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Takako Endo

Publications and source records attributed to Takako Endo.

13 recordsLinked to original sources
arXiv2025

Spectrum for a non-unitary one-dimensional two-state quantum walk with one defect

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization. Also, for the study of open quantum systems, non-Hermitian systems have attracted much attention. As mathematical models for such systems, non-unitary quantum walks with the chiral symmetry are essential for the study of the topological insulator. In this paper, we give the whole picture of the eigenvalues of a non-unitary one-dimensional two-state quantum walks with one defect and the chiral symmetry.

math-ph↗
arXiv2021

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 $x\geq0$ and $x\leq-1$. We call the QW the {\it complete two-phase QW} to discern from the two-phase QW with one defect\cite{endosan,maman}. 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 {\it localization} and the {\it ballistic spreading}, 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, including dependences on initial states, 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 {\it topological protections}, we clarify that localization of the two-phase QW with one defect, studied in the previous work\cite{endosan}, can be related to localization of the complete two-phase QW under symmetry preserving perturbations.

math-ph↗
arXiv2019

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.

math.SP↗
arXiv2019

Stationary measure for three-state quantum walk

We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.

math-ph↗
arXiv2017

Sensitivity of quantum walks to boundary of two-dimensional lattices: approaches from the CGMV method and topological phases

In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys.Rev. A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly,we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.

math-ph↗
arXiv2016

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.

math-ph↗
arXiv2015

Weak convergence of the Wojcik model

We study "the Wojcik model" which is a discrete-time quantum walk (QW) with one defect in one dimension, introduced by Wojcik et al.. For the Wojcik model, we give the weak convergence theorem describing the ballistic behavior of the walker in the probability distribution in a rescaled position-space. In our previous studies, we obtained the time-averaged limit and stationary measures concerning localization for the Wojcik model. As a result, we get the mathematical expression of the whole picture of the behavior of the walker for the Wojcik model. Here the coexistence of localization and the ballistic spreading is one of the peculiar properties of one-dimensional QWs with one defect. Due to the coexistence, it has been strongly expected to utilize QWs to quantum search algorithms. In order to derive the weak convergence theorem, we take advantage of the generating function method. We emphasize that the time-averaged limit measure is symmetric for the origin, however, the weight function in the weak limit measure is asymmetric in general, which implies that the weak convergence theorem represents the asymmetry of the probability distribution. Furthermore, the weak limit measure heavily depends on the phase of the defect and initial state of the walker. Comparing with our previous studies, we also show some numerical results of the probability distribution to confirm that our result is relevant mathematically, and consider the effect of changing the phase and initial coin state on the probability distribution, or the ballistic spreading, which is one of the motivations of our study.

math-ph↗
arXiv2015

Limit theorems of a two-phase quantum walk with one defect

We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model "the two-phase QW" here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods "the splitted generating function method" and "the generating function method", respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.

math-ph↗
arXiv2015

Weak limit theorem of a two-phase quantum walk with one defect

We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW "the two-phase QW", which we treated concerning localization theorems [10]. The two-phase QW has been expected to be a mathematical model of the topological insulator [16] which is an intense issue both theoretically and experimentally [3,5,11]. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the time-averaged limit measure is symmetric for the origin [10], however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.

math-ph↗
arXiv2014

A one-dimensional Hadamard walk with one defect

We consider a one-dimensional space-inhomogeneous discrete time quantum walk. This model is the Hadamard walk with one defect at the origin which is different from the model introduced by Wojcik et al. [14]. We obtain a stationary measure of the model by solving the eigenvalue problem and an asymptotic behaviour of the return probability by the path counting approach. Moreover, we get the time-averaged limit measure using the space-time generating function method. The measure is symmetric for the origin and independent of the initial coin state at the starting point. So localization depends only on the parameter which determines the model.

math-ph↗
arXiv2014

The time-averaged limit measure of the Wojcik model

We investigate "the Wojcik model" introduced and studied by Wojcik et al., which is a one-defect quantum walk (QW) having a single phase at the origin. They reported that giving a phase at one point causes an astonishing effect for localization. There are three types of measures having important roles in the study of QWs: time-averaged limit measure, weak limit measure, and stationary measure. The first two measures imply a coexistence of localized behavior and the ballistic spreading in the QW. As Konno et al. suggested, the time-averaged limit and stationary measures are closely related to each other for some models. In this paper, we focus on a relation between the two measures for the Wojcik model. The stationary measure was already obtained by our previous work. Here, we get the time-averaged limit measure by several methods. Our results show that the stationary measure is a special case of the time-averaged limit measure.

math-ph↗
arXiv2014

The stationary measure of a space-inhomogeneous quantum walk on the line

We study a discrete-time quantum walk (QW) on the line with a single phase at the origin which was introduced and studied by Wojcik et al.[1]. We call the model "Wojcik model" here. Konno et al.[2] investigated other types of QWs with one defect at the origin. They presented a method which gives the stationary measure corresponding to localization for the QWs by use of the generating functions splitted in positive and negative parts respectively. In this paper, we call the method "the splitted generating function method (the SGF method)". To clarify in detail which QW is appropriate for the method in the case of study is one of the important challenges to investigate localization properties for various QWs. As for the Wojcik model, we solve the eigenvalue problem by the SGF method and our results agree with Ref.[1]. From the solution of the problem, we derive a stationary measure with an exponential decay for the position. The explicit expression for the stationary measure is symmetric for the origin and ensures localization depending on the initial coin state.

quant-ph↗