Perfect state transfer on weighted graphs of the Johnson scheme

Abstract

We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme \({\mathcal {J}}(n,k)\), which represent spin networks with non-nearest neighbor couplings. Given \({\mathcal {J}}(n,k)=\{A_{1},A_{2},\ldots ,A_{k}\}\) and \(A(X) = w_0A_0 + \cdots + w_m A_m\), we show that X has perfect state transfer at time \(\tau \) if and only if \(n=2k\), \(m\ge 2^{\lfloor {\log _2(k)} \rfloor }\), and there are integers \(c_{1},c_{2},\ldots ,c_{m}\) such that

  1. (i)

    \(c_j\) is odd if and only if j is a power of 2, and

  2. (ii)

    for \(r=1,2,\ldots ,m\),

    $$\begin{aligned} w_r = \frac{\pi }{\tau } \sum _{j=r}^m \frac{c_j}{\left( {\begin{array}{c}2j\\ j\end{array}}\right) } \left( {\begin{array}{c}k-r\\ j-r\end{array}}\right) . \end{aligned}$$

We then characterize perfect state transfer on unweighted graphs of \({\mathcal {J}}(n,k)\). In particular, we obtain a simple construction that generates all graphs of \({\mathcal {J}}(n,k)\) with perfect state transfer at time \(\pi /2\).

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    Ahmadi, B., Shirdareh Haghighi, M.H., Mokhtar, A.: Perfect quantum state transfer on the Johnson scheme (2017). arXiv:1710.09096

  2. 2.

    Albanese, C., Christandl, M., Datta, N., Ekert, A.: Mirror inversion of quantum states in linear registers. Phys. Rev. Lett. 93(23), 230502 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Bannai, E., Ito, T.: Algebraic Combinatorics. I. The Benjamin/Cummings Publishing Co. Inc, Menlo Park, CA (1984)

    MATH  Google Scholar 

  4. 4.

    Bhargava, M.: The factorial function and generalizations. Technical report 9 (2000)

  5. 5.

    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)

    Book  Google Scholar 

  6. 6.

    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Universitext, Springer, New York (2012)

    Book  Google Scholar 

  7. 7.

    Chan, A.: Complex Hadamard matrices, instantaneous uniform mixing and cubes (2013). arXiv:1305.5811

  8. 8.

    Godsil, C.: Periodic graphs. Electron. J. Combin. 18(1), 23 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92(18), 187902 (2004)

    ADS  Article  Google Scholar 

  10. 10.

    Christandl, M., Vinet, L., Zhedanov, A.: Analytic next-to-nearest-neighbor X X models with perfect state transfer and fractional revival. Phys. Rev. A 96(3), 032335 (2017)

    ADS  Article  Google Scholar 

  11. 11.

    Coutinho, G., Godsil, C., Guo, K., Vanhove, F.: Perfect state transfer on distance-regular graphs and association schemes. Linear Algebra Appl. 478, 108–130 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dickson, L.E.: History of the Theory of Numbers: Divisibility and Primality, vol. 1. Chelsea Publishing Co., New York (1966)

    MATH  Google Scholar 

  13. 13.

    Godsil, C.: When can perfect state transfer occur? Electron. J. Linear Algebra 23(1), 877–890 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Jafarizadeh, M.A., Sufiani, R.: Perfect state transfer over distance-regular spin networks. Phys. Rev. A 77(2), 022315 (2008)

    ADS  Article  Google Scholar 

  15. 15.

    Kay, A.: Perfect state transfer: beyond nearest-neighbor couplings. Phys. Rev. A 73(3), 032306 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Kay, A.: Perfect, efficient, state transfer and its application as a constructive tool. Int. J. Quantum Inf. 08(04), 641–676 (2010)

    Article  Google Scholar 

  17. 17.

    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics. Springer, Berlin (2010)

    Book  Google Scholar 

  18. 18.

    Vinet, L., Zhedanov, A.: How to construct spin chains with perfect state transfer. Phys. Rev. A 85(1), 012323 (2012)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their helpful comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hanmeng Zhan.

Ethics declarations

Conflicts of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Vinet, L., Zhan, H. Perfect state transfer on weighted graphs of the Johnson scheme. Lett Math Phys 110, 2491–2504 (2020). https://doi.org/10.1007/s11005-020-01298-6

Download citation

Keywords

  • Quantum walks
  • Perfect state transfer
  • Beyond nearest neighbor couplings
  • Johnson scheme
  • Dual Hahn polynomial

Mathematics Subject Classification

  • 05C50
  • 05E30
  • 11A07
  • 11C20