A conjecture about the orthogonal vector equations associated with the Hadamard matrix
DOI:
https://doi.org/10.54097/q1zz1g24Keywords:
complex number; orthogonal vector; plane and analytic geometry.Abstract
The investigation of Hadamard matrices provides an ancient problem in algebra which has extensive links to various real-world applications, such as signal transformation. In this work, we investigate a conjecture on the form of the solution corresponding to a special case of 6×6 complex Hadamard matrices. Establishing this conjecture will substantially influence the development of so-called non-H2-reducible Hadamard matrices. To build our case, we first establish two encompassing lemmas based on the geometric properties of unit modulus complex numbers with sum zero. Then, using these lemmas we establish the conjecture for various special cases. Finally, we take the general form of the problem and simply it down to a system of trigonometric equations. The numerical evidence presented give strong evidence of validity suggested a potential path for a complete proof.
Downloads
References
[1] Brierley S. Mutually unbiased bases in low dimensions. University of York, 2009.
[2] Brierley S, Weigert S. Maximal sets of mutually unbiased quantum states in dimension 6. Physical Review A, 2008, 78(4): 042312. DOI: https://doi.org/10.1103/PhysRevA.78.042312
[3] Brierley S, Weigert S. Constructing mutually unbiased bases in dimension six. Physical Review A, 2009, 79(5): 052316. DOI: https://doi.org/10.1103/PhysRevA.79.052316
[4] Brierley S, Weigert S. Mutually unbiased bases and semi-definite programming. Journal of Physics: Conference Series, 2010, 254: 012008. DOI: https://doi.org/10.1088/1742-6596/254/1/012008
[5] Maxwell A S, Brierley S. On properties of Karlsson Hadamards and sets of mutually unbiased bases in dimension six. Linear Algebra and its Applications, 2015, 466: 296-306. DOI: https://doi.org/10.1016/j.laa.2014.10.017
[6] Szöllusi F. Complex hadamard matrices of order 6: a four-parameter family. Journal of the London Mathematical Society, 2012, 85(3): 616-632. DOI: https://doi.org/10.1112/jlms/jdr052
[7] Turek O, Goyeneche D. A generalization of circulant Hadamard and conference matrices. Linear Algebra and its Applications, 2019, 569: 241-265. DOI: https://doi.org/10.1016/j.laa.2019.01.018
[8] Nicoara R, Worley C. A finiteness result for circulant core complex Hadamard matrices. Linear Algebra and its Applications, 2019, 571: 143-153. DOI: https://doi.org/10.1016/j.laa.2019.02.016
[9] Szöllusi F. Parametrizing complex hadamard matrices. European Journal of Combinatorics, 2008, 29(5): 1219-1234. DOI: https://doi.org/10.1016/j.ejc.2007.06.009
[10] Goyeneche D. Mutually unbiased triplets from non-affine families of complex hadamard matrices in dimension 6. Journal of Physics A: Mathematical and Theoretical, 2013, 46(10): 105301. DOI: https://doi.org/10.1088/1751-8113/46/10/105301
[11] Jaming P, Matolcsi M, Móra P, et al. A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6. Journal of Physics A: Mathematical and Theoretical, 2009, 42(24): 245305. DOI: https://doi.org/10.1088/1751-8113/42/24/245305
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Highlights in Science, Engineering and Technology
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
