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The Maximum Number of Mutually Unbiased Bases in Dimension 14 is Exactly Three

Pompetzki, Christopher

Published Dec 16, 2025• 0 citations
Publisher

Abstract

AbstractWe prove that the maximum number of mutually unbiased bases (MUBs) in C14 is exactlythree: M(14) = 3. The proof proceeds by constructing three product MUBs using the tensorstructure C14 = C2 ⊗ C7, then showing via finite coupling enumeration in the μ7 phase modelthat every triply-unbiased state is separable. This extends the tensor saturation paradigm fromdimensions 6, 10, and 12 to the case d = 2p with p = 7.

DOI
10.5281/zenodo.17949989

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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+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.
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