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Computing Equiangular Lines in Complex Space

  • Markus Grassl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5393)

Abstract

We consider the problem of finding equiangular lines in complex space, i. e., sets of unit vectors such that the modulus of the inner product between any two vectors is constant. We focus on the case of d 2 such vectors in a space of dimension d which corresponds to so-called SIC-POVMs. We discuss how symmetries can be used to simplify the problem and how the corresponding system of polynomial equations can be solved using techniques of modular computation.

Keywords

Complex Space Polynomial Equation Computer Algebra System Hilbert Series Tight Frame 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Appleby, D.M.: SIC-POVMs and the Extended Clifford Group. Journal of Mathematical Physics 46, 052107 (2005); Preprint quant-ph/0412001MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnold, E.A.: Modular Algorithms for Computing Gröbner Bases. Journal of Symbolic Computation 35(4), 403–419 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J.J., Playoust, C.: The Magma Algebra System I: The User Language. Journal of Symbolic Computation 24(3–4), 235–266 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buchberger, B.: A Theoretical Basis for the Reduction of Polynomials to Canonical Forms. ACM SIGSAM Bulletin 10(3), 19–29 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cox, D.A., Little, J.B., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    Faugère, J.C.: A New Efficient Algorithm for Computing Gröbner Bases (F4). Journal of Pure and Applied Algebra 139(1–3), 61–88 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Faugère, J.C., Gianni, P., Lazard, D., Mora, T.: Efficient Computations of Zero-dimensional Gröbner Bases by Change of Ordering. Journal of Symbolic Computation 16(4), 329–344 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Flammia, S.T.: On SIC-POVMs in Prime Dimensions. Journal of Physics A 39(43), 13483–13493 (2006); Preprint quant-ph/0605050MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Flammia, S.T., Silberfarb, A., Caves, C.M.: Minimal Informationally Complete Measurements for Pure States. Foundations of Physics 35(12), 1985–2006 (2005); Preprint quant-ph/0404137MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grassl, M.: On SIC-POVMs and MUBs in Dimension 6. In: Proceedings ERATO Conference on Quantum Information Science 2004 (EQIS 2004), Tokyo, pp. 60–61 (September 2004); Preprint quant-ph/0406175Google Scholar
  11. 11.
    Grassl, M.: Tomography of Quantum States in Small Dimensions. Electronic Notes in Discrete Mathematics 20, 151–164 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grassl, M.: Finding Equiangular Lines in Complex Space. Talk at the MAGMA 2006 Conference, Technische Universität Berlin (July 2006)Google Scholar
  13. 13.
    Hoggar, S.G.: t-Designs in Projective Spaces. European Journal of Combinatorics 3, 233–254 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khatirinejad, M.: On Weyl-Heisenberg Orbits of Equiangular Lines. Journal of Algebraic Combinatorics 28(3), 333–349 (2007) (Published online November 6, 2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lemmens, P.H.W., Seidel, J.J.: Equiangular Lines. Journal of Algebra 24(3), 494–512 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric Informationally Complete Quantum Measurements. Journal of Mathematical Physics 45(6), 2171–2180 (2004); Preprint quant-ph/0310075MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, P.S., Guy, M.J.T., Davenport, J.H.: P-adic Reconstruction of Rational Numbers. ACM SIGSAM Bulletin 16(2), 2–3 (1982)CrossRefzbMATHGoogle Scholar
  18. 18.
    Zauner, G.: Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie. Dissertation, Universität Wien (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Markus Grassl
    • 1
  1. 1.Institute for Quantum Optics and Quantum InformationAustrian Academy of SciencesInnsbruckAustria

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