The maximum number of mutually unbiased bases in dimension six
Two orthonormal bases $\{u_1, \dots, u_6\}$ and $\{w_1, \dots, w_6\}$ of $\mathbb{C}^6$ are mutually unbiased if $$|\langle u_i, w_j \rangle|^2 = \frac{1}{6} \qquad (1 \le i, j \le 6).$$ Let $M(6)$ be the largest integer $r$ for which there are $r$ pairwise mutually unbiased orthonormal bases of $\mathbb{C}^6$. In $\mathbb{C}^d$ there are at most $d + 1$ pairwise mutually unbiased bases, and $d + 1$ of them exist when $d$ is a prime power (Ivanović; Wootters and Fields). Tensor products give at least three in every dimension $d \ge 2$ (Klappenecker and Rötteler), so $3 \le M(6) \le 7$; six is the smallest dimension that is not a prime power.
Problem. Determine $M(6)$. In particular, do four pairwise mutually unbiased orthonormal bases of $\mathbb{C}^6$ exist?
Source: Zauner (1999), as Conjecture 1 of Klappenecker and Rötteler (2004); Horodecki, Rudnicki and Życzkowski (2022), Problem 2, ask for four bases or for a proof that there are not seven. Stated by AIM — Open Applied Problems as the determination of M(6).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, complex Hadamard matrices, mutually unbiased bases
References
@phdthesis{zauner1999,
author = {Zauner, Gerhard},
title = {Quantendesigns: {G}rundz{\"u}ge einer nichtkommutativen {D}esigntheorie},
school = {Universit{\"a}t Wien},
year = {1999},
}
@incollection{klappenecker2004,
author = {Klappenecker, Andreas and R{\"o}tteler, Martin},
title = {Constructions of mutually unbiased bases},
booktitle = {Finite Fields and Applications},
series = {Lecture Notes in Comput. Sci.},
volume = {2948},
publisher = {Springer},
address = {Berlin},
year = {2004},
pages = {137--144},
doi = {10.1007/978-3-540-24633-6_10},
eprint = {quant-ph/0309120},
archivePrefix = {arXiv},
}
@article{horodecki2022,
author = {Horodecki, Pawe{\l} and Rudnicki, {\L}ukasz and {\.Z}yczkowski, Karol},
title = {Five open problems in quantum information theory},
journal = {PRX Quantum},
volume = {3},
number = {1},
year = {2022},
pages = {010101},
doi = {10.1103/PRXQuantum.3.010101},
eprint = {2002.03233},
archivePrefix = {arXiv},
}
@article{ivanovic1981,
author = {Ivanovi{\'c}, I. D.},
title = {Geometrical description of quantal state determination},
journal = {J. Phys. A},
volume = {14},
number = {12},
year = {1981},
pages = {3241--3245},
doi = {10.1088/0305-4470/14/12/019},
}
@article{wootters1989,
author = {Wootters, William K. and Fields, Brian D.},
title = {Optimal state-determination by mutually unbiased measurements},
journal = {Ann. Physics},
volume = {191},
number = {2},
year = {1989},
pages = {363--381},
doi = {10.1016/0003-4916(89)90322-9},
}
@misc{aim-mub-dimension-six,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The maximum number of mutually unbiased bases in dimension six}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/031-mub-dimension-six.md},
note = {Entry 031 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that the maximum number of mutually unbiased orthonormal bases in $\C^6$ is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: math @ adc7f12, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00699, "The maximum number of mutually unbiased bases in dimension six", clopen.solutions.