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Title: Maximum number of mutually unbiased bases in composite dimensions
Plain: 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: 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 $1 + \operatorname{min}_i p_i^{a_i}$ in some dimensions, e.g. five MUBs in $d = 12$ (Cardenes Wuttig and Tindall, arXiv:2609.40311, 2026).
Settled by: Proofs of upper bounds below d + 1 together with matching constructions in specific composite dimensions.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: A September 2026 preprint gives constructions beyond the tensor-product bound in many composite dimensions.
Posed since: (empty)