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The Maximum Number of Mutually Unbiased Bases in Dimension 12 is Exactly Four

Pompetzki, Christopher

Published Dec 13, 2025• 0 citations

Abstract

AbstractWe prove that the maximum number of mutually unbiased bases (MUBs) in C12 is exactlyfour: M(12) = 4. The proof proceeds by constructing four product MUBs using the tensorstructure C12 = C4 ⊗ C3, then showing via a finite exhaustive search over 12th roots of unitythat no vector exists unbiased to all four. The key insight is that the column orthogonalityconstraint from the qutrit marginal condition reduces the search space to exactly 96 candidatematrices, none of which satisfy the transform unimodularity requirements."Towards thee I roll, thou all-destroying but unconquering whale; to the last I grapple with thee; from hell’s heart I stab at thee; for hate’s sake I spit my last breath at thee. Sink all coffins and all hearses to one common pool! and since neither can be mine, let me then tow to pieces, while still chasing thee, though tied to thee, thou damned whale!"- Captain Ahab

DOI: 10.5281/zenodo.17949275

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We prove that the maximum number of mutually unbiased bases in complex dimension sixis three. The proof is structural and does not rely on exhaustive classification, numerical search,or probabilistic arguments. The key ingredients are: (1) a Laurent–polynomial obstructionblocking the Haagerup family of complex Hadamard matrices, (2) an analogous obstructionblocking the 2-circulant family, and (3) a structural Bridge Lemma showing that any complexHadamard matrix arising as a transition matrix in a MUB triplet must exhibit an opposite-pairphenomenon in its row-ratio multisets. Combined with the recognition theorem of Matszangoszand Sz¨oll˝osi, this forces all MUB-triplet transition matrices in dimension six into families alreadyobstructed, yielding a contradiction to the existence of a fourth mutually unbiased basis.