Time operators for quantum walks
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Time operators for quantum walks Daiju Funakawa1 · Yasumichi Matsuzawa2 · Itaru Sasaki3 · Akito Suzuki4 · Noriaki Teranishi5 Received: 21 September 2019 / Revised: 18 March 2020 / Accepted: 6 June 2020 © Springer Nature B.V. 2020
Abstract We study time operators for discrete-time quantum systems. Quantum walks are typical examples. We construct time operators for one-dimensional homogeneous quantum walks and determine their deficiency indices and spectra. Our time operators always have self-adjoint extensions. This is in contrast to the fact that time operators for continuous-time quantum systems generally have no self-adjoint extensions. The uniqueness of the extensions relates to the winding numbers corresponding to the system. If it is unique, its spectrum becomes a discrete set of real numbers, i.e., the time operator is quantized. Keywords Quantum walks · Time operators · Self-adjoint extensions · Deficiency indices · Spectra
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Yasumichi Matsuzawa [email protected] Daiju Funakawa [email protected] Itaru Sasaki [email protected] Akito Suzuki [email protected] Noriaki Teranishi [email protected]
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Department of Electronics and Information Engineering, Hokkai-Gakuen University, Sapporo 062-8605, Japan
2
Department of Mathematics, Faculty of Education, Shinshu University, 6-Ro, Nishi-nagano, Nagano 380-8544, Japan
3
Department of Mathematics, Shinshu University, Matsumoto 390-8621, Japan
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Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato, Nagano 380-8553, Japan
5
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
123
D. Funakawa et al.
Mathematics Subject Classification 46N50 · 47B25 · 81Q10
1 Introduction The time evolution of a continuous-time quantum system is described by a oneparameter unitary group {e−it H }t∈R , and its generator H is a self-adjoint operator called Hamiltonian. A time operator of the system is formally defined as a Hermitian operator T which satisfies the canonical commutation relation with H , i.e., T H − H T = i. It was widely believed for a long time that in quantum theory there exists no time operator, as Pauli pointed out in his famous textbook [11, p. 63, footnote 2]. On the other hand, Aharonov and Bohm constructed a concrete time operator of the one-dimensional free Hamiltonian [1]. This apparently contradicts to Pauli’s claim. This contradiction comes from not distinguishing between self-adjoint operators and symmetric operators. This suggests that we must study time operators in a mathematically rigorous way. It is important to pay attention to domains of time operators. Such a study was initiated by Miyamoto [9]. He introduced a strong time operator, which is a symmetric operator satisfying the following operator equality eit H T e−it H = T + t, ∀t ∈ R. This relation leads T H − H T = i on some dense subspace. In this sense, a strong time operator is a time operator. Arai [4] investigated the spectra of strong time operators. A purely mathematical study of such pairs (T , H
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