Do four mutually unbiased bases exist in dimension six
In plain words
In dimension 6, the smallest that is not a power of a prime, three mutually unbiased bases are known and a perfect set would have seven. Whether even a fourth exists is unknown.
Precise statement
Do there exist four orthonormal bases of $C^6$ such that $\left|\langle a \mid b\rangle\right|^2 = 1/6$ for any vectors a, b taken from different bases? Three are known; numerics and partial classifications of $6 x 6$ complex Hadamard matrices suggest that four do not exist.
What would settle it
A proof of non-existence (for example via a complete classification of order-6 complex Hadamard matrices), or explicit four bases.
Status in the literature
Unverified note
Still described as open in September 2026; a claimed complete classification of order-6 complex Hadamard matrices (arXiv:2608.18053) has not closed it.