HEXAGONbeta

Regular complex Hadamard matrices of order 7,8

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsquant-ph — Quantum Physics

Contributed by Teo Banica ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

A complex Hadamard matrix H∈MN(C)H\in M_N(\mathbb C) is called regular when the scalar products between its rows ∑kHikHˉjk=0\sum_kH_{ik}\bar{H}_{jk}=0 decompose as trivially vanishing sums of roots of unity, rotated by scalars. We review the general theory of such matrices, including the known classification at N=6N=6, and then we work out classification results at N=7N=7 and N=8N=8.

Provenance statement

Written by Chat GPT, with help from Teo Banica.

Tools used

OpenAI
ChatGPT

References

  1. J. Avan, T. Fonseca, L. Frappat, P. Kulish, E. Ragoucy and G. Rollet, Temperley-Lieb R-matrices from generalized Hadamard matrices, Theor. Math. Phys. 178 (2014), 223--240.
  2. T. Banica, The defect of generalized Fourier matrices, Linear Algebra Appl. 438 (2013), 3667--3688.
  3. T. Banica, Invitation to Hadamard matrices, allcolors=black arXiv:1910.06911 (https://arxiv.org/abs/1910.06911) (2024).arXiv
  4. T. Banica, J. Bichon and J.-M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864--2910.
  5. T. Banica, I. Nechita and K. Życzkowski, Almost Hadamard matrices: general theory and examples, Open Syst. Inf. Dyn. 19 (2012), 1--26.
  6. T. Banica, D. Özteke and L. Pittau, Isolated partial Hadamard matrices and related topics, Open Syst. Inf. Dyn. 25 (2018), 1--27.
  7. K. Beauchamp and R. Nicoara, Orthogonal maximal abelian ∗*-subalgebras of the 6×66\times 6 matrices, Linear Algebra Appl. 428 (2008), 1833--1853.
  8. G. Björck, Functions of modulus 11 on Zn{\rm Z}_n whose Fourier transforms have constant modulus, and cyclic nn-roots, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 315 (1990), 131--140.
  9. G. Björck and R. Fröberg, A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic nn-roots, J. Symbolic Comput. 12 (1991), 329--336.
  10. A.T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894--898.
Show all 34 referencesShow fewer references
  1. W. de Launey and D.A. Levin, A Fourier-analytic approach to counting partial Hadamard matrices, Cryptogr. Commun. 2 (2010), 307--334.
  2. P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (2004), 5355--5374.
  3. T. Durt, B.-G. Englert, I. Bengtsson and K. Życzkowski, On mutually unbiased bases, Int. J. Quantum Inf. 8 (2010), 535--640.
  4. U. Haagerup, Orthogonal maximal abelian ∗*-subalgebras of the n×nn\times n matrices and cyclic nn-roots, in ``Operator algebras and quantum field theory'', International Press (1997), 296--323.
  5. U. Haagerup, Cyclic pp-roots of prime lengths pp and related complex Hadamard matrices, preprint 2008.
  6. M. Hall, Integral matrices AA for which AAt=mIAA^t=mI, in ``Number Theory and Algebra'', Academic Press (1977), 119--134.
  7. G. Hiranandani and J.-M. Schlenker, Small circulant complex Hadamard matrices of Butson type, European J. Combin. 51 (2016), 306--314.
  8. V.F.R. Jones, Planar algebras I, preprint 1999.
  9. R. Kannan and A. Bachem, Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. Comput. 8 (1979), 499--507.
  10. T.Y. Lam and K.H. Leung, On vanishing sums of roots of unity, J. Algebra 224 (2000), 91--109.
  11. P. H. J. Lampio, P. R. J. Östergå rd, and F. Szőllősi, Orderly generation of Butson Hadamard matrices , Mathematics of Computation 89 (2020), 313--331.
  12. D. McNulty and S. Weigert, Isolated Hadamard matrices from mutually unbiased product bases, J. Math. Phys. 53 (2012), 1--21.
  13. F.J. Murray and J. von Neumann, On rings of operators, Ann. of Math. 37 (1936), 116--229.
  14. R. Nicoara and J. White, Analytic deformations of group commuting squares and complex Hadamard matrices, J. Funct. Anal. 272 (2017), 3486--3505.
  15. R. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933), 311--320.
  16. M. Petrescu, Existence of continuous families of complex Hadamard matrices of certain prime dimensions and related results, Ph.D. Thesis, UCLA (1997).
  17. S. Popa, Orthogonal pairs of ∗*-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), 253--268.
  18. F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264--286.
  19. F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun. 2 (2010), 187--198.
  20. W. Tadej, Defect of a Kronecker product of unitary matrices, Linear Algebra Appl. 436 (2012), 1924--1959.
  21. W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133--177.
  22. W. Tadej and K. Życzkowski, Defect of a unitary matrix, Linear Algebra Appl. 429 (2008), 447--481.
  23. T. Tao, Fuglede's conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), 251--258.
  24. S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195--211.

Version history

  1. v1 · 64 pagesSubmitted by Teo BanicaInitial depositCurrentNov 04, 2026