Quantum Information & Computation
Volume 26
RESEARCH ARTICLE Open access

Eigenvalues of Two-Parameter Complex Hadamard Matrices of Order Six

Yanzu Huang, Lin Chen

Author affiliations
  • Yanzu Huang1
  • Lin Chen1
  1. Ministry of Education, and School of Mathematical Sciences, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing, China

pp. 192-209 · 10.2478/qic-2026-0010

Read PDF View on Reference Global
On this page

Abstract

Characterizing complex Hadamard matrices (CHMs) of order six is an open problem in both algebra and quantum information theory. Analyzing the eigenvalues of CHMs provides a novel approach to this problem. We construct a class of two-parameter CHMs of order six with two sorts of eigenvalues. All matrices in this class are H 2-reducible, and the Diţă matrix, Haagerup matrices and Hermitian matrices are both included. This class turns out to be complex equivalent to the class of all CHMs in the dephased form whose eigenvalues are of the form 6\sqrt 6, – 6\sqrt 6, λ1, λ1, λ2, λ2 with λj of modulus 6\sqrt 6 excluding Tao matrix S6(0)S_6^{(0)}. We further show that some CHMs including all Hermitian CHMs in the former class have Schmidt rank at most three up to complex equivalence, and they are actually controlled unitary gates implementable in experiments. Our result demonstrates the feasibility of using eigenvalues for characterizing order-six CHMs and their application in quantum circuits.
Open accessCC BY-NC-ND 4.0

Keywords

  • complex hadamard matrices
  • eigenvalues and eigenvectors
  • controlled unitary gates

Suggested citation

Yanzu Huang, Lin Chen (2026). Eigenvalues of Two-Parameter Complex Hadamard Matrices of Order Six. Quantum Information & Computation, 26(1), 192-209. https://doi.org/10.2478/qic-2026-0010

Download BibTeX

References (34)

