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Continuous-Time Quantum Walk on Locally Infinite Graph

Ce Wang

Abstract

Time-reversal symmetry is of fundamental importance to physics. In the classical theory of time-reversal symmetry, the time-reversal symmetry of a quantum system is described by an anti-unitary operator, which is known as the time-reversal operator of the system. In this paper, we introduce and study a model of continuous-time quantum walk on a special locally infinite graph. After examining its spectral property, we investigate the time-reversal symmetry of the model. To our surprise, we find that its time-reversal symmetry can be described directly by a unitary operator, which contrasts sharply with that in the classical theory of time-reversal symmetry. Some other related results are also proven.

Continuous-Time Quantum Walk on Locally Infinite Graph

Abstract

Time-reversal symmetry is of fundamental importance to physics. In the classical theory of time-reversal symmetry, the time-reversal symmetry of a quantum system is described by an anti-unitary operator, which is known as the time-reversal operator of the system. In this paper, we introduce and study a model of continuous-time quantum walk on a special locally infinite graph. After examining its spectral property, we investigate the time-reversal symmetry of the model. To our surprise, we find that its time-reversal symmetry can be described directly by a unitary operator, which contrasts sharply with that in the classical theory of time-reversal symmetry. Some other related results are also proven.
Paper Structure (4 sections, 16 theorems, 62 equations)

This paper contains 4 sections, 16 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Model of continuous-time quantum walk
  3. Spectral property
  4. Time-reversal symmetry

Key Result

Theorem 2.1

Let $w$ be a weight on $\mathbb{N}$. Then $A_w$ is self-adjoint and moreover $\|A_w\| = |w|$.

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Remark 2.1
  • ...and 23 more