188 research outputs found
Relation between two-phase quantum walks and the topological invariant
We study a position-dependent discrete-time quantum walk (QW) in one
dimension, whose time-evolution operator is built up from two coin operators
which are distinguished by phase factors from and . We call
the QW the - to discern from the
two-phase QW with one defect[13,14]. Because of its localization properties,
the two-phase QWs can be considered as an ideal mathematical model of
topological insulators which are novel quantum states of matter characterized
by topological invariants. Employing the complete two-phase QW, we present the
stationary measure, and two kinds of limit theorems concerning and the , which are the
characteristic behaviors in the long-time limit of discrete-time QWs in one
dimension. As a consequence, we obtain the mathematical expression of the whole
picture of the asymptotic behavior of the walker in the long-time limit. We
also clarify relevant symmetries, which are essential for topological
insulators, of the complete two-phase QW, and then derive the topological
invariant. Having established both mathematical rigorous results and the
topological invariant of the complete two-phase QW, we provide solid arguments
to understand localization of QWs in term of topological invariant.
Furthermore, by applying a concept of , we
clarify that localization of the two-phase QW with one defect, studied in the
previous work[13], can be related to localization of the complete two-phase QW
under symmetry preserving perturbations.Comment: 50 pages, 13 figure
The stationary measure for diagonal quantum walk with one defect
This study is motivated by the previous work [14]. We treat 3 types of the
one-dimensional quantum walks (QWs), whose time evolutions are described by
diagonal unitary matrix, and diagonal unitary matrices with one defect. In this
paper, we call the QW defined by diagonal unitary matrices, "the diagonal QW",
and we consider the stationary distributions of generally 2-state diagonal QW
with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW
with one defect. One of the purposes of our study is to characterize the QWs by
the stationary measure, which may lead to answer the basic and natural
question, "What the stationary measure is for one-dimensional QWs ?". In order
to analyze the stationary distribution, we focus on the corresponding
eigenvalue problems and the definition of the stationary measure.Comment: 10 page
Eigenvalues of two-state quantum walks induced by the Hadamard walk
Existence of the eigenvalues of the discrete-time quantum walks is deeply
related to localization of the walks. We revealed the distributions of the
eigenvalues given by the splitted generating function method (the SGF method)
of the quantum walks we had treated in our previous studies. In particular, we
focused on two kinds of the Hadamard walk with one defect models and the
two-phase QWs that have phases at the non-diagonal elements of the unitary
transition operators. As a result, we clarified the characteristic parameter
dependence for the distributions of the eigenvalues with the aid of numerical
simulation.Comment: 9 pages, 4 figure
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