Limit theorems for the discrete-time quantum walk on a graph with joined half lines

We consider a discrete-time quantum walk Wt,κ at time t on a graph with joined half lines Jκ, which is composed of κ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that Wt,κ can be reduced to the walk on a half line even if the initial state is asymmetric. For Wt,κ, we obtain two types of limit theorems. The first one is an asymptotic behavior of Wt,κ which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for Wt,κ. On each half line, Wt,κ converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.