symmetry we show that P-det <= 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 bound works well even in large systems, and we show how to tighten the lower bound in this case.