Site recurrence of open and unitary quantum walks on the line

Abstract

We study the problem of site recurrence of discrete time nearest neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks our discussion concerns two notions of recurrence, one given by a monitoring procedure [9,13], another in terms of Pólya numbers [22], and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks are related by an additive interference term in a simple way. Based on a previous result of positive recurrence we describe an open quantum version of Kac's lemma for the expected return time to a site.

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

“…Among the different notions of recurrence appearing in the quantum literature, we will refer here to a recent one based on a monitoring process, developed for unitary quantum walks [1,9,19,24], and later extended to open quantum walks [5,14,18,26]. Consider a discrete time evolution given by iterating a quantum channel Φ on a Hilbert space H. Given a subspace H 0 ⊂ H, we will identify I(H 0 ) with the subspace constituted by those operators ρ ∈ I(H) with ran ρ ⊂ H 0 and ker ρ ⊃ H ⊥ 0 .…”

“…In recent years the problem of finding quantum versions of this lemma has been investigated. In the context of OQWs, versions of Kac's Lemma for the mean return time to some given vertex have been proved in [5,14] (the result is essentially the same in both works, but the proofs employ different techniques). In such a context the OQW is assumed to be irreducible and the mean return time to a vertex |i is conditioned on starting with the i-th positive matrix of the stationary density χ = i χ i ⊗ |i i| of the walk.…”

“…We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].This work provides a detailed study of the consequences of the splitting properties for FR-functions regarding recurrence in quantum Markov chains. Two kinds of FR-function splittings, related to factorizations and decompositions into sums of the underlying operator, yield two types of splitting rules for quantum Markov chains and, thus, for the particular case of classical Markov chains.…”

“…walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].…”

Recurrence and Transience of Continuous-Time Open Quantum Walks

“…From a mathematical point of view, their properties can been studied in analogy with those of classical Markov chain. In particular, usual notions such as ergodicity, central limit theorem, irreducibility, period [1,9,10,8,11,20] have been investigated. For example, the notions of transience and recurrence have been studied in [5], proper definitions of these notions have been developed in this context and the analogues of transient or recurrent points have been characterized.…”