Detection of edge defects by embedded eigenvalues of quantum walks

TL;DR

This paper introduces a method to detect edge defects in position-dependent quantum walks on integers by analyzing embedded eigenvalues of the system's time evolution operator, linking defect locations to spectral properties.

Contribution

It establishes a novel detection technique for edge defects using spectral analysis of the quantum walk's evolution operator, specifically through embedded eigenvalues.

Findings

Edge defects correspond to embedded eigenvalues in the spectrum.

Detection method is applicable under certain assumptions.

Finite edge defects relate to specific spectral features.

Abstract

We consider a position-dependent quantum walk on ${\bf Z}$. In particular, we derive a detection method for edge defects by embedded eigenvalues of its time evolution operator. In the present paper, the set of edge defects is that of points in ${\bf Z}$ on which the coin operator is an anti-diagonal matrix. In fact, under some suitable assumptions, the existence of a finite number of edge defects is equivalent to the existence of embedded eigenvalues of the time evolution operator.