Site recurrence of open and unitary quantum walks on the line

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 (Grünbaum et al. in Commun Math Phys 320:543–569, 2013; Lardizabal and Souza in J Stat Phys 159:772–796, 2015), 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 is 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.