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Weak limit theorem for a nonlinear quantum walk

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Abstract

This paper continues the study of large time behavior of a nonlinear quantum walk begun in Maeda et al. (Discrete Contin Dyn Syst 38:3687–3703, 2018). In this paper, we provide a weak limit theorem for the distribution of the nonlinear quantum walk. The proof is based on the scattering theory of the nonlinear quantum walk, and the limit distribution is obtained in terms of its asymptotic state.

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References

  1. Di Molfetta, G., Debbasch, F., Brachet, M.: Nonlinear optical Galton board: thermalization and continuous limit. Phys. Rev. E 92, 042923 (2015)

    Article  ADS  Google Scholar 

  2. Endo, S., Endo, T., Konno, N., Segawa, E., Takei, M.: Weak limit theorem of a two-phase quantum walk with one defect, Interdiscip. Inform. Sci. J-STAGE Advance, Graduate School of Information Sciences, Tohoku University (2016)

  3. Endo, T., Konno, N., Obuse, H., Segawa, E.: Sensitivity of quantum walks to a boundary of two-dimensional lattices: approaches based on the CGMV method and topological phases. J. Phys. A Math. Theor. 50(45), 455302 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Fuda, T., Funakawa, D., Suzuki, A.: Weak limit theorem for a one-dimensional split-step quantum walk, arXiv:1804.05125 (2018)

  5. Gerasimenko, Y., Tarasinski, B., Beenakker, C.W.J.: Attractor-repeller pair of topological zero modes in a nonlinear quantum walk. Phys. Rev. A 93, 022329 (2016)

    Article  ADS  Google Scholar 

  6. Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)

    Article  ADS  Google Scholar 

  7. Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Spectral and asymptotic properties of Grover walks on crystal lattices. J. Funct. Anal. 267, 4197–4235 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Higuchi, Yu., Segawa, E.: The spreading behavior of quantum walks induced by drifted random walks on some magnifier graph. Quantum Inf. Comput. 17, 0399–0414 (2017)

    MathSciNet  Google Scholar 

  9. Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345–354 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Konno, N.: A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Jpn. 57(4), 1179–1195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Konno, N., Łuczak, T., Segawa, E.: Limit measures of inhomogeneous discrete-time quantum walks in one dimension. Quantum Inf. Process. 12(1), 33–53 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Lee, C.-W., Kurzyński, P., Nha, H.: Quantum walk as a simulator of nonlinear dynamics: nonlinear Dirac equation and solitons. Phys. Rev. A 92, 052336 (2015)

    Article  ADS  Google Scholar 

  13. Machida, T., Segawa, E.: Trapping and spreading properties of quantum walk in homological structure. Quantum Inf. Process. 14, 1539–1558 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Maeda, M., Sasaki, H., Segawa, E., Suzuki, A., Suzuki, K.: Scattering and inverse scattering for nonlinear quantum walks. Discrete Contin. Dyn. Syst. 38, 3687–3703 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Maeda, M., Sasaki, H., Segawa, E., Suzuki, A., Suzuki, K.: Dynamics of solitons to a nonlinear quantum walk (in preparation)

  16. Navarrete-Benlloch, C., Pérez, A., Roldán, E.: Nonlinear optical Galton board. Phys. Rev. A 75, 062333 (2007)

    Article  ADS  Google Scholar 

  17. Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin I: spectral theory. Lett. Math. Phys. 108, 331–357 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Richard, S., Suzuki, A., Tiedra de Aldecoa, R.: Quantum walks with an anisotropic coin II: scattering theory. Lett. Math. Phys. (2018). https://doi.org/10.1007/s11005-018-1100-1

  19. Shikano, Y., Wada, T., Horikawa, J.: Discrete-time quantum walk with feed-forward quantum coin. Sci. Rep. 4, 4427 (2014)

    Article  ADS  Google Scholar 

  20. Suzuki, A.: Asymptotic velocity of a position-dependent quantum walk. Quantum Inf. Process. 15(1), 103–119 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

M.M. was supported by the JSPS KAKENHI Grant Numbers JP15K17568, JP17H02851, and JP17H02853. H.S. was supported by JSPS KAKENHI Grant Number JP17K05311. E.S. acknowledges the financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant No. 16K17637 and No. 16K03939). A.S. was supported by JSPS KAKENHI Grant Numbers JP26800054 and JP18K03327. K.S acknowledges JSPS Grant-in-Aid for Scientific Research (C) 26400156 and 18K03354.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba, 263-8522, Japan

    Masaya Maeda & Hironobu Sasaki

  2. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Etsuo Segawa

  3. Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Nagano, 380-8553, Japan

    Akito Suzuki

  4. College of Science, Ibaraki University, 2-1-1 Bunkyo, Mito, 310-8512, Japan

    Kanako Suzuki

Authors
  1. Masaya Maeda
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  2. Hironobu Sasaki
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  3. Etsuo Segawa
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  4. Akito Suzuki
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  5. Kanako Suzuki
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Corresponding author

Correspondence to Akito Suzuki.

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Maeda, M., Sasaki, H., Segawa, E. et al. Weak limit theorem for a nonlinear quantum walk. Quantum Inf Process 17, 215 (2018). https://doi.org/10.1007/s11128-018-1981-z

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  • DOI: https://doi.org/10.1007/s11128-018-1981-z

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