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arXiv:1804.05127 · Journal of Mathematical Physics

Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

Toru Fuda, Daiju Funakawa, Akito Suzuki

Published Aug 1, 2018• 23 citations
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Abstract

For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)]. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there are no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±1 exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.

DOI
10.1063/1.5035300

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For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)]. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there are no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±1 exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.
Given two Hilbert spaces, H\mathcal{H}H and K\mathcal{K}K, we introduce an abstract unitary operator UUU on H\mathcal{H}H and its discriminant TTT on K\mathcal{K}K induced by a coisometry from H\mathcal{H}H to K\mathcal{K}K and a unitary involution on H\mathcal{H}H. In a particular case, these operators UUU and TTT become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between UUU and TTT via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.
Suzuki's split-step quantum walk on the one-dimensional integer lattice can be naturally viewed as a chirally symmetric quantum walk. Given the unitary time-evolution of such a chirally symmetric quantum walk, we can separately introduce well-defined indices for the eigenvalues ±1.\pm 1.±1. The bulk-edge correspondence for Suzuki's split-step quantum walk is twofold. Firstly, we show that the multiplicities of the eigenvalues ±1\pm 1±1 coincide with the absolute values of the associated indices. Note that this can be viewed as the symmetry protection of bound states, and that the indices we consider are robust in the sense that these depend only on the asymptotic behaviour of the parameters of the given model. Secondly, we show that that such bound states exhibit exponential decay at spatial infinity.
The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.
In this paper, we derive sufficient conditions for the localization of two-dimensional split-step quantum walks with a strong shift. For this purpose, we analyze the zero points of the function fff introduced by Fuda et. al. (Quantum Inf Process 16(8) 203, 2017) and make these zero points explicit. These zeros provide a concrete representation of the eigenvalues and eigenvectors of the evolution operator, and in particular, clarify where localization occurs. In addition, the eigenvalues obtained here asymptotically approach threshold resonance in special cases. We also describe the display of threshold resonances and generalized eigenfunctions.