Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line

Abstract

In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line ($\mathbb{Z}$) and on the grid ($\mathbb{Z}^2$). Two versions are considered: discrete-time open quantum walks (OQW) and continuous-time open quantum walks (CTOQW). We present three distinct recurrence criteria for OQWs on $\mathbb{Z}$, each adapted to different types of coins. The first criterion applies to coins whose auxiliary map has a unique invariant state, resulting in the first recurrence criterion for Lazy OQWs. The second one applies to Lazy OQWs of dimension 2, where we provide a complete characterization of the recurrence for this low-dimensional case. The third one presents a general criterion for finite-dimensional coins in the non-lazy case, which generalizes many of the previously known results for OQWs on $\mathbb{Z}$. Also, we present a general recurrence criterion for OQWs on $\mathbb{Z}^2$ via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on $\mathbb{Z}^2$.

Figures (3)

• Figure 1: Classical Homogeneous Lazy Random Walk on $\mathbb{Z}$.
• Figure 2: Lazy Quantum Walk on $\mathbb{Z}$.
• Figure 3: Homogeneous Open Quantum walk on $\mathbb{Z}^2$.

Theorems & Definitions (52)

• Theorem 3.1
• proof
• Corollary 3.2
• Corollary 3.3
• Remark 3.4
• Proposition 3.5
• proof
• Theorem 3.6
• proof
• Theorem 3.7: Chung-Fuchs Theorem for HOQWs with ergodic auxiliary map