## Abstract

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.

Original language | English |
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Article number | 1650025 |

Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics |

Volume | 19 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 Dec 1 |

### Bibliographical note

Funding Information:E.S. was supported by the Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 25800088). The research by H.J.Y. was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2013285).

Publisher Copyright:

© 2016 World Scientific Publishing Company.

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Applied Mathematics