Asymptotic velocity of a position-dependent quantum walk

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• 作者：Akito Suzuki
• 关键词：Quantum walk ; Asymptotic velocity ; Weak limit
• 刊名：Quantum Information Processing
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：15
• 期：1
• 页码：103-119
• 全文大小：451 KB
• 参考文献：1.Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: One-dimensional quantum walks with one defect. Rev. Math. Phys 24, 52 (2012)CrossRef MathSciNet MATH
  2.Endo, T., Konno, N.: Weak Convergence of the Wojcik Model. arXiv:​1412.​7874v3
  3.Endo, S., Endo, T., Konno, N., Segawa, E., Takei, M.: Weak Limit Theorem of a Two-phase Quantum Walk with One Defect. arXiv:​1412.​4309v2
  4.Grimmett, G., Janson, S., Scudo, P.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)CrossRef ADS
  5.Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2002)MathSciNet CrossRef MATH
  6.Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)MathSciNet CrossRef MATH
  7.Konno, N.: Quantum Walks. Quantum Potential Theory. Lecture Notes in Mathematics, vol. 1954, pp. 309–452. Springer, Berlin (2008)
  8.Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9, 405–418 (2010)MathSciNet CrossRef MATH
  9.Konno, N., Łuczak, T., Segawa, E.: Limit measure of inhomogeneous discrete-time quantum walks in one dimension. Quantum Inf. Process. 12, 33–53 (2013)
  10.Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Academic Press, New York (1972)
  11.Simon, B.: Trace Ideals and Their Applications: Second Edition, Mathematical Surveys and Monographs., vol. 120. American Mathematical Society, Providence (2005)
  12.Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015–1106 (2012)MathSciNet CrossRef MATH
  13.Wójcik, A., Łuczak, T., Kurzyński, P., Grudka, A., Gdala, T., Bednarska-Bzdega, M.: Trapping a particle of a quantum walk on the line. Phys. Rev. A 85, 012329 (2012)CrossRef ADS
  
• 作者单位：Akito Suzuki (1)
  
  1. Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato, Nagano, 380-8553, Japan
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Mathematics
  Engineering, general
  Computer Science, general
  Characterization and Evaluation Materials
  
• 出版者：Springer Netherlands
• ISSN：1573-1332

文摘

We consider a position-dependent coined quantum walk on \(\mathbb {Z}\) and assume that the coin operator C(x) satisfies $$\begin{aligned} \Vert C(x) - C_0 \Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z}\setminus \{0\} \end{aligned}$$with positive \(c_1\) and \(\epsilon \) and \(C_0 \in U(2)\). We show that the Heisenberg operator \(\hat{x}(t)\) of the position operator converges to the asymptotic velocity operator \(\hat{v}_+\) so that $$\begin{aligned} \text{ s- }\lim _{t \rightarrow \infty } \mathrm{exp}\left( i \xi \frac{\hat{x}(t)}{t} \right) = \Pi _\mathrm{p}(U) + \mathrm{exp}(i \xi \hat{v}_+) \Pi _\mathrm{ac}(U) \end{aligned}$$provided that U has no singular continuous spectrum. Here \(\Pi _\mathrm{p}(U)\) (resp., \(\Pi _\mathrm{ac}(U)\)) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable \(X_t\) denoting the position of a quantum walker at time \(t \in \mathbb {N}\), \(X_t/t\) converges in law to a random variable V with the probability distribution $$\begin{aligned} \mu _V = \Vert \Pi _\mathrm{p}(U)\Psi _0\Vert ^2\delta _0 + \Vert E_{\hat{v}_+}(\cdot ) \Pi _\mathrm{ac}(U)\Psi _0\Vert ^2, \end{aligned}$$where \(\Psi _0\) is the initial state, \(\delta _0\) the Dirac measure at zero, and \(E_{\hat{v}_+}\) the spectral measure of \(\hat{v}_+\). Keywords Quantum walk Asymptotic velocity Weak limit Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (13) References1.Cantero, M.J., Grünbaum, F.A., Moral, L., Velázquez, L.: One-dimensional quantum walks with one defect. Rev. Math. Phys 24, 52 (2012)CrossRefMathSciNetMATH2.Endo, T., Konno, N.: Weak Convergence of the Wojcik Model. arXiv:​1412.​7874v3 3.Endo, S., Endo, T., Konno, N., Segawa, E., Takei, M.: Weak Limit Theorem of a Two-phase Quantum Walk with One Defect. arXiv:​1412.​4309v2 4.Grimmett, G., Janson, S., Scudo, P.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)CrossRefADS5.Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1, 345–354 (2002)MathSciNetCrossRefMATH6.Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57, 1179–1195 (2005)MathSciNetCrossRefMATH7.Konno, N.: Quantum Walks. Quantum Potential Theory. Lecture Notes in Mathematics, vol. 1954, pp. 309–452. Springer, Berlin (2008)8.Konno, N.: Localization of an inhomogeneous discrete-time quantum walk on the line. Quantum Inf. Process. 9, 405–418 (2010)MathSciNetCrossRefMATH9.Konno, N., Łuczak, T., Segawa, E.: Limit measure of inhomogeneous discrete-time quantum walks in one dimension. Quantum Inf. Process. 12, 33–53 (2013)10.Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Academic Press, New York (1972)11.Simon, B.: Trace Ideals and Their Applications: Second Edition, Mathematical Surveys and Monographs., vol. 120. American Mathematical Society, Providence (2005)12.Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015–1106 (2012)MathSciNetCrossRefMATH13.Wójcik, A., Łuczak, T., Kurzyński, P., Grudka, A., Gdala, T., Bednarska-Bzdega, M.: Trapping a particle of a quantum walk on the line. Phys. Rev. A 85, 012329 (2012)CrossRefADS About this Article Title Asymptotic velocity of a position-dependent quantum walk Journal Quantum Information Processing Volume 15, Issue 1 , pp 103-119 Cover Date2016-01 DOI 10.1007/s11128-015-1183-x Print ISSN 1570-0755 Online ISSN 1573-1332 Publisher Springer US Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Quantum Information Technology, Spintronics Quantum Computing Data Structures, Cryptology and Information Theory Quantum Physics Mathematical Physics Keywords Quantum walk Asymptotic velocity Weak limit Industry Sectors Aerospace IT & Software Telecommunications Authors Akito Suzuki (1) Author Affiliations 1. Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato, Nagano, 380-8553, Japan Continue reading... To view the rest of this content please follow the download PDF link above.