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

Localization of Quantum Walks Induced by Recurrence Properties of Random Walks

“…(35): the density is a convex combination of a delta measure at the origin corresponding to localization, and a product of a rational polynomial and the Konno density function. This result of [18] suggests that a small number of perturbations generates edge state of the QW, and removes the missing value corresponding to bulk state ballistically to infinite places keeping a shape characterized by the Konno density functions as is the second term of RHS of Eq. (35).…”

“…Now how about disordered quantum walk case? In the case of one defect on the doubly infinite line [18], the missing value, which spreads away ballistically, appears in the weak limit theorem which is expressed by a similar formulas of Eq. (35): the density is a convex combination of a delta measure at the origin corresponding to localization, and a product of a rational polynomial and the Konno density function.…”

Localization of the Grover Walks on Spidernets and Free Meixner Laws

“…From a mathematical viewpoint sharp contrast between quantum walks and random walks is of particular importance. For example, the ballistic spreading is observed in a wide class of quantum walks [2,10,17,21,22,24,35], i.e., the speed of a quantum walker's spreading is proportional to the time n while the typical scale for a random walk is √ n. Moreover, the limit distributions of quantum walks are obtained [10,17,21,22,24,31,35] with a significant contrast with the normal Gaussian law in the case of random walks. In this paper we focus on the phenomenon called localization, which is also considered as a typical property of quantum walks, see [8,17,25] among others.…”

“…Then we formulate two concepts of localization, that is, initial point localization and exponential localization. Several quantum walks are known to exhibit the localization, see e.g., [8,10,17,24,25,35]. For relevant discussion see also [29].…”

“…Our result is an extension to an infinite system, where the orthogonal polynomials with respect to the free Meixner play a key role. Conservation of probability is an interesting question for a quantum walk, see e.g., [10,17,24,30,35]. The quantum walks studied in [10,17,24] are non-conservative and the "missing probability" is found through the weak convergence theorem in such a way that the limit distribution is a convex combination of a point mass at the origin corresponding to localization and the Konno density function [21,22] coming from ballistic spreading, see [24] for details.…”