Mutually unbiased bases in dimension six

Either construct at least four mutually unbiased bases (MUBs) in the six-dimensional complex Hilbert space C^6, or prove that a complete set of seven mutually unbiased bases cannot exist in C^6.

Background

Mutually unbiased bases are sets of orthonormal bases in which the squared overlaps between vectors from different bases equal 1/N. In prime-power dimensions N = pk, complete sets of N+1 MUBs exist, but in composite dimensions, including N = 6, the situation is unresolved.

Despite extensive work—including classifications in low dimensions and connections to complex Hadamard matrices—only three MUBs are known in dimension six. Determining whether more exist (at least four) or whether seven are impossible would clarify fundamental structure in finite-dimensional quantum theory.

References

but otherwise the number of existing MUBs remains unknown.

— Five open problems in quantum information  (2002.03233 - Horodecki et al., 2020) in Section: Discrete structures in the Hilbert space; Subsection: MUBs in dimension six

Determining how many mutually unbiased bases exist in $6$ is one of the central open problems of quantum information theory.

— A Complete Classification of Complex Hadamard Matrices of Order Six  (2608.18053 - Wuttig et al., 18 Aug 2026) in Introduction, Section 1