pith. sign in

arxiv: 2604.04164 · v1 · submitted 2026-04-05 · 🪐 quant-ph

Generalized Numerical Construction of MUBs: A Group Theoretical Investigation

Bu\u{g}ra G\"ultekin , Solomon B. Samuel , Zafer Gedik This is my paper

Pith reviewed 2026-05-13 16:41 UTC · model grok-4.3

classification 🪐 quant-ph
keywords mutually unbiased basesMUBsGram matrixphase space optimizationClifford groupWeyl-Heisenberg groupquantum tomographyBargmann invariants
0
0 comments X

The pith

A numerical search reduces construction of mutually unbiased bases to phase-space optimization of Gram matrix projection constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to build complete sets of d+1 mutually unbiased bases in dimension d by treating their Gram matrix as a projection and enforcing only third- and fourth-order trace conditions. This turns the search into an optimization problem over phase space without assuming any underlying group or field structure in advance. In dimensions 3, 4, and 5 every solution found is isomorphic to the standard construction, occupies an isolated point, and has an automorphism group that matches the Clifford group. The same procedure finds no valid configurations in dimension 6 within the parameter space explored.

Core claim

Formulating the MUB problem at the level of the Gram matrix reduces the search for a complete set of d+1 bases in dimension d to a phase-space optimization problem whose solutions are certified by the third- and fourth-order trace constraints that characterize projection matrices. Numerical runs in dimensions 3, 4, and 5 produce only mutually isomorphic configurations whose automorphism groups coincide exactly with the Clifford group, the normalizer of the Weyl-Heisenberg group. In dimension 6 the search yields no MUBs.

What carries the argument

The Gram matrix of a complete MUB set, treated as a projection matrix whose validity is guaranteed by third- and fourth-order trace constraints.

If this is right

  • All numerically constructed MUBs in dimensions 3, 4, and 5 are isolated points in phase space.
  • Their automorphism groups coincide exactly with the Clifford group.
  • A classification based on third-order Bargmann invariants and automorphism groups distinguishes the geometric structure of the resulting configurations.
  • Limited searches in dimension 6 detect no maximal sets of MUBs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the optimization to wider parameter ranges or higher dimensions could test existence of MUBs where analytic constructions are unavailable.
  • The same Gram-matrix constraint approach might apply to other equiangular configurations such as SIC-POVMs.
  • If the trace conditions prove exhaustive, the method offers a computational route to proving non-existence in specific dimensions.

Load-bearing premise

The third- and fourth-order trace constraints on the Gram matrix are necessary and sufficient to guarantee a valid set of MUBs.

What would settle it

Discovery of a set of seven MUBs in dimension 6 that satisfies the trace constraints but was missed by the optimization, or an explicit MUB configuration in dimension 5 whose automorphism group differs from the Clifford group.

Figures

Figures reproduced from arXiv: 2604.04164 by Bu\u{g}ra G\"ultekin, Solomon B. Samuel, Zafer Gedik.

Figure 1
Figure 1. Figure 1: Commuting classes in dimension 3: points ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Commuting classes in dimension 4. The intersection points are marked with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Generating set elements of the triple product tensor plotted on the unit circle (left [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
read the original abstract

Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on a priori group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of $d+1$ MUBs in dimension $d$ to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a valid projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions $3$, $4$, and $5$ demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. Though the scope of the search was limited, in dimension $d = 6$ our numerical search yielded no MUBs within explored parameter space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a numerical method for constructing maximal sets of mutually unbiased bases (MUBs) in dimension d by optimizing over the Gram matrix of d(d+1) vectors, reducing the problem to phase-space search under the claim that third- and fourth-order trace constraints tr(G^3) and tr(G^4) are necessary and sufficient to ensure G is a valid rank-d projection matrix. Numerical searches recover the known MUBs in d=3,4,5, all of which are shown to be isolated points isomorphic to the standard Clifford constructions with automorphism groups matching the normalizer of the Weyl-Heisenberg group; a limited search in d=6 yields no solutions within the explored parameter space. A classification scheme based on third-order Bargmann invariants is also developed.

