Open quantum jump chain for a class of continuous-time open quantum walks

In this work, we study the recurrence for a class of continuous-time open quantum walks (CTOQWs). Those random walks are continuous-time version of the open quantum walks (OQWs) introduced by Attal et al. and generalize the classical continuous-time Markov chains. We introduce the open quantum jump chain for a class of CTOQWs and show that this chain is a discrete-time OQW and that it is recurrent if and only if the CTOQW is recurrent. Moreover, we show that a vertex is recurrent with respect to some density ρ for the OQW if and only if it is recurrent with respect to ρ for the CTOQW, therefore this statistic is connected by the initial quantum state, and thus the discrete-time and continuous-time versions of open quantum walks perform an extended property of the classical one for each initial state. Finally, we give a complete recurrence criterion for this class of CTOQWs induced by a coin of dimension 2.