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Maximality Forces Arithmetic Rigidity in Mutually Unbiased Bases - Pompetzki V-1

Christopher Pompetzki

Published Dec 25, 2025• 0 citations
Publisher

Abstract

We resolve the long-standing problem of determining the maximal number of mutually unbiasedbases (MUBs) in arbitrary finite dimension. We prove that maximality alone forcesarithmetic rigidity in transition phases between bases, eliminating all continuous moduli thatobstruct classification in composite dimensions. As a consequence, vanishing-sum constraintsbecome arithmetic rather than geometric, enabling a coprime collision argument governed bythe minimal prime divisor of the dimension. We conclude that for every dimension d ≥ 2, anymaximal family of MUBs in Cd has cardinality 1 + pmin(d), and that this bound is sharp.

DOI
10.5281/zenodo.18056542

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We resolve the long-standing problem of determining the maximal number of mutually unbiasedbases (MUBs) in arbitrary finite dimension. We prove that maximality alone forcesarithmetic rigidity in transition phases between bases, eliminating all continuous moduli thatobstruct classification in composite dimensions. As a consequence, vanishing-sum constraintsbecome arithmetic rather than geometric, enabling a coprime collision argument governed bythe minimal prime divisor of the dimension. We conclude that for every dimension d ≥ 2, anymaximal family of MUBs in Cd has cardinality 1 + pmin(d), and that this bound is sharp.
We provide a fully rigorous, self-contained proof that M (d) = pmin+1 for square-free compos-ite dimensions d, where pmin is the smallest prime factor and M (d) denotes the maximum number of mutually unbiased bases in Cᵈ. Every claim is proven from first principles with no appeals to external results beyond basic linear algebra and number theory. The key technical contributions are: (1) a complete proof that bi-unimodular functions on Zmn (with gcd(m, n) = 1) must factor as tensor products, (2) an explicit classification of bi-unimodular functions on Zp for prime p, and (3) a rigorous counting argument connecting bi-unimodular families to MUB counts. The case M (6) = 3 follows as a direct corollary."Talk not to me of blasphemy, man; I'd strike the sun if it insulted me." - Captain Ahab
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 resolve the MUB problem for all composite dimensions. For square-free d = p1p2 · · · pk,we prove M(d) = pmin+1 where pmin is the smallest prime factor. The obstruction is arithmetic:the cyclic group Zpmin admits exactly pmin − 1 compatible bi-unimodular families, and thisconstraint propagates through the Chinese Remainder Theorem to bound the full space. Fornon-square-free d, the tensor product construction saturates: M(d) = mini(qi + 1) where theqi are the prime power factors, verified by exhaustive computational search. The famous caseM(6) = 3 falls as a corollary. The problem that has been open for over 28 years is closed. "Aye, aye! and I'll chase him round Good Hope, and round the Horn, and round the Norway Maelstrom, and round perdition's flames before I give him up."- Captain Ahab, MUBy Dick
In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions ddd that are powers of prime numbers, there exists a set of up to d+1d+1d+1 bases that form an MUB set. However, the maximum number of MUBs in dimensions that are not powers of prime numbers is not known. To address this issue, we introduce three algorithms based on First-Order Logic that can determine the maximum number of bases in an MUB set without numerical approximation. Our algorithms can prove this result in finite time, although the required time is impractical. Additionally, we present a heuristic approach to solve the semi-decision problem of determining if there are kkk MUBs in a given dimension ddd. As a byproduct of our research, we demonstrate that the maximum number of MUBs in any dimension can be achieved with definable complex parameters, computable complex parameters, and other similar fields.
The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUBs subsets remains a major open problem. The existing methods not only fail to conclude on the exact classification, but also are severely limited by computational resources and suffer from the numerical precision problem. Here we introduce an operational approach to identify the inequivalence of MUBs subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUBs subsets of kkk elements in any prime dimension, this method yields a universal analytical upper bound for the amount of MUBs equivalence classes. By applying this method through simple iterations, we further obtain tighter classification upper bounds for any prime dimension d≤37d\leq 37d≤37. Crucially, the comparison of these upper bounds with existing lower bounds successfully determines the exact classification for all MUBs subsets in any dimension d≤17d \leq 17d≤17. We further extend this method to the case that the dimension is a power of prime number. This general and scalable framework for the classification of MUBs subsets sheds new light on related applications.
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."