Significance. If the trace constraints are rigorously validated as sufficient for exact MUB properties, the work supplies a group-structure-independent numerical tool for probing MUB existence and uniqueness in non-prime-power dimensions, together with a concrete classification framework. The recovery of known solutions and the isolation/automorphism results in d=3-5 provide useful confirmation, while the d=6 negative result, though qualified, adds to the accumulating evidence against maximal MUBs in that dimension.

major comments (2)
  1. [§2] §2 (Gram-matrix formulation): the assertion that the third- and fourth-order trace conditions on G are necessary and sufficient to guarantee a valid MUB configuration (i.e., d+1 orthonormal bases with exact |<e_i|f_j>|^2 = 1/d) is load-bearing for all numerical claims, yet the text does not demonstrate that global trace equalities automatically enforce the required block-diagonal structure (identity blocks for each basis and constant-modulus off-block entries). Post-optimization verification that each d×d block is exactly a rank-d projector with unit diagonal and that inter-basis inner products satisfy the unbiasedness condition exactly is needed; without it, extraneous projection matrices could be accepted.
  2. [§5] §5 (d=6 search): the statement that no MUBs were found is qualified as 'limited' and 'within explored parameter space,' but no quantitative details are supplied on phase-space sampling density, number of random initializations, optimizer convergence tolerances, or the precise bounds of the explored region. These parameters are required to evaluate whether the negative result is robust or merely reflects incomplete coverage.
minor comments (2)
  1. [§3] The notation for the Bargmann invariants and the precise definition of the automorphism group action should be stated explicitly in §3 rather than left to the reader to reconstruct from the classification results.
  2. [Figures 2-4] Figure captions for the phase-space plots in d=3-5 should include the numerical tolerance used to declare a point 'isolated' and the method by which isomorphism between solutions was verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We have carefully considered each point and will make revisions to address the concerns raised. Our point-by-point responses are provided below.

read point-by-point responses

  1. Referee: [§2] §2 (Gram-matrix formulation): the assertion that the third- and fourth-order trace conditions on G are necessary and sufficient to guarantee a valid MUB configuration (i.e., d+1 orthonormal bases with exact |<e_i|f_j>|^2 = 1/d) is load-bearing for all numerical claims, yet the text does not demonstrate that global trace equalities automatically enforce the required block-diagonal structure (identity blocks for each basis and constant-modulus off-block entries). Post-optimization verification that each d×d block is exactly a rank-d projector with unit diagonal and that inter-basis inner products satisfy the unbiasedness condition exactly is needed; without it, extraneous projection matrices could be accepted.

    Authors: We agree with the referee that an explicit verification step is necessary to confirm that the optimized Gram matrices correspond to valid MUB configurations rather than other projectors. Although the trace conditions tr(G^3) = d(d+1) and tr(G^4) = d(d+1)^2 ensure that G is a rank-d projector (as derived from the properties of projection matrices), they do not automatically guarantee the block structure specific to MUBs. In the revised manuscript, we will add post-optimization checks: for each converged solution, we verify that the d×d blocks are orthonormal projectors (unit diagonal, rank d) and that the off-block inner products satisfy |⟨ψ|φ⟩|^2 = 1/d exactly (within numerical precision). These checks were performed internally and confirmed the known MUBs in dimensions 3–5; we will document them in the text. revision: yes

  2. Referee: [§5] §5 (d=6 search): the statement that no MUBs were found is qualified as 'limited' and 'within explored parameter space,' but no quantitative details are supplied on phase-space sampling density, number of random initializations, optimizer convergence tolerances, or the precise bounds of the explored region. These parameters are required to evaluate whether the negative result is robust or merely reflects incomplete coverage.

    Authors: We accept that the lack of quantitative details on the numerical search parameters limits the interpretability of the d=6 result. In the revised version of the manuscript, we will include a detailed description of the search protocol, specifying the number of random initializations, the sampling method and density in phase space, the convergence criteria of the optimizer, and the exact bounds of the parameter space explored. This will allow readers to better assess the extent of the search and the strength of the negative finding, while we continue to qualify the result as limited due to computational constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical search under explicit constraints is self-contained

full rationale

The derivation reduces MUB search to phase-space optimization of the Gram matrix G subject to the explicit conditions that G is a projection matrix (tr(G^3) and tr(G^4) constraints stated as necessary and sufficient). These are applied directly to generate candidate configurations in d=3,4,5; the reported isomorphism to Clifford-group solutions and absence in d=6 are outputs of the numerical procedure, not inputs or self-referential definitions. Post-hoc classification via Bargmann invariants and automorphism groups does not feed back into the search or existence claims. No fitted parameters are renamed as predictions, no self-citations justify the core constraints, and the method contains no ansatz or uniqueness theorem imported from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the domain assumption that any valid MUB set corresponds to a projection matrix whose third- and fourth-order traces obey specific equalities; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The Gram matrix of a set of MUBs is a projection matrix
    Invoked to reduce the construction problem to trace constraints.
  • domain assumption Third- and fourth-order trace constraints are necessary and sufficient for a valid MUB Gram matrix
    Used to turn the search into a phase-space optimization problem.

