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The Maximum Number of Mutually Unbiased Bases in Dimension Six - Pompetzki, Chris

Pompetzki, Christopher

Published Dec 13, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.17918687

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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.
We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.
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.
We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F ( a , b ). However, in the main result of this paper we also prove that for any values of the parameters ( a , b ), the standard basis and F ( a , b ) cannot be extended to a MUB-quartet . The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.
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.