QI In the literature: open

Maximum number of mutually unbiased bases in composite dimensions

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 $1 + \operatorname{min}_i p_i^{a_i}$ in some dimensions, e.g. five MUBs in $d = 12$ (Cardenes Wuttig and Tindall, arXiv:2609.40311, 2026).

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

A September 2026 preprint gives constructions beyond the tensor-product bound in many composite dimensions.