MUBs from bent functions
Pith reviewed 2026-05-19 22:10 UTC · model grok-4.3
The pith
Bent functions yield complete sets of mutually unbiased bases by defining vectors as explicit linear combinations of the standard basis.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Bent functions can be used to construct complete sets of mutually unbiased bases by writing each new basis vector as a linear combination of the standard basis vectors, where the coefficients are taken from the bent function in such a way that the flat Fourier spectrum guarantees the inner-product condition between any two distinct bases.
What carries the argument
Bent functions with flat Fourier spectrum, employed to supply the coefficients in the explicit linear combinations that define the vectors of each new basis.
If this is right
- Complete sets of MUBs become constructible whenever bent functions exist over the relevant field or ring.
- The basis vectors are given by concrete formulas rather than abstract existence arguments.
- The method directly produces the required number of bases to form a complete set.
- The construction applies in the dimensions where bent functions are known to exist.
Where Pith is reading between the lines
- The same bent-function technique might be tested in other algebraic settings where similar spectral flatness holds.
- Explicit formulas could allow direct computation of quantities such as the number of bases or the field size needed for a given dimension.
- Connections to coding theory or combinatorial designs may appear once the linear combinations are written out.
Load-bearing premise
The flat Fourier spectrum of the bent functions is enough to force the inner products between vectors from different constructed bases to have constant magnitude.
What would settle it
An explicit bent function for which the resulting vectors from two different bases fail to satisfy the constant-magnitude inner-product requirement.
read the original abstract
This note contains a simple construction of complete sets of MUBs, using bent functions to write the new basis vectors as explicit linear combinations of the standard basis.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents a simple explicit construction of complete sets of mutually unbiased bases (MUBs) in dimension d. New basis vectors are formed as linear combinations of the standard basis vectors, with coefficients determined by a bent function f over an appropriate domain.
Significance. If the construction holds, it supplies an algebraic, parameter-free method for producing MUBs that directly exploits the flat Fourier spectrum of bent functions. This could streamline explicit constructions in quantum information and combinatorial designs, especially when bent functions are already tabulated or constructed for other purposes.
major comments (1)
- [Main construction (following the abstract)] The central step—that the indicated linear combinations yield |⟨u|v⟩|² = 1/d for any vector u from one constructed basis and v from a distinct basis—rests on the bent property of f. The note invokes the flat Walsh/Hadamard spectrum but does not carry out or cite the explicit character-sum evaluation over the additive group that would establish the constant-magnitude inner-product condition for the specific embedding into ℂ^d. This verification is load-bearing for the main claim.
Simulated Author's Rebuttal
We thank the referee for their thoughtful review and for recognizing the potential utility of our construction in quantum information and combinatorial designs. We address the major comment point by point below.
read point-by-point responses
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Referee: [Main construction (following the abstract)] The central step—that the indicated linear combinations yield |⟨u|v⟩|² = 1/d for any vector u from one constructed basis and v from a distinct basis—rests on the bent property of f. The note invokes the flat Walsh/Hadamard spectrum but does not carry out or cite the explicit character-sum evaluation over the additive group that would establish the constant-magnitude inner-product condition for the specific embedding into ℂ^d. This verification is load-bearing for the main claim.
Authors: We agree that providing an explicit verification strengthens the manuscript. The flat Walsh spectrum of bent functions is a defining property, but to make the argument complete for the specific linear combinations in ℂ^d, we will add a detailed computation of the inner product using the character sum over the additive group. This will be included as a new section or appendix in the revised version, citing standard references on bent functions where appropriate. revision: yes
Circularity Check
No circularity: construction invokes standard bent-function properties from external literature
full rationale
The note presents an explicit construction of MUBs by writing basis vectors as linear combinations using a bent function f. Bent functions are defined and their flat Fourier/Walsh spectrum properties are established in the prior literature (independent of this paper). The paper does not define any quantity in terms of the target MUB inner-product condition, nor does it fit parameters to a subset of data and relabel the fit as a prediction. No self-citation chain is load-bearing for the central claim, and no uniqueness theorem or ansatz is imported from the author's own prior work in a way that collapses the derivation. The derivation chain therefore remains self-contained against external algebraic benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Bent functions possess a flat Fourier transform that can be used to enforce constant inner-product magnitudes between bases.
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
If ℶ is a mubent set ... then {M_∞} ∪ {M_B | B∈ℶ} is a complete set of MUBs ... |(e_{a,B}, e_{a',B'})|^2 = N
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
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[1]
S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury and F. Vatan, A new proof for the existence of mutually unbiased bases. Algorithmica 34 (2002) 512--528
work page 2002
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[2]
A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, _4 --Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. LMS 75 (1997) 436–480
work page 1997
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[3]
R. S. Coulter and R. W. Matthews, Planar functions and planes of Lenz-Barlotti class II. Des. Codes Crypt. 10 (1997) 167--184
work page 1997
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[4]
P. Dembowski, Finite Geometries. Springer, Berlin-Heidelberg-NY 1968
work page 1968
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[5]
W. M. Kantor, MUBs and affine planes. J. Mathematical Physics 53 (2012) 032204
work page 2012
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[6]
Mesnager, Bent Functions: Fundamentals and Results
S. Mesnager, Bent Functions: Fundamentals and Results. Springer, Switzerland 2016
work page 2016
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[7]
W. K. Wootters, Quantum measurements and finite geometry. Found. Phys. 36 (2006) 112–126
work page 2006
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[8]
W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191 (1989) 363--381
work page 1989
discussion (0)
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