Quantum Information & Computation
Volume 26
RESEARCH ARTICLE Open access

Order-Six CHMs Containing Exactly Three Distinct Elements

Yanzu Huang, Mengfan Liang, Lin Chen

Author affiliations
  • Yanzu Huang1
  • Mengfan Liang1
  • Lin Chen1
  1. LMIB(Beihang University), Ministry of education, and School of Mathematical Sciences, Beihang University, Beijing, China

pp. 1-19 · 10.2478/qic-2026-0001

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Abstract

Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show that such CHMs can only be complex equivalent to two kinds of matrices, one is H2 -reducible and the other is the Tao matrix. Using our result one can further narrow the range of MUB trio (a set of four MUBs in ℂ6 consists of an MUB trio and the identity) since we find that neither of the two CHMs belong to any MUB trio. Our results may lead to the more complete classification of 6 × 6 CHMs whose elements in the first row are all 1.
Open accessCC BY-NC-ND 4.0

Keywords

  • Complex Hadamard matrices
  • MUB problem

Suggested citation

Yanzu Huang, Mengfan Liang, Lin Chen (2026). Order-Six CHMs Containing Exactly Three Distinct Elements. Quantum Information & Computation, 26(1), 1-19. https://doi.org/10.2478/qic-2026-0001

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References (32)

  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. Philippe Jaming, Máté Matolcsi, Péter Móra, Ferenc Szöllösi and Mihály Weiner (2009). A generalized pauli problem and an infinite family of mub-triplets in dimension 6. Journal of Physics A: Mathematical and Theoretical, 42(24):245305.DOI: 10.1088/1751-8113/42/24/245305Google Scholar
  4. 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
  5. 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
  6. Sébastien Designolle, Paul Skrzypczyk, Florian Fröwis and Nicolas Brunner (2018). Quantifying measurement in compatibility of mutually unbiased bases.DOI: 10.1103/PhysRevLett.122.050402Google Scholar
  7. 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
  8. 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
  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. 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
  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. 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
  15. 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
  16. A. T. Butson (1962). Generalized Hadamard matrices. Proceedings of the American Mathematical Society, 13(6):894–898.DOI: 10.1090/S0002-9939-1962-0142557-0Google Scholar
  17. 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
  18. Bjö rck Gran and Frö berg Ralf (1991). A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots. Journal of Symbolic Computation, 12(3):329–336.DOI: 10.1016/S0747-7171(08)80153-8Google Scholar
  19. R. Bengt Karlsson (2011). Three-parameter complex Hadamard matrices of order 6. Linear Algebra and its Applica tions, 434(1):247–258.DOI: 10.1016/j.laa.2010.08.020Google Scholar
  20. U Haagerup (1997). Orthogonal maximal abelian *-subalgebras of the n × n matrices and cyclic n-roots. International Press.Google Scholar
  21. 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
  22. 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
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. A. Bondal and I. Zhdanovskiy (2016). Orthogonal pairs and mutually unbiased bases. Journal of Mathematical Sciences, 216:23–40.DOI: 10.1007/s10958-016-2885-zGoogle Scholar
  29. Mengfan Liang, Lin Chen, Fengyue Long and Xinyu Qiu (2024). The non-h< sub > 2 < / sub >-reducible matrices and some special complex hadamard matrices. QUANTUM INFORMATION PROCESSING, 23(7).DOI: 10.1007/s11128-024-04482-yGoogle Scholar
  30. Stephen Brierley and Stefan Weigert (2009). Constructing mutually unbiased bases in dimension six. Phys. Rev. A, 79:052316.DOI: 10.1103/PhysRevA.79.052316Google Scholar
  31. Mengfan Liang, Mengyao Hu, Yize Sun, Lin Chen and Xiaoyu Chen (2019). Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six. Linear and Multilinear Algebra, 0(0):1–18.Google Scholar
  32. Mengfan Liang, Mengyao Hu, Lin Chen and Xiaoyu Chen (2019). The H2-reducible matrix in four sixdimensional mutually unbiased bases. Quantum Information Processing, 18(11):352.DOI: 10.1007/s11128-019-2467-3Google Scholar