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An exact MUB triple in dimension six with two Fourier edges

Adian Jaupi

Published Oct 3, 2026• 0 citations
Publisher

Abstract

In this research, I constructed an exact algebraic triple of mutually unbiased bases (MUBs) in ℂ⁶ consisting of the computational basis, a member D(ρ) of the Diţă slice from Karlsson's three-parameter family, and an explicit third basis B. Through exact symbolic computation over a degree-96 number field extension, I proved that both remaining transition matrices in the triple belong precisely to the two-parameter Fourier family F(a,b) up to equivalence and transposition, revealing that the Diţă-slice matrix factors directly into a product of two Fourier-type Hadamard matrices. Applying the theorem of Jaming et al., I established that both Fourier edges are non-extendable, proving the triple cannot be extended to any MUB quartet. Additionally, I derived a closed-form expression D(z) for the entire Diţă slice, resolved degeneracies in Karlsson's Möbius parametrization, and computed its 2×2 Hadamard submatrices exactly (51 generically and 75 at z=1), while identifying a second exact Fourier-Fourier triangle at z=1. Finally, using homotopy continuation to track witness systems across seven slice parameters, I recovered 40 third bases that all preserved this Fourier-Fourier pattern to residuals within 5×10⁻¹⁵, providing strong numerical backing for a global structural conjecture along the full Diţă slice.

DOI
10.5281/zenodo.23111923

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In this research, I constructed an exact algebraic triple of mutually unbiased bases (MUBs) in ℂ⁶ consisting of the computational basis, a member D(ρ) of the Diţă slice from Karlsson's three-parameter family, and an explicit third basis B. Through exact symbolic computation over a degree-96 number field extension, I proved that both remaining transition matrices in the triple belong precisely to the two-parameter Fourier family F(a,b) up to equivalence and transposition, revealing that the Diţă-slice matrix factors directly into a product of two Fourier-type Hadamard matrices. Applying the theorem of Jaming et al., I established that both Fourier edges are non-extendable, proving the triple cannot be extended to any MUB quartet. Additionally, I derived a closed-form expression D(z) for the entire Diţă slice, resolved degeneracies in Karlsson's Möbius parametrization, and computed its 2×2 Hadamard submatrices exactly (51 generically and 75 at z=1), while identifying a second exact Fourier-Fourier triangle at z=1. Finally, using homotopy continuation to track witness systems across seven slice parameters, I recovered 40 third bases that all preserved this Fourier-Fourier pattern to residuals within 5×10⁻¹⁵, providing strong numerical backing for a global structural conjecture along the full Diţă slice.
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.
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.
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.
Which pure states give uniform outcomes in several mutually unbiased measurements, and when can they form another measurement basis? We distinguish common unbiased states, their orthonormal completion, and further extension. For a displayed phase-deformed Fourier pair in dimension 2m2m2m, a clock symmetry and skew antiunitary complete every prescribed common unbiased state to an explicit orthonormal basis. In dimension six, an amplitude-profile obstruction proves strong unextendibility of a given mutually unbiased triple when its original third-basis rays are invariant under the ternary clock. We give an exact nonproduct example and an application to a companion whose columns are localized at single Fourier frequencies. For every exact Hadamard matrix in a specified closed entrywise neighborhood of a fixed reference matrix, the coordinate and Hadamard bases have exactly 48 common unbiased rays but no third measurement basis, with a quantitative lower bound on the squared Gram defect of every six-column candidate. We also study auxiliary complex equations for common states, retaining algebraic multiplicities. For a flat unitary of order six with all 3×33\times33×3 minors nonzero, the projective ray scheme is finite; each singular two-minor contributes two reduced isolated boundary points, and the affine coordinate algebra is finite-dimensional over C\mathbb{C}C. In every order, a critical-algebra identity preserves multiplicities; in dimension six it gives finiteness for a fixed phase fibre. Specialization over a discrete valuation ring controls complete reciprocal-orthogonal collections. A nonphysical Tao/Potts collection separates algebraic from physical solutions. These results do not determine the global maximum number of mutually unbiased bases or classify all triples.
Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased 3×33 \times 33×3 corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.
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.