pith-pipeline@v0.9.0 · 5583 in / 1375 out tokens · 63508 ms · 2026-05-13T16:41:34.360841+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Three ways to look at mutually unbiased bases.AIP Conference Proceedings, 889(1):40–51, 02 2007

    Ingemar Bengtsson. Three ways to look at mutually unbiased bases.AIP Conference Proceedings, 889(1):40–51, 02 2007

    work page 2007

  2. [2]

    Cambridge University Press, 2006

    Ingemar Bengtsson and Karol Zyczkowski.Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, 2006

    work page 2006

  3. [3]

    Optimal state-determination by mutually un- biased measurements.Annals of Physics, 191(2):363–381, 1989

    William K Wootters and Brian D Fields. Optimal state-determination by mutually un- biased measurements.Annals of Physics, 191(2):363–381, 1989

    work page 1989

  4. [4]

    Unitary operator bases.Proceedings of the National Academy of Sci- ences of the United States of America, 46(4):570–579, 1960

    Julian Schwinger. Unitary operator bases.Proceedings of the National Academy of Sci- ences of the United States of America, 46(4):570–579, 1960

    work page 1960

  5. [5]

    Geometrical description of quantal state determination.Journal of Physics A: Mathematical and General, 14(12):3241, dec 1981

    I D Ivonovic. Geometrical description of quantal state determination.Journal of Physics A: Mathematical and General, 14(12):3241, dec 1981

    work page 1981

  6. [6]

    On mutually unbiased bases.International Journal of Quantum Information, 08(04):535–640, 2010

    Thomas Durt, Berthold-Georg Englert, Ingemar Bengtsson, and Karol ˙Zyczkowski. On mutually unbiased bases.International Journal of Quantum Information, 08(04):535–640, 2010

    work page 2010

  7. [7]

    Numerical evidence for the maximum number of mu- tually unbiased bases in dimension six.Physics Letters A, 369(1):5–8, 2007

    Paul Butterley and William Hall. Numerical evidence for the maximum number of mu- tually unbiased bases in dimension six.Physics Letters A, 369(1):5–8, 2007. 16

    work page 2007

  8. [8]

    Maximal sets of mutually unbiased quantum states in dimension 6.Phys

    Stephen Brierley and Stefan Weigert. Maximal sets of mutually unbiased quantum states in dimension 6.Phys. Rev. A, 78:042312, Oct 2008

    work page 2008

  9. [9]

    On sic-povms and mubs in dimension 6, 2009

    Markus Grassl. On sic-povms and mubs in dimension 6, 2009

    work page 2009

  10. [10]

    Mutually unbiased bases in six dimensions: The four most distant bases.Phys

    Philippe Raynal, Xin L¨ u, and Berthold-Georg Englert. Mutually unbiased bases in six dimensions: The four most distant bases.Phys. Rev. A, 83:062303, Jun 2011

    work page 2011

  11. [11]

    A. J. Scott and M. Grassl. Symmetric informationally complete positive-operator-valued measures: A new computer study.Journal of Mathematical Physics, 51(4):042203, 04 2010

    work page 2010

  12. [12]

    Fuchs, Michael C

    Christopher A. Fuchs, Michael C. Hoang, and Blake C. Stacey. The sic question: History and state of play.Axioms, 6(3), 2017

    work page 2017

  13. [13]

    Shayne F. D. Waldron.An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkh¨ auser, New York, NY, 2018

    work page 2018

  14. [14]

    On discrete structures in finite hilbert spaces, 2017

    Ingemar Bengtsson and Karol Zyczkowski. On discrete structures in finite hilbert spaces, 2017

    work page 2017

  15. [15]

    W. K. Wootters. Quantum measurements and finite geometry.Foundations of Physics, 36(1):112–126, 01 2006

    work page 2006

  16. [16]

