Localization of a multi-dimensional quantum walk with one defect

For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in [14]. 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 is 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.

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“…Proof. The proof proceeds along the same lines as the proof of [6][Lemma 3.2]. Identifying H with K ⊕ K, we observe that S = p qL qL * −p .…”

Section: Spectral Mapping Theoremmentioning

confidence: 85%

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

Fuda

Funakawa

Suzuki

2018

Journal of Mathematical Physics

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“…For this topic for position-independent cases, see Tate [36]. We also mention Segawa-Suzuki [34], Fuda et al [11] and [12]. The authors used the discriminant operator which is a kind of Hamiltonians generating the time evolution operators of some DTQWs.…”

Section: 2mentioning

confidence: 99%

Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

Morioka

2019

Rev. Math. Phys.

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We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ℓ ∞ -space on Z. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

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“…More recently, Komatsu and Konno [ 4 ] investigated stationary amplitudes of quantum walks on the higher-dimensional integer lattice. There are other works about quantum walks on higher dimensional integer lattices (see e.g., [ 13 , 14 , 15 ]).…”

Section: Introductionmentioning

confidence: 99%

Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Wang

2020

Entropy

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Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relations (CAR) in equal-time. In this paper, we use quantum Bernoulli noises to introduce a model of open quantum walk on the ddimensional integer lattice Z d for a general positive integer d ≥ 2, which we call the d-dimensional open QBN walk. We obtain a quantum channel representation of the d-dimensional open QBN walk, and find that it admits the "separability-preserving" property. We prove that, for a wide range of choices of its initial state, the d-dimensional open QBN walk has a limit probability distribution of d-dimensional Gauss type. Finally we unveil links between the d-dimensional open QBN walk and the unitary quantum walk recently introduced in [Ce Wang and Caishi Wang, Higher-dimensional quantum walk in terms of quantum Bernoulli noises, Entropy 2020, 22, 504].

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Localization of a multi-dimensional quantum walk with one defect

Cited by 32 publications

References 47 publications

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

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

Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

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