Mutually unbiased bases in dimension 6
Decide whether four (or seven) mutually unbiased bases exist in C^6; only three are known, and a complete set of seven is widely believed not to exist.
Cite
@misc{cairn-mutually-unbiased-bases-dimension-6,
title = {Mutually unbiased bases in dimension 6},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/mutually-unbiased-bases-dimension-6}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
} Also: CITATION.cff · Atom feed of results
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Current state
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The problem
The question. Two orthonormal bases of C^d are mutually unbiased if |⟨e_i|f_j⟩|^2 = 1/d for all i, j. At most d + 1 such bases exist, and d + 1 are known when d is a prime power. In d = 6, the smallest other case, does a set of four mutually unbiased bases exist? (A complete set would have seven.)
Known status. McNulty and Weigert's review (Quantum, 2026) states that no more than three MUBs have been found in d = 6 and that the existence of seven remains unproven. Brierley and Weigert (2008) found numerically only 18 of 35 possible "MU constellations", which they call the strongest numerical evidence that no seven MUBs exist. Jaming, Matolcsi and Móra (2010) proposed a discretisation approach for a computer-assisted proof. Searches using complex Hadamard matrices and numerical optimisation have failed to find a fourth basis.
What counts as progress
- Reproducible numerical searches (with code, random seeds, optimiser settings and the best residual reached) for four MUBs or for MU constellations, including documented negative results.
- Rigorous non-existence results for restricted families (e.g. sets containing a given Hadamard family or a product basis), ideally as computer-assisted proofs with verifiable certificates (interval arithmetic, exact algebra) or Lean formalisations.
- Syntheses mapping known partial results and the barriers of each method.
How it is checked. Numerical searches are re-run and residuals recomputed. A claimed set of four MUBs would be checked directly: the submitted 6×6 unitaries must satisfy all overlap conditions to within a stated tolerance, followed by an exact or interval-arithmetic verification. Computer-assisted non-existence proofs are checked by re-running the certificate verification.