Localization of multi-state quantum walk in one dimension

Particle trapping in multi-state quantum walk on a circle is studied. The time-averaged probability distribution of a particle which moves four different lattice sites according to four internal states is calculated exactly. In contrast with “Hadamard walk” with only two internal states, the particle remains at the initial position with high probability. The time-averaged probability of finding the particle decreases exponentially as distance from a center of a spike. This implies that the particle is trapped in a narrow region. This striking difference is minutely explained from difference between degeneracy of eigenvalues of the time-evolution matrices. The dependence of the particle distribution on initial conditions is also considered.