Maximum number of MUBs in C^6 (smallest open case)

Prove that there do not exist seven mutually unbiased bases in C^6, i.e., establish MUB(6) < 7.

Background

In dimension d, the number of mutually unbiased bases satisfies MUB(d) ≤ d + 1, with equality known for prime powers. For non–prime-power dimensions the exact values are unknown; the smallest unresolved case is d = 6.

Resolving MUB(6) would be a landmark in finite frame theory and quantum information, with implications for SIC-POVMs and combinatorial design structures.

References

For $d$ not a prime power, the problem of determining $\mathrm{MUB}(d)$ is still open. The smallest open instance is particularly well known for being a tantalizing open problem that has remained open (see Open Problem 6.2 in). Let $\mathrm{MUB}(d)$ denote the largest possible number of (simultaneously) mutually unbiased bases (MUBs) in $\mathbb{C}d$. We have $\mathrm{MUB}(6)<7$.

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “Mutually Unbiased Bases, ETFs, and Zauner’s Conjecture (ASB)” (Entry 11)