Home | About Journal | Web Links | E-mail Alerts | RSS RSS Icon | Browse
Previous Article Next Article

Hadamard matrices from mutually unbiased bases

Source: J. Math. Phys. 51, 072202 (2010); doi:10.1063/1.3456082

Published 19 July 2010

KEYWORDS and PACS
Keywords
PACS
  • 03.67.Dd
    Quantum cryptography and communication security
  • 02.10.Yn
    Matrix theory
  • YEAR: 2010
RELATED DATABASES

To view database links for this article,
you need to log in.
To view database links for this article,
you need to log in.
PUBLICATION DATA
Publisher:
AIP is a member of CrossRef AIP
P. Dită
National Institute of Physics and Nuclear Engineering, P.O. Box MG6, 077125 Bucharest, Romania
An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional Hadamard matrices that depend on five arbitrary parameters whose modulus is equal to unity. ©2010 American Institute of Physics
History: Received 25 March 2010; accepted 1 June 2010; published 19 July 2010
Permalink: http://link.aip.org/link/?JMAPAQ/51/072202/1

REFERENCES (30)

For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. Bandyopadhyay, S., Boykin, P. O., Roychowdhuri, V., and Vatan, F., “A new proof of the existence of mutually unbiased bases,” Algorithmica 34, 512 (2002).
  2. Bengtsson, I., Bruzda, W., Ericsson, A., Larsson, J. -A., Tadej, W., and Życzkowski, K., “Mutually unbiased bases and Hadamard matrices of order six,” J. Math. Phys. 48, 052106 (2007).
  3. Björck, G. and Fröberg, R., “A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots,” J. Symb. Comput. 12, 329 (1991).
  4. Bruzda, W., Tadej, W. and Życzkowsky, K., online catalog of known Hadamard matrices, http://chaos.if.uj.edu.pl/~karol/hadamard.
  5. Brierley, S., Weigert, S., and Bengtsson, I., “All mutually unbiased bases in dimensions two to five,” e-print arXiv:0907.4097, Int. J. Quant. Inform. (to be published).
  6. Bouguezel, S., Ahmed, M. O., and Swami, M. N. S., “A new class of reciprocal-orthogonal parametric transforms,” IEEE Trans. Circuits Syst., I: Regul. Pap. 56, 795 (2009).
  7. Combescure, M., “The mutually unbiased bases revisited,” Contemp. Math. 447, 29 (2008)
    8. e-print arXiv:quant-ph/0605090
  8. Di[t-cedilla]ă, P., “Some results on the parametrization of complex Hadamard matrices,” J. Phys. A 37, 5355 (2004).
  9. Dită, P., “Separation of unistochastic matrices from the double stochastic ones: Recovery of a 3×3 unitary matrix from experimental data,” J. Math. Phys. 47, 083510 (2006).
  10. Haagerup, U., Operator Algebras and Quantum Field Theory (Rome) (MA International, Cambridge, 1996), pp. 296–322.
  11. Hadamard, J., “Résolution d'une question rélative aux déterminants,” Bull. Sci. Math. 17, 240 (1893).
  12. Horadam, K. J., “A generalized Hadamard transform,” Proc. IEEE ISIT 2005, 1006 (2005).
  13. Ivonovic, I. D., “Geometrical description of quantal state determination,” J. Phys. A 14, 3241 (1981).
  14. Jarlskog, C., “Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP violation,” Phys. Rev. Lett. 55, 1039 (1985).
  15. Jarlskog, C. and Stora, R., “Unitarity polygons and CP violation areas and phases in the standard electroweak model,” Phys. Lett. B 208, 268 (1988).
  16. Karlsson, B. R., “H2-reducible Hadamard matrices of order 6,” e-print arXiv:1003.4133.
  17. Karlsson, B. R., “Three-parameter complex Hadamard matrices of order 6,” e-print arXiv:1003.4177.
  18. Keller, J. B., “Multiple eigenvalues,” Linear Algebr. Appl. 429, 2209 (2008).
  19. Klappenecker, A. and Rötteler, M., “Constructions of mutually unbiased bases,” Lect. Notes Comput. Sci. 2948, 137 (2004)
    20. e-print arXiv:quant-ph/0309120.
  20. Lee, M. H., “A reverse jacket transform and its fast algorithm,” IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 47, 39 (2000).
  21. Matolcsi, M., Réffy, J., and Szöllősi, F., “Constructions of complex Hadamard matrices via tiling Abelian groups,” Open Syst. Inf. Dyn. 14, 247 (2007).
  22. Popa, S., “Orthogonal pairs of *-subalgebras in finite von Neumann algebras,” J. Oper. Theory 9, 253 (1983).
  23. Putnam, C. R., “On Normal Operators in Hilbert Space,” Amer. J. Math. 73, 357 (1951).
  24. Schwinger, J., “Unitary operator bases,” Proc. Natl. Acad. Sci. U.S.A. 46, 570 (1960).
  25. Szöllősi, F., “Exotic complex Hadamard matrices and their equivalence,” e-print arXiv:1001.3062, Cryptography Commun. (to be published).
  26. Sylvester, J. J., “Thoughts on inverse orthogonal matrices, simultaneous sign-succession, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile work, and the theory of numbers,” Philos. Mag. 34, 461 (1867).
  27. Tadej, W. and Życzkowski, K., “A concise guide to complex Hadamard matrices,” Open Syst. Inf. Dyn. 13, 133 (2006).
  28. Wallis, W. D., Combinatorial Designs, Pure and Applied Mathematics Series (Dekker, New York, 1988).
  29. Wootters, W. K. and Fields, B. D., “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363 (1989).
  30. Zauner, G., “Quantendesigns-Grundzüge einer nichtkommutativen designtheorie,” Ph.D. thesis, Universität Wien, 1999; available online: http://www.mat.univie.ac.at/~neum/ms/zauner.pdf.

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.
ADVERTISEMENT