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Detection of edge defects by embedded eigenvalues of quantum walks

Hisashi Morioka, Etsuo Segawa

TL;DR

A detection method for edge defects by embedded eigenvalues of its time evolution operator is derived from a position-dependent quantum walk on Z on which the coin operator is an anti-diagonal matrix.

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.

Detection of edge defects by embedded eigenvalues of quantum walks

TL;DR

A detection method for edge defects by embedded eigenvalues of its time evolution operator is derived from a position-dependent quantum walk on Z on which the coin operator is an anti-diagonal matrix.

Abstract

We consider a position-dependent quantum walk on . 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 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.

Paper Structure

This paper contains 11 sections, 14 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Continuous spectrum
  3. Spectral decomposition of unitary operators
  4. Essential spectrum
  5. Absence of embedded eigenvalues
  6. Thresholds
  7. Absence of embedded eigenvalues
  8. Existence of embedded eigenvalues
  9. Finite support of eigenfunctions
  10. Embedded eigenvalues
  11. Summary and discussion

Key Result

Theorem 1.2

Let $p\in (0,1]$. We assume that there is no edge defect i.e. there exists a constant $\delta >0$ such that $|a(x)| \geq \delta$ for all $x\in {\bf Z}$. Moreover, suppose that $C$ and $C_0$ satisfy the condition (S1_eq_exp). Then the continuous spectrum of $U$ is $\sigma_{ess} (U) = \{ e^{i\theta } Moreover, there is no eigenvalue in $\sigma_{ess} (U) \setminus \mathcal{T}$ where $\mathcal{T} = \

Figures (4)

  • Figure 1: The distribution of $P_v (X_t =x )$ at $t=100$.
  • Figure 2: The distribution of $P_e ( X_t = x)$ at $t=100$.
  • Figure 3: The distribution of $\sigma (U_v)$.
  • Figure 4: The distribution of $\sigma (U_e)$.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Theorem 3.2
  • ...and 5 more