    Group theoretical classification of sic-povms.Journal of Physics A: Mathematical and Theoretical, 57(29):295304, jul 2024

    S B Samuel and Z Gedik. Group theoretical classification of sic-povms.Journal of Physics A: Mathematical and Theoretical, 57(29):295304, jul 2024

    work page 2024

  17. [17]

    Oscar Boykin, Vwani Roychowdhury, and Farrokh Vatan

    Somshubhro Bandyopadhyay, P. Oscar Boykin, Vwani Roychowdhury, and Farrokh Vatan. A new proof for the existence of mutually unbiased bases.Algorithmica, 34(4):512– 528, 2002

    work page 2002

  18. [18]

    Equiangular lines, mutually unbiased bases, and spin models

    Chris Godsil and Aidan Roy. Equiangular lines, mutually unbiased bases, and spin models. European Journal of Combinatorics, 30(1):246–262, January 2009

    work page 2009

  19. [19]

    Z4-kerdock codes, orthogo- nal spreads, and extremal euclidean line-sets.Proceedings of the London Mathematical Society, 75(2):436–480, 1997

    AR Calderbank, PJ Cameron, WM Kantor, and JJ Seidel. Z4-kerdock codes, orthogo- nal spreads, and extremal euclidean line-sets.Proceedings of the London Mathematical Society, 75(2):436–480, 1997

    work page 1997

  20. [20]

    Constructions of mutually unbiased bases

    Andreas Klappenecker and Martin Roetteler. Constructions of mutually unbiased bases. 2003

    work page 2003

  21. [21]

    Quantum measurements with prescribed symmetry.Phys

    Wojciech Bruzda, Dardo Goyeneche, and Karol ˙Zyczkowski. Quantum measurements with prescribed symmetry.Phys. Rev. A, 96:022105, Aug 2017

    work page 2017

  22. [22]

    A characterization of projective unitary equiv- alence of finite frames and applications.SIAM Journal on Discrete Mathematics, 30(2):976–994, 2016

    Tuan-Yow Chien and Shayne Waldron. A characterization of projective unitary equiv- alence of finite frames and applications.SIAM Journal on Discrete Mathematics, 30(2):976–994, 2016

    work page 2016

  23. [23]

    The projective symmetry group of a finite frame

    Tuan-Yow Chien and Shayne Waldron. The projective symmetry group of a finite frame. New Zealand Journal of Mathematics, 48:55–81, Dec. 2018

    work page 2018

  24. [24]

    D. M. Appleby, Steven T. Flammia, and Christopher A. Fuchs. The lie algebraic signif- icance of symmetric informationally complete measurements.Journal of Mathematical Physics, 52(2):022202, 02 2011. 17

    work page 2011

  25. [25]

    All mutually unbiased bases in dimensions two to five.Quantum Inf

    Stephen Brierley, Stefan Weigert, and Ingemar Bengtsson. All mutually unbiased bases in dimensions two to five.Quantum Inf. Comput., 10:803–820, 2009

    work page 2009

  26. [26]

    Orbits of mutually unbiased bases.Journal of Physics A: Mathematical and Theoretical, 47(13):135303, March 2014

    Kate Blanchfield. Orbits of mutually unbiased bases.Journal of Physics A: Mathematical and Theoretical, 47(13):135303, March 2014

    work page 2014

  27. [27]

    D. M. Appleby. Symmetric informationally complete–positive operator valued measures and the extended clifford group.Journal of Mathematical Physics, 46(5):052107, 04 2005

    work page 2005

  28. [28]

    D. M. Appleby. Properties of the extended clifford group with applications to sic-povms and mubs. 2009

    work page 2009

  29. [29]

    Mutually unbiased bases as minimal clifford covariant 2-designs.Physical Review A, 91(6), June 2015

    Huangjun Zhu. Mutually unbiased bases as minimal clifford covariant 2-designs.Physical Review A, 91(6), June 2015

    work page 2015

  30. [30]

    Morvan, V

    A. Morvan, V. V. Ramasesh, M. S. Blok, J. M. Kreikebaum, K. O’Brien, L. Chen, B. K. Mitchell, R. K. Naik, D. I. Santiago, and I. Siddiqi. Qutrit randomized benchmarking. Physical Review Letters, 126(21), May 2021. 18

    work page 2021