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.
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Acknowledgements
The authors would like to thank the referees for constructive comments, which have been helpful to improve this paper. This work was supported by JSPS KAKENHI (Grant Number JP18K03327, JP16K17612 and 26800055), and by the Research Institute for Mathematical Sciences, a Joint Usage/ Research Center located in Kyoto University.
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A Fundamental properties of time operators of a unitary operator
A Fundamental properties of time operators of a unitary operator
The following are fundamental properties of time operators of a unitary operator.
Proposition A.1
Let T be a strong time operator of a unitary operator U. Then the following hold.
-
(1)
Its closure \({\bar{T}}\) is a strong time operator of U as well.
-
(2)
\([T,U]=U\) holds on a dense subspace D(T). In particular, T is a time operator of U. The converse is not true, i.e., not every time operator of U is a strong time operator of U.
-
(3)
\(\sigma (T)=\sigma (T+1)\) holds. In particular, T is unbounded.
-
(4)
If T is essentially self-adjoint, then \(\sigma (U)={\mathbb {T}}\)
Proof
(1) follows from a simple limiting argument. The first half of (2) is straightforward. For the second half of (2), see Example 4.6. (3) is obvious. We show (4). It follows that
for any \(t\in {\mathbb {R}}\). Thus, we obtain \(e^{it{\bar{T}}}Ue^{-it{\bar{T}}}=e^{it}U\). This means that \(\sigma (U)=\sigma (e^{it}U)\) for all \(t\in {\mathbb {R}}\). Hence, \(\sigma (U)={\mathbb {T}}\) holds. \(\square \)
Theorem A.2
Let U be a unitary operator admitting a strong time operator T. Then U has no eigenvalues.
Proof
The proof is same as the proof of [9, Corollary 4.3], and thus we omit it. \(\square \)
A strong time operator governs the decay rate of the transition probability as follows:
Theorem A.3
Let T be a strong time operator of a unitary operator U. Then for any \(n\in {\mathbb {N}}\), \(\psi \in D(T^n)\) and \(\phi \in D\bigl ((T^*)^n\bigr )\), there exists a constant \(C_n(\phi ,\psi )>0\) such that
holds.
Proof
The proof is same as the proof of [3, Theorem 8.5], and thus we omit it. \(\square \)
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Funakawa, D., Matsuzawa, Y., Sasaki, I. et al. Time operators for quantum walks. Lett Math Phys 110, 2471–2490 (2020). https://doi.org/10.1007/s11005-020-01299-5
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DOI: https://doi.org/10.1007/s11005-020-01299-5