  1. William K Wootters and Brian D Fields (1989). Optimal state-determination by mutually unbiased measurements. Annals of Physics, 191(2): 363–381.DOI: 10.1016/0003-4916(89)90322-9Google Scholar
  2. Philippe Raynal, Xin Lü and Berthold-Georg Englert (2011). Mutually unbiased bases in six dimensions: The four most distant bases. Phys. Rev. A, 83: 062303.DOI: 10.1103/PhysRevA.83.062303Google Scholar
  3. Daniel McNulty and Stefan Weigert (2012). The limited role of mutually unbiased product bases in dimension 6. Journal of Physics A: Mathematical and Theoretical, 45(10): 102001.DOI: 10.1088/1751-8113/45/10/102001Google Scholar
  4. Lin Chen and Li Yu (2018). Mutually unbiased bases in dimension six containing a product-vector basis. Quantum Information Processing, 17(8): 198.DOI: 10.1007/s11128-018-1964-0Google Scholar
  5. Daniel McNulty and Stefan Weigert (2012). All mutually unbiased product bases in dimension 6. Journal of Physics A: Mathematical and Theoretical, 45(13): 135307.DOI: 10.1088/1751-8113/45/13/135307Google Scholar
  6. Daniel McNulty, Bogdan Pammer and Stefan Weigert (2016). Mutually unbiased product bases for multiple qudits. Journal of Mathematical Physics, 57(3).DOI: 10.1063/1.4943301Google Scholar
  7. DANIEL MCNULTY and STEFAN WEIGERT (2012). On the impossibility to extend triples of mutually unbiased product bases in dimension six. International Journal of Quantum Information, 10(05): 1250056.DOI: 10.1142/S0219749912500566Google Scholar
  8. Lin Chen and Li Yu (2017). Product states and Schmidt rank of mutually unbiased bases in dimension six. Journal of Physics A Mathematical General, 50(47): 475304.DOI: 10.1088/1751-8121/aa8f9eGoogle Scholar
  9. P. O. Boykin, M. Sitharam, P. H. Tiep and P. Wocjan (2005). Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras.Google Scholar
  10. Stephen Brierley and Stefan Weigert (2008). Maximal sets of mutually unbiased quantum states in dimension 6. Phys. Rev. A, 78: 042312.DOI: 10.1103/PhysRevA.78.042312Google Scholar
  11. Stephen Brierley and Stefan Weigert (2010). Mutually Unbiased Bases and Semi-definite Programming. Journal of Physics: Conference Series, 254: 012008.DOI: 10.1088/1742-6596/254/1/012008Google Scholar
  12. Sébastien Designolle, Paul Skrzypczyk, Florian Fröwis and Nicolas Brunner (2018). Quantifying measurement incompatibility of mutually unbiased bases.DOI: 10.1103/PhysRevLett.122.050402Google Scholar
  13. M. Wiesniak, T. Paterek and A. Zeilinger (2011). Entanglement in mutually unbiased bases. New Journal of Physics, 13(5): 053047.DOI: 10.1088/1367-2630/13/5/053047Google Scholar
  14. T. Durt, B.-G. Englert, I. Bengtsson and K. Zyczkowski (2010). On mutually unbiased bases. Int. J. Quantum Information, 8(4): 535–640.DOI: 10.1142/S0219749910006502Google Scholar
  15. Bengt R. Karlsson (2011). Three-parameter complex Hadamard matrices of order 6. Linear Algebra and its Applications,434(1): 247–258.DOI: 10.1016/j.laa.2010.08.020Google Scholar
  16. U. Haagerup (1997). Orthogonal maximal abelian *-subalgebras of the n × n matrices and cyclic n-roots. International Press.Google Scholar
  17. Terence Tao (2004). Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett., 11: 251.DOI: 10.4310/MRL.2004.v11.n2.a8Google Scholar
  18. Ferenc Szöllösi (2012). Complex Hadamard matrices of order 6: a four-parameter family. Journal of the London Mathematical Society, 85(3): 616–632.DOI: 10.1112/jlms/jdr052Google Scholar
  19. Mengfan Liang and Lin Chen (2024). Eigenvalues and eigenvectors of complex hadamard matrices.Google Scholar
  20. Beauchamp Kyle and Nicoara Remus (2006). Orthogonal maximal abelian *-subalgebras of the 6 × 6 matrices. Linear Algebra and Its Applications, 428(8): 1833–1853.DOI: 10.1016/j.laa.2007.10.023Google Scholar
  21. Tadej, Wojciech and K.Zyczkowski (2006). A concise guide to complex Hadamard matrices. Open Systems and Information Dynamics, 13(2): 133–177.DOI: 10.1007/s11080-006-8220-2Google Scholar
  22. T. Banica, J. Bichon and J.M. Schlenker (2009). Representations of quantum permutation algebras. Journal of Functional Analysis, 257(9): 2864–2910.DOI: 10.1016/j.jfa.2009.04.013Google Scholar
  23. W.D. Launey and D.M. Gordon (2001). A comment on the Hadamard conjecture. Journal of Combinatorial Theory, Series A, 95(1): 180–184.DOI: 10.1006/jcta.2000.3155Google Scholar
  24. P. Dita (2002). New results on the parametrisation of complex Hadamard matrices. Journal of Physics A General Physics, 37(20): 5355.DOI: 10.1088/0305-4470/37/20/008Google Scholar
  25. G. Hiranandani and J.M. Schlenker (2014). Small circulant complex Hadamard matrices of Butson type. European Journal of Combinatorics, 51: 306–314.DOI: 10.1016/j.ejc.2015.05.010Google Scholar
  26. Harold Ollivier and Wojciech H. Zurek (2001). Quantum discord: A measure of the quantumness of correlations. Phys. Rev. Lett., 88: 017901, Dec.DOI: 10.1103/PhysRevLett.88.017901Google Scholar
  27. J. I. Cirac and P. Zoller (1995). Quantum computations with cold trapped ions. Phys. Rev. Lett., 74: 4091–4094, May.DOI: 10.1103/PhysRevLett.74.4091Google Scholar
  28. Lin Chen and Li Yu (2014). Nonlocal and controlled unitary operators of Schmidt rank three. Phys. Rev. A, 89: 062326.DOI: 10.1103/PhysRevA.89.062326Google Scholar
  29. Lin Chen and Li Yu (2014). On the Schmidt-rank-three bipartite and multipartite unitary operator. Annals of Physics, 351: 682–703.DOI: 10.1016/j.aop.2014.09.026Google Scholar
  30. P. Dita (2004). Some results on the parametrization of complex hadamard matrices. Journal of Physics A: Mathematical and General, 37(20): 5355.DOI: 10.1088/0305-4470/37/20/008Google Scholar
  31. Terence Tao (2004). Fuglede’s conjecture is false in 5 and higher dimensions. Mathematical Research Letters, 11(2): 251–258.DOI: 10.4310/MRL.2004.v11.n2.a8Google Scholar
  32. A. T. Butson (1962). Generalized hadamard matrices. Proceedings of the American Mathematical Society, 13: 894–898.DOI: 10.1090/S0002-9939-1962-0142557-0Google Scholar
  33. Kyle Beauchamp and Remus Nicoara (2008). Orthogonal maximal abelian *-subalgebras of the 6 × 6 matrices. Linear Algebra and its Applications, 428(8): 1833–1853.DOI: 10.1016/j.laa.2007.10.023Google Scholar
  34. Lin Chen and Li Yu (2015). Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates. Phys. Rev. A, 91: 032308.DOI: 10.1103/PhysRevA.91.032308Google Scholar