Maximum number of mutually unbiased bases in composite dimensions
This text mentions work from 2024 on. Those parts were partly written without a live literature search and have not been checked by an editor yet.
In plain words
In dimensions that are prime powers, the largest possible number of mutually unbiased bases is always reached. For every other dimension the maximum is unknown.
Precise statement
Let $N(d)$ be the maximal number of MUBs in $C^d$, with $N(d) = d + 1$ for prime powers. Determine whether $N(d) < d + 1$ for every d that is not a prime power, and find $N(d)$ for $d = 10\text{ and }12$. Lower bounds exceed the tensor-product bound $1 + \min_i p_i^{a_i}$ in some dimensions, e.g. five MUBs in $d = 12$ (Cardenes Wuttig and Tindall, arXiv:2609.40311, 2026).
What would settle it
Proofs of upper bounds below d + 1 together with matching constructions in specific composite dimensions.
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.
A September 2026 preprint gives constructions beyond the tensor-product bound in many composite dimensions.
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