pith. sign in

arxiv: 2410.23997 · v2 · submitted 2024-10-31 · 🪐 quant-ph

Mutually Unbiased Bases in Composite Dimensions -- A Review

Daniel McNulty , Stefan Weigert This is my paper

Pith reviewed 2026-05-23 18:49 UTC · model grok-4.3

classification 🪐 quant-ph
keywords mutually unbiased basescomposite dimensionsexistence problemquantum informationHilbert spacesprime power dimensions
0
0 comments X

The pith

Fourteen mathematically equivalent formulations recast the open question of whether complete sets of mutually unbiased bases exist in composite dimensions.

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

This review shows that it remains unknown whether maximal collections of mutually unbiased bases exist in Hilbert spaces whose dimension is not a prime power. It organises the literature around fourteen different but mathematically equivalent ways to state the existence problem. All published analytic constructions, computer searches and numerical evidence for dimensions such as six and ten are collected in one place, together with variants of the original question and suggested attack routes. A sympathetic reader cares because the answer governs which quantum measurements and information-processing tasks can be realised exactly in those dimensions.

Core claim

The existence of a complete set of mutually unbiased bases in a Hilbert space of composite dimension is an open problem that admits fourteen mathematically equivalent formulations; the review gathers every known partial result, modification and potential solution strategy under this organising principle.

What carries the argument

Mutually unbiased bases (sets of orthonormal bases in which vectors from distinct bases have constant inner-product modulus 1/sqrt(d)), together with the fourteen equivalent reformulations of the maximal-existence question that allow results from different mathematical domains to be compared directly.

If this is right

  • Any result obtained in one of the fourteen formulations immediately transfers to all the others.
  • Known non-existence results or exhaustive computer searches in small composite dimensions apply uniformly across all equivalent statements.
  • Modifications such as approximate unbiasedness or incomplete sets inherit the same equivalence structure and can be studied interchangeably.

Where Pith is reading between the lines

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

  • If complete sets do not exist in dimension six, certain quantum tomography and cryptography protocols that presuppose them cannot be realised exactly with six-level systems.
  • The equivalence suggests that algebraic-geometry or combinatorial-design techniques may be as effective as direct linear-algebra attacks.
  • A general non-existence theorem would imply that the maximum number of mutually unbiased bases is strictly less than d+1 whenever d has at least two distinct prime factors.

Load-bearing premise

The fourteen formulations of the existence problem are mathematically equivalent.

What would settle it

An explicit construction of six mutually unbiased bases in dimension six, or a rigorous proof that no such set exists, would decide the central open question.

Figures

Figures reproduced from arXiv: 2410.23997 by Daniel McNulty, Stefan Weigert.

Figure 1.1
Figure 1.1. Figure 1.1: Number of preprints published annually in the sections computer sciences, mathematics and phys [PITH_FULL_IMAGE:figures/full_fig_p005_1_1.png] view at source ↗
read the original abstract

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

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

1 major / 1 minor

Summary. The manuscript is a review on mutually unbiased bases (MUBs) in quantum information. It states that the existence of complete sets of MUBs remains unknown in Hilbert spaces whose dimension is not a prime power (composite dimensions such as 6 or 10). The central organizing device is the presentation of fourteen mathematically equivalent formulations of the existence problem, followed by a comprehensive summary of analytic, computer-aided and numerical results relevant to composite dimensions, a review of known modifications of the problem, and an outline of potential solution strategies.

Significance. If the claimed equivalences among the fourteen formulations hold uniformly (including for composite dimensions) and the summary of results is accurate and complete, the review would provide a useful consolidation of the literature on this long-standing open question. Explicit credit is due for the review's attempt to organize disparate results around equivalent formulations rather than a single presentation; this structure, if rigorously supported, could help identify cross-formulation insights or obstructions.

major comments (1)
  1. [Introduction and the section(s) presenting the fourteen formulations] The claim that fourteen formulations are mathematically equivalent is load-bearing for the entire review structure (abstract and introduction). The manuscript must explicitly verify or re-derive the equivalences rather than relying solely on citations, particularly checking that no formulation introduces extra assumptions valid only for prime-power dimensions and that the equivalences survive in regimes with number-theoretic obstructions (e.g., d=6, d=10). Without such verification, the organizing principle of the review risks being incomplete.
minor comments (1)
  1. Ensure all cited analytic and numerical results for specific composite dimensions are accompanied by clear pointers to the original sources and any limitations of the methods used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recognizing the potential value of organizing the literature around equivalent formulations. We address the single major comment below and agree that strengthening the presentation of the equivalences will improve the manuscript.

read point-by-point responses

  1. Referee: [Introduction and the section(s) presenting the fourteen formulations] The claim that fourteen formulations are mathematically equivalent is load-bearing for the entire review structure (abstract and introduction). The manuscript must explicitly verify or re-derive the equivalences rather than relying solely on citations, particularly checking that no formulation introduces extra assumptions valid only for prime-power dimensions and that the equivalences survive in regimes with number-theoretic obstructions (e.g., d=6, d=10). Without such verification, the organizing principle of the review risks being incomplete.

