No seven mutually unbiased bases in ℂ⁶
Two orthonormal bases $\{v_i\}$ and $\{w_j\}$ of $\mathbb{C}^d$ are mutually unbiased if $|\langle v_i, w_j \rangle| = 1/\sqrt d$ for all $i, j$. Let $\mathrm{MUB}(d)$ be the largest number of pairwise mutually unbiased bases of $\mathbb{C}^d$. Then $\mathrm{MUB}(d) \le d + 1$, with equality when $d$ is a prime power (see Bandyopadhyay, Boykin, Roychowdhury and Vatan), and $\mathrm{MUB}(6) \ge 3$; dimension $6$ is the smallest open case.
Problem. Prove or disprove that $\mathrm{MUB}(6) < 7$, that is, that $\mathbb{C}^6$ has no seven pairwise mutually unbiased bases.
Source: A well-known open problem; Open Problem 6.2 of Bandeira's Ten Lectures and Forty-Two Open Problems (2016). Conjecture 22 of Randomstrasse101, in Bandeira's “Mutually Unbiased Bases, ETFs, and Zauner's Conjecture” (2025); Entry 11 of arXiv:2603.29571.
The statement is adapted from randomstrasse101.math.ethz.ch, whose text is available under CC BY 4.0.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MP
- Keywords
- frames, mutually unbiased bases, quantum information, Randomstrasse101
- Literature checked
- Oct. 4, 2026 by clemens-admin
References
@misc{randomstrasse2025,
author = {Bandeira, Afonso S. and Dmitriev, Daniil and Lucca, Kevin and Nizi{\'c}-Nikolac, Petar and R{\"o}dder, Almut},
title = {{Randomstrasse101}: Open Problems of 2025},
year = {2026},
eprint = {2603.29571},
archivePrefix = {arXiv}
}
@article{bandyopadhyay2002mubs,
author = {Bandyopadhyay, Somshubhro and Boykin, P. Oscar and Roychowdhury, Vwani P. and Vatan, Farrokh},
title = {A new proof for the existence of mutually unbiased bases},
journal = {Algorithmica},
volume = {34},
number = {4},
pages = {512--528},
year = {2002},
doi = {10.1007/s00453-002-0980-7}
}
@misc{mcnulty2024mubs,
author = {McNulty, Daniel and Weigert, Stefan},
title = {Mutually unbiased bases in composite dimensions -- a review},
year = {2024},
eprint = {2410.23997},
archivePrefix = {arXiv}
}
@article{bandeira2022dualmub,
author = {Bandeira, Afonso S. and Doppelbauer, Nikolaus and Kunisky, Dmitriy},
title = {Dual bounds for the positive definite functions approach to mutually unbiased bases},
journal = {Sampling Theory, Signal Processing, and Data Analysis},
volume = {20},
number = {2},
pages = {18},
year = {2022}
}
@misc{bandeira2016fortytwo,
author = {Bandeira, Afonso S.},
title = {Ten Lectures and Forty-Two Open Problems in the Mathematics of Data Science},
year = {2016},
url = {https://people.math.ethz.ch/~abandeira/TenLecturesFortyTwoProblems.pdf}
}
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 |
|---|---|---|---|
| 2026-10-07 | Open → Solution claimed | The registry's curators | The maximum number of mutually unbiased bases in dimension six v1 recorded from 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-00075, "No seven mutually unbiased bases in ℂ⁶", clopen.solutions.