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 $6 \times 6$ 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: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Still described as open in September 2026; a claimed complete classification of order-6 complex Hadamard matrices (arXiv:2608.18053) has not closed it.
Parent problems
- Special case of Maximum number of mutually unbiased bases in composite dimensions · open · from revision 1 The parent was revised since revision 1: it now states another question.
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