Complete MUB sets outside prime-power dimensions

Complete sets of $d+1$ mutually unbiased bases are known in prime-power dimensions. Whether any non-prime-power dimension supports a complete set is the sharp existence question here; finding every value $M(d)$ is a broader programme. N2 records the dimension-six benchmark.

Writing $d=\prod_i p_i^{k_i}$ gives the tensor-product lower bound $M(d)\geq1+\min_i p_i^{k_i}$, while $M(d)\leq d+1$. Links with finite geometry motivate conjectures, but complete complex MUBs are not known to be equivalent to finite affine/projective planes. Classical complete mutually orthogonal Latin squares have their own established relation to planes. In particular, projective planes of order 10 were proved not to exist by Lam–Thiel–Swiercz in 1989; that result does not by itself exclude complete MUBs in dimension 10.

The 2026-published composite-dimension review provides a current baseline. Numerical evidence (August 2026): Wu–Liu–Wu report standard tensor-product constructions in 64 non-prime-power dimensions up to 100 and extensive extension searches in such dimensions up to 30. No extension was found. This tests chosen search procedures and starting families, not all possible MUBs, so it does not resolve any universal nonexistence claim.

Source review dated 4 September 2026. “Improved” records partial progress and may include preprint claims; it does not mean the problem has been solved.

numerical: Extension searches for tensor-product MUB constructions found no extension; this is not a nonexistence proof. Evidence