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M(2p) = 3 for All Primes p ≥ 3 A Complete Proof via Tensor Saturation and Double Flatness Rigidity

Pompetzki, Christopher

Published Dec 16, 2025• 0 citations
Publisher

Abstract

AbstractWe prove that the maximum number of mutually unbiased bases in dimension d = 2p, wherep ≥ 3 is prime, equals exactly three: M(2p) = 3. The proof combines tensor product saturationwith a new rigidity theorem: any unimodular sequence on Zp that is flat in both the Fourier basisFp and the Weyl-Heisenberg basis Cp must be a quadratic chirp. This “double flatness” rigidity,proved via character-theoretic arguments on autocorrelation functions, forces all triply-unbiasedstates to be product states, contradicting the existence of a fourth MUB. No restriction to rootsof unity is required. This resolves the MUB problem for infinitely many composite dimensions

DOI
10.5281/zenodo.17951290

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The Maximum Number of Mutually Unbiased Bases in Dimension Ten is Three - PDF

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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

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Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct $q+1$ bases in dimension $d = q(q+1)$ for every prime power $q\equiv3\pmod4$. This count grows as $\sqrt d$ and cannot be exceeded by tensor products of smaller sets.

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AbstractWe prove that the maximum number of mutually unbiased bases in dimension d = 2p, wherep ≥ 3 is prime, equals exactly three: M(2p) = 3. The proof combines tensor product saturationwith a new rigidity theorem: any unimodular sequence on Zp that is flat in both the Fourier basisFp and the Weyl-Heisenberg basis Cp must be a quadratic chirp. This “double flatness” rigidity,proved via character-theoretic arguments on autocorrelation functions, forces all triply-unbiasedstates to be product states, contradicting the existence of a fourth MUB. No restriction to rootsof unity is required. This resolves the MUB problem for infinitely many composite dimensions
AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.
AbstractWe prove that M(10) = 3, where M(d) denotes the maximum number of mutually unbiased bases in Cd. The proof uses the tensor product structure C10 ∼=C2 ⊗ C5 and a rigidity analysis for vectors in C5 that are flat in multiple bases. We show that any vector unbiased to three canonical product MUBs must be a product vector, yielding a contradiction with the product saturation bound.The author continues to blast Hungarian music from the 16th-17th century btw.https://x.com/mathCObot/status/1791341944125489463?s=20The author continues to meme on heretics in Japan btw. Visions of The Dreamworks Intergalactic Empire btw.Visions of Swamps btw. Shrek is love btw.Shrek is life btw.
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
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1q+1q+1 bases in dimension d=q(q+1)d = q(q+1)d=q(q+1) for every prime power q≡3(mod4)q\equiv3\pmod4q≡3(mod4). This count grows as d\sqrt dd​ and cannot be exceeded by tensor products of smaller sets.