Limit measures of inhomogeneous discrete-time quantum walks in one dimension

We treat three types of measures of the quantum walk (QW) with the spatial perturbation at the origin, which was introduced by Konno (Quantum Inf Proc 9:405, 2010): time averaged limit measure, weak limit measure, and stationary measure. From the first two measures, we see a coexistence of the ballistic and localized behaviors in the walk as a sequential result following (Konno in Quantum Inf Proc 9:405, 2010; Quantum Inf Proc 8:387-399, 2009). We propose a universality class of QWs with respect to weak limit measure. It is shown that typical spatial homogeneous QWs with ballistic spreading belong to the universality class. We find that the walk treated here with one defect also belongs to the class. We mainly consider the walk starting from the origin. However when we remove this restriction, we obtain a stationary measure of the walk. As a consequence, by choosing parameters in the stationary measure, we get the uniform measure as a stationary measure of the Hadamard walk and a time averaged limit measure of the walk with one defect respectively.