Localization of quantum walks induced by recurrence properties of random walks

Details Localization of quantum walks induced by recurrence properties of random walks Localization of quantum walks induced by recurrence properties of random walks

2011-12-21

arxiv.org/abs/1112.4982v1

Physics

Quantum Physics

12 pages

Scientific paper

We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization of the corresponding QW. We find the following two fundamental derivations of localization of the QW. The first one is the set of all the $\ell^2$ summable eigenvectors of the corresponding RW. The second one is the orthogonal complement, whose eigenvalue is -1, of the subspace induced by the RW. In particular, as a consequence, for an infinite half line, we show that localization of the QW can be ensured by the positive recurrence of the corresponding RWs, and also that the existence of only one self loop affects localization properties.

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