### Abstract

We study a generalized Hadamard walk in one dimension with three inner states. The particle governed by the three-state quantum walk moves, in superposition, both to the left and to the right according to the inner state. In addition to these two degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle. In contrast with the Hadamard walk with two inner states on a line, the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states. This implies that the particle is trapped near the origin after a long time with high probability.

Original language | English |
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Article number | 056112 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 72 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2005 Nov 1 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

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## Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*72*(5), [056112]. https://doi.org/10.1103/PhysRevE.72.056112