Skip to main content

Mutually Unbiased Biangular Vectors and Association Schemes

  • Conference paper
Algebraic Design Theory and Hadamard Matrices

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 133))

Abstract

A class of unbiased (−1, 1)-matrices extracted from a single Hadamard matrix is shown to provide uniform imprimitive association schemes of four class and six class.

This paper is in final form and no similar paper has been or is being submitted elsewhere.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    MATH  Google Scholar 

  2. Best, D., Kharaghani, H.: Unbiased complex Hadamard matrices and bases. Cryptogr. Commun. 2(2), 199–209 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boykin, P.O., Sitharam, M., Tarifi, M., Wocjan, P.: Real mutually unbiased bases. Preprint. arXiv:quant ph/0502024v2

    Google Scholar 

  4. Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: Z 4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. (3) 75(2), 436–480 (1997)

    Google Scholar 

  5. Cameron, P.J.: On groups with several doubly-transitive permutation representations. Math. Z. 128, 1–14 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. van Dam, E.R., Martin, W.J., Muzychuk, M.: Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems. J. Comb. Theory Ser. A 120(7), 1401–1439 (2013)

    Article  Google Scholar 

  7. Durt, T., Englert, B.H., Bengtsson, I., Życzkowski, K.: On mutually unbiased bases. Int. J. Quantum Inf. 8, 535–640 (2010)

    Article  MATH  Google Scholar 

  8. Ge, G.: Group divisible designs. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications (Boca Raton), 2nd edn, pp. xxii+984. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  9. Godsil, C., Roy, A.: Equiangular lines, mutually unbiased bases, and spin models. Eur. J. Comb. 30(1), 246–262 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Godsil, C.D., Song, S.Y.: Association schemes. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 325–330. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  11. Holzmann, W.H., Kharaghani, H., Orrick, W.: On the real unbiased Hadamard matrices. In: Combinatorics and Graphs. Contemporary Mathematics, vol. 531, pp. 243–250. American Mathematical Society, Providence, RI (2010)

    Google Scholar 

  12. Kharaghani, H.: New class of weighing matrices. Ars Comb. 19, 69–72 (1985)

    MathSciNet  MATH  Google Scholar 

  13. LeCompte, N., Martin, W.J., Owens, W.: On the equivalence between real mutually unbiased bases and a certain class of association schemes. Eur. J. Comb. 31(6), 1499–1512 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mathon, R.: 3-class association schemes. In: Proceedings of the Conference on Algebraic Aspects of Combinatorics (University Toronto, Toronto, Ontario, 1975), pp. 123–155. Congressus Numerantium, No. XIII. Utilitas Math., Winnipeg, Man. (1975)

    Google Scholar 

  15. Mathon, R.: The systems of linked 2 − (16, 6, 2) designs. Ars Comb. 11, 131–148 (1981)

    Google Scholar 

  16. Wocjan, P., Beth, T.: New construction of mutually unbiased bases in square dimensions. Quantum Inf. Comput. 5(2), 93–101 (2005)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Kharaghani thanks NSERC for the continuing support of his research.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to H. Kharaghani .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Holzmann, W.H., Kharaghani, H., Suda, S. (2015). Mutually Unbiased Biangular Vectors and Association Schemes. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_12

Download citation

Publish with us

Policies and ethics