Existence of complete MUB sets in non-prime-power dimensions (notably d=6)

Determine whether a complete set of d+1 mutually unbiased bases exists in the complex Hilbert space C^d for dimensions d that are not prime powers; in particular, ascertain whether seven mutually unbiased bases exist in dimension d=6.

Background

Mutually unbiased bases (MUBs) are well understood in prime-power dimensions, where finite-field constructions guarantee the existence of a complete set of d+1 MUBs. In contrast, dimensions that are not prime powers lack these algebraic tools, and the general existence problem remains unresolved.

The paper emphasizes the special status of dimension d=6, the smallest non-prime-power case, as the best-studied example where only three MUBs are known explicitly. The authors contrast the flexibility in prime-power dimensions (e.g., d=4) with the structural rigidity observed in d=6, underscoring why the existence of a complete set is still undetermined.

References

However, in dimensions that are not prime powers, the existence of maximal sets of MUBs remains an open problem, with the six-dimensional case d=6 being the most famous and extensively studied example [bengtsson,brierley].

— Explicit constructions of mutually unbiased bases via Hadamard matrices  (2604.02234 - Pain, 2 Apr 2026) in Introduction (Section 1)

More generally, it is known that d + 1 MUBs exist whenever d is a prime power , while the case of general d remains an open problem.

— Computing key rates for one-sided device-independent quantum key distribution  (2608.25915 - Bluhm et al., 26 Aug 2026) in Appendix, Section "Additional protocols", subsection "MUB protocol"