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Recurrence criteria for reducible homogeneous open quantum walks on the line and on the grid

Newton Loebens

Published Apr 29, 2026• 0 citations
Publisher

Abstract

Abstract In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line ( $$\mathbb {Z}$$ Z ) and on the grid ( $$\mathbb {Z}^2$$ 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 $$\mathbb {Z}$$ 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 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 several of the previously known results for OQWs on $$\mathbb {Z}$$ Z . Also, we present a general recurrence criterion for OQWs on $$\mathbb {Z}^2$$ Z 2 via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on $$\mathbb {Z}^2$$ Z 2 .

DOI
10.1007/s11128-026-05180-7

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Abstract In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line ( Z\mathbb {Z}Z Z ) and on the grid ( Z2\mathbb {Z}^2Z2 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 Z\mathbb {Z}Z 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 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 several of the previously known results for OQWs on Z\mathbb {Z}Z Z . Also, we present a general recurrence criterion for OQWs on Z2\mathbb {Z}^2Z2 Z 2 via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on Z2\mathbb {Z}^2Z2 Z 2 .
In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line (Z\mathbb{Z}Z) and on the grid (Z2\mathbb{Z}^2Z2). 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 Z\mathbb{Z}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 Z\mathbb{Z}Z. Also, we present a general recurrence criterion for OQWs on Z2\mathbb{Z}^2Z2 via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on Z2\mathbb{Z}^2Z2.
In this work, we study open quantum random walks, as described by S. Attal et al.. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties of the operators involved. This allows us to obtain a complete criterion for site recurrence in the case for which the internal degree of freedom of each site (coin space) is of dimension 2. We also present the analogous result for irreducible walks with an internal degree of arbitrary finite dimension and the absorption problem for walks on the semi-infinite line.
In recent years, several properties and recurrence criteria of discrete-time open quantum walks (OQWs) have been presented. Recently, Pellegrini introduced continuous-time open quantum walks (CTOQWs) as continuous-time natural limits of discrete-time OQWs. In this work, we study semifinite CTOQWs and some of their basic properties concerning statistics, such as transition probabilities and site recurrence. The notion of SJK-recurrence for CTOQWs is introduced, and it is shown to be equivalent to the traditional concept of recurrence. This statistic arises from the definition of δ\deltaδ-skeleton of CTOQWs, which is a dynamic that allows us to obtain a discrete-time OQW in terms of a CTOQW. We present a complete criterion for site recurrence in the case of CTOQW induced by a coin of finite dimension with a set of vertices Z\mathbb{Z}Z such that its auxiliary Lindblad operator has a single stationary state. Finally, we present a similar criterion that completes the case in which the internal degree of freedom of each site is of dimension 2.
The P\'olya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. \ifmmode \check{S}\else \v{S}\fi{}tefa\ifmmode \check{n}\else \v{n}\fi{}\'ak et al., Phys. Rev. Lett. 100, 020501 (2008)] which is based on a specific measurement scheme. The P\'olya number of a quantum walk depends, in general, on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by Hadamard tensor-product coins, the P\'olya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an estimation of the P\'olya number for this class of quantum walks. Second, we examine the two-dimensional Grover walk, which exhibits localization and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with classical walks which are recurrent only for the dimensions d=1,2d=1,2d=1,2. Finally, we analyze the recurrence of the 2D Fourier walk. This quantum walk is recurrent except for a two-dimensional subspace of the initial states. We provide an estimation of the P\'olya number in its dependence on the initial state.
Recurrence of quantum walks on lattices can be characterized by the generalized Pólya number. Its value reflects the difference between a classical and a quantum system. The dimension of the lattice is not a unique parameter in the quantum case; both the coin operator and the initial quantum state of the coin influence the recurrence in a nontrivial way. In addition, the definition of the Pólya number involves measurement of the system. Depending on how measurement is included in the definition, the recurrence properties vary. We show that in the limiting case of frequent, strong measurements, one can approach the classical dynamics. Comparing various cases, we have found numerical indication that our previous definition of the Pólya number provides an upper limit.
We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.