A Complete Classification of Complex Hadamard Matrices of Order Six
Abstract
Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased 3 × 3 corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.
Related
BibTeX
@misc{wuttig2026complete,
title = {{A Complete Classification of Complex Hadamard Matrices of Order Six}},
author = {Mateo Cárdenes Wuttig and Joseph Tindall},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.18053},
eprint = {2608.18053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Complete Classification of Complex Hadamard Matrices of Order Six
AU - Mateo Cárdenes Wuttig
AU - Joseph Tindall
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.18053
UR - https://arxiv.org/abs/2608.18053
AB - Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased 3 × 3 corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.
ER -