Stability of point spectrum for three-state quantum walks on a line

TL;DR

This paper investigates the stability of eigenvalues in three-state quantum walks on a line, identifying classes of coin operators that preserve stationary states and contribute to understanding quantum walk dynamics.

Contribution

It introduces new classes of coin operators that maintain the point spectrum in three-state quantum walks, expanding the known spectrum of possible quantum walk behaviors.

Findings

Two classes of coin operators preserve the point spectrum.

Generalizations of previously known coin operators are presented.

The results enhance understanding of stationary states in quantum walks.

Abstract

Evolution operators of certain quantum walks possess, apart from the continuous part, also point spectrum. The existence of eigenvalues and the corresponding stationary states lead to partial trapping of the walker in the vicinity of the origin. We analyze the stability of this feature for three-state quantum walks on a line subject to homogenous coin deformations. We find two classes of coin operators that preserve the point spectrum. These new classes of coins are generalizations of coins found previously by different methods and shed light on the rich spectrum of coins that can drive discrete-time quantum walks.