OP-3061Partially solvedQuantum information

Mutually unbiased bases in dimension six

Determine the maximal number $M(6)$ of mutually unbiased bases in $\mathbb{C}^6$ ; in particular decide whether $M(6)=3$ or whether four or more exist.

Source-reported statusAs of 2026-09-04Source & evidence ↓

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Problem statement

Determine the maximal number $M(6)$ of mutually unbiased bases in $\mathbb{C}^6$ ; in particular decide whether $M(6)=3$ or whether four or more exist.

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Background

Two orthonormal bases are mutually unbiased if $|\langle e_i|f_j\rangle|^2=1/d$ for every pair of vectors. Complete sets of $d+1$ bases exist in prime-power dimensions. Dimension six is the smallest remaining case and is retained as the flagship benchmark of N3.

Tensor-product constructions give $M(6)\geq3$ , while the general dimension bound gives $M(6)\leq7$ . Extensive numerical, computer-algebra and Gröbner-basis investigations have not produced a fourth basis. Restrictions on particular complex Hadamard families provide rigorous partial exclusions; numerical search failure is not a proof of global nonexistence.

Preprint progress (August 2026): Cárdenes Wuttig and Tindall claim a complete exact finite-incidence classification of order-six complex Hadamard matrices, including Szöllősi's conjecture. This concerns individual matrices up to standard equivalence. A MUB set additionally requires simultaneous mutual unbiasedness between several matrices, so this claimed auxiliary classification does not determine $M(6)$ .

The sharp target remains whether four or more MUBs exist, and ultimately the exact value of $M(6)$ .

Source disclosures

Cárdenes Wuttig and Tindall disclose interactive assistance from ChatGPT Sol 5.6 Pro, Codex 5.6 Sol and Claude Opus 5.0 in proofs, searches, algebra and verification code. They state that they repaired or rejected incorrect proposals and independently checked retained material. Source

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Claimed classification of order-six complex Hadamard matrices; simultaneous mutual-unbiasedness constraints remain additional and M(6) is not determined. Evidence

References

[1]Durt, Englert, Bengtsson, Życzkowski, 'On mutually unbiased bases', Int. J. Quantum Inf. 8 (2010) link
[2]Brierley & Weigert, 'Maximal sets of mutually unbiased quantum states in dimension six', PRA 78 (2008) link
[3]Cárdenes Wuttig & Tindall, 'A Complete Classification of Complex Hadamard Matrices of Order Six' (2026 preprint) link

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