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Title: Do four mutually unbiased bases exist in dimension six
Plain: 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: 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.
Settled by: A proof of non-existence (for example via a complete classification of order-6 complex Hadamard matrices), or explicit four bases.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status 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.
Posed since: (empty)