    Authors: We agree that the equivalences are central to the review's structure. While the manuscript presents the fourteen formulations and cites the literature establishing their equivalence, we acknowledge that an explicit verification tailored to composite dimensions would strengthen the exposition. In the revised version we will insert a short dedicated subsection (immediately following the list of formulations) that (i) sketches the chain of equivalences with references to the original derivations, (ii) confirms that none of the steps invoke the assumption that the dimension is a prime power, and (iii) explicitly checks the relevant number-theoretic conditions for the representative composite cases d=6 and d=10. This addition will be kept concise while making the load-bearing claim self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: review summarizes external literature without self-referential derivations

full rationale

This review paper presents no derivation chain, predictions, or fitted parameters. The statement that fourteen formulations are mathematically equivalent serves as an organizing principle drawn from the cited literature rather than a claim proven within the paper itself. No load-bearing step reduces to a self-citation, ansatz, or input by construction. The central observation (open existence question for non-prime-power dimensions) is reported as the current state of external results, not derived here.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; no new free parameters, axioms, or invented entities are introduced by the authors beyond the standard mathematical background of quantum information theory.

pith-pipeline@v0.9.0 · 5602 in / 1034 out tokens · 38366 ms · 2026-05-23T18:49:32.277600+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. All pure entangled states can lead to fully nonlocal correlations

    quant-ph 2026-04 unverdicted novelty 7.0

    Non-maximally entangled states exhibit full nonlocality under simple Schmidt coefficient conditions, and all pure entangled states can be activated to full nonlocality with multiple copies.

Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages · cited by 1 Pith paper · 23 internal anchors

  1. [1]

    The new ten most annoy- ing questions in quantum computing,

    Aaronson, S. (2014), “The new ten most annoy- ing questions in quantum computing,” Shtetl- Optimized Blog

    work page 2014

  2. [2]

    (2015), J

    Abdukhalikov, K. (2015), J. Algebr. Comb. 41, 1055

    work page 2015

  3. [3]

    Bannai, and S

    Abdukhalikov, K., E. Bannai, and S. Suda (2009), J. Comb. Theory, Ser. A116, 434

    work page 2009

  4. [4]

    (1984), in Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, edited by L

    Accardi, L. (1984), in Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, edited by L. Accardi, A. Frigerio, and V . Gorini (Springer) pp. 1–19

    work page 1984

  5. [5]

    Adamson, R. B. A., and A. M. Steinberg (2010), Phys. Rev. Lett. 105, 030406

    work page 2010

  6. [6]

    Agaian, S. S. (1985), Hadamard Matrices and Ap- plications (Springer LNM)

    work page 1985

  7. [7]

    Aguilar, E. A., J. J. Borkała, P . Mironowicz, and M. Pawłowski (2018), Phys. Rev. Lett. 121, 050501

    work page 2018

  8. [8]

    Charnes (2018), Phys

    Alber, G., and C. Charnes (2018), Phys. Scr. 94, 014007

    work page 2018

  9. [9]

    (1980), IEEE Trans

    Alltop, W. (1980), IEEE Trans. Inform. Theory 26, 350

    work page 1980

  10. [10]

    Emerson (2007), in Twenty- Second Annual Conference on IEEE Computational Complexity (IEEE) pp

    Ambainis, A., and J. Emerson (2007), in Twenty- Second Annual Conference on IEEE Computational Complexity (IEEE) pp. 129–140

    work page 2007

  11. [11]

    Nayak, A

    Ambainis, A., A. Nayak, A. Ta-Shma, and U. Vazirani (1999), in Proceedings of the thirty- first annual ACM symposium on Theory of comput- ing (Association for Computing Machinery) pp. 376–383

    work page 1999

  12. [12]

    Sharma, D

    Amburg, I., R. Sharma, D. M. Sussman, and W. K. Wootters (2014), J. Math. Phys.55, 122206

    work page 2014

  13. [13]

    Bengtsson (2017), Rep

    Andersson, O., and I. Bengtsson (2017), Rep. Math. Phys. 79, 33

    work page 2017

  14. [14]

    Appleby, D. M. (2007), Opt. Spectrosc. 103, 416

    work page 2007

  15. [15]

    Appleby, D. M., I. Bengtsson, and H. B. Dang (2014), arXiv:1409.7987

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  16. [16]

    Appleby, D. M., H. B. Dang, and C. A. Fuchs (2014), Entropy 16, 1484

    work page 2014

  17. [17]

    Bengtsson, M

    Appleby, M., I. Bengtsson, M. Grassl, M. Har- rison, and G. McConnell (2022), J. Math. Phys. 63, 112205

    work page 2022

  18. [18]

    Chien, S

    Appleby, M., T.-Y. Chien, S. Flammia, and S. Waldron (2018), J. Phys. A: Math. Theor. 51, 165302

    work page 2018

  19. [19]

    Aravind, P . K. (2003), Z. Naturforsch. A58, 85

    work page 2003

  20. [20]

    (2005), J

    Archer, C. (2005), J. Math. Phys. 46, 022106

    work page 2005

  21. [21]

    Aschbacher, M., A. M. Childs, and P . Wocjan (2007), J. Algebr. Comb. 25, 111

    work page 2007

  22. [22]

    Avella, V . G., J. Czartowski, D. Goyeneche, and K. ˙Zyczkowski (2024), arXiv:2404.18847

    work page arXiv 2024

  23. [23]

    Entropic uncertainty relations for incomplete sets of mutually unbiased observables

    Azarchs, A. (2004), arXiv:quant-ph/0412083

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  24. [24]

    Square multiples n give in- finitely many cyclic n-roots,

    Backelin, J. (1989), “Square multiples n give in- finitely many cyclic n-roots,” Stockholms Uni- versitet

    work page 1989

  25. [25]

    Bae, J., A. Bera, D. