A degree-4 Sum-of-Squares proof that ℂ⁶ has no seven mutually unbiased bases
Seven pairwise mutually unbiased bases of $\mathbb{C}^6$ are vectors $v^{(k)}_i \in \mathbb{C}^6$, $i = 1, \dots, 6$, $k = 1, \dots, 7$, with $|\langle v^{(k)}_i, v^{(\ell)}_j \rangle|^2$ equal to $1$ if $k = \ell$ and $i = j$, to $0$ if $k = \ell$ and $i \ne j$, and to $\frac16$ if $k \ne \ell$. These can, for example, be encoded as quartic equations in the real and imaginary parts of the coordinates. Alternatively, since $1/\sqrt6$ is the smallest possible value of the largest overlap between two bases, the $k \ne \ell$ conditions can be rewritten as $|\langle v^{(k)}_i, v^{(\ell)}_j \rangle| \le 1/\sqrt6$, which are semidefinite constraints on $2 \times 2$ Hermitian matrices involving quadratic quantities. Bandeira, Doppelbauer and Kunisky showed that a relaxation related to degree-2 Sum-of-Squares cannot prove that no seven exist.
Problem. Is there a degree-4 Sum-of-Squares proof that $\mathbb{C}^6$ has no seven mutually unbiased bases?
Source: Bandeira, Open Problem 23 of Randomstrasse101, in “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
- OC — Optimization and Control
- Secondary
- AG, CO
- Keywords
- mutually unbiased bases, Randomstrasse101, semidefinite programming, Sum-of-Squares
- Literature checked
- Oct. 2, 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{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{mcnulty2024mubs,
author = {McNulty, Daniel and Weigert, Stefan},
title = {Mutually unbiased bases in composite dimensions -- a review},
year = {2024},
eprint = {2410.23997},
archivePrefix = {arXiv}
}
Resolution claims
We prove a degree-four Sum-of-Squares lower bound for the standard vector-coordinate formulations of mutually unbiased bases. For every dimension $d$ and every proposed number $m$ of bases, we construct a degree-four pseudoexpectation satisfying the orthonormality constraints and the cross-unbiasedness constraints in the quartic equality formulation. The construction is expectation over $m$ independent Haar-random orthonormal bases. We also prove that the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-09 | Open → Partially resolved | The registry's curators | Degree-Four Vector-Coordinate SoS Cannot Detect the MUB Upper Bound v1 recorded from arXiv:2606.13903v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00038, "A degree-4 Sum-of-Squares proof that ℂ⁶ has no seven mutually unbiased bases", clopen.solutions.