Decay Parameter for Continuous-Time Open Quantum Walks

Abstract In this paper we study the exponential decay parameter for continuous-time open quantum walks. This statistic is obtained by a composed function that associates the natural logarithm to the transition probability of the walk, being inspired by the classical decay parameter of continuous-time Markov chains. The existence of this statistic is proven for finite graphs where each vertex has a finite degree of freedom, and it is intimately connected to the spectrum of the Lindblad generator describing the dynamics. We show that faithful states yield the slowest decay, whereas non-faithful states can accelerate decay and modify the communication structure of the graph, a phenomenon with no classical analogue. Extensions to infinite graphs are discussed as open problems and conjectures. An equivalence theorem relating different notions of recurrence is established under the hypothesis that the decay parameter exists.