Abstract
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.
Highlights
P Var[H ]d | d|[H, D ]|ψin |2, where P = Pdet − | ψin|d |2 and D = |d d| is the measurement projector
In this paper we provide three insights on Pdet: an uncertainty principle, symmetry arguments, and an exact solution
The exact solution relies on decomposing the Hilbert space into mutually orthogonal dark and bright subspaces
Summary
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. The investigation of a single particle on a finite graph, i.e., a quantum walker [20,21,22,23,24,25,26,27,28,29], prepared and detected in the states |ψin and |d , respectively, was promoted as a basic model in the context of quantum search [1,2,4,20,30,31,32,33,34,35,36,37,38].
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have