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We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limit. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0, Comment: 14 pages, 1 figure

Quantum walks are powerful tools not only to construct the quantum speedup algorithms but also to describe specific models in physical processes. Furthermore, the discrete time quantum walk has been experimentally realized in various setups. We apply the concept of the quantum walk to the problems in quantum foundations. We show that randomness and the arrow of time in the quantum walk gradually emerge by periodic projective measurements from the mathematically obtained limit distribution under the time scale transformation., 6 pages, 3 figures

We consider a discrete-time quantum walk W_t given by the Grover transformation on the Cayley tree. We reduce W_t to a quantum walk X_t on a half line with a wall at the origin. This paper presents two types of limit theorems for X_t. The first one is X_t as t\to\infty, which corresponds to a localization in the case of an initial qubit state. The second one is X_t/t as t\to\infty, whose limit density is given by the Konno density function [1-4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the Cayley tree., 10 pages, 4 figures

We consider a discrete-time quantum walk $W_{t,\kappa}$ at time $t$ on a graph with joined half lines $\mathbb{J}_\kappa$, which is composed of $\kappa$ 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 \textit{symmetric} initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that $W_{t,\kappa}$ can be reduced to the walk on a half line even if the initial state is \textit{asymmetric}. For $W_{t,\kappa}$, we obtain two types of limit theorems. The first one is an asymptotic behavior of $W_{t,\kappa}$ which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for $W_{t,\kappa}$. On each half line, $W_{t,\kappa}$ 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., Comment: 18 pages, 7 figures