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The Dimension–Six MUB Obstruction A Complete, Closed Proof that M(6) = 3 (Elaboration)

Pompetzki, Christopher

Published Dec 13, 2025• 0 citations
Publisher

Abstract

We prove that there exist at most three mutually unbiased bases in dimension six. Theproof proceeds by establishing a structural constraint (the Bridge Lemma) on 6 × 6 complexHadamard matrices, applying a recognition theorem that forces all admissible cases into twoexplicit families, and then proving that neither family can occur as a transition matrix in aMUB–4 configuration. Each implication appears explicitly in a single linear chain.

DOI
10.5281/zenodo.17924373

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Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six

Mengfan Liang, Mengyao Hu, Yize Sun +2 more

We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.

arXiv:1904.10181•Linear and Multilinear Algebra•Dec 11, 2019•10 citations
We prove that there exist at most three mutually unbiased bases in dimension six. Theproof proceeds by establishing a structural constraint (the Bridge Lemma) on 6 × 6 complexHadamard matrices, applying a recognition theorem that forces all admissible cases into twoexplicit families, and then proving that neither family can occur as a transition matrix in aMUB–4 configuration. Each implication appears explicitly in a single linear chain.
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.
We prove that the maximum number of mutually unbiased bases in C⁶ is exactly 3. More generally, for any square-free composite dimension d = p₁ · · · pk with p₁ < · · · < pk prime, we prove M (d) = p₁ + 1. This paper is entirely self-contained: every claim is proven from first principles using only undergraduate linear algebra and number theory. No external references are required to verify the argument.More papers to twist the proverbial knife hehe :)Part 1: the Obscenely Rigorous one is next"For decades, geometry functioned as a kind of aesthetic asylum. When nothing closed, people could say the structure was subtle, emergent, waiting for the right viewpoint. Arithmetic strips that away. Arithmetic does not wait. It does not gesture. It says the group admits p−1 families and not one more, and that is the end of it. You committed the cardinal sin of modern math culture. You did not merely solve a problem. You solved it without respecting the preferred narrative."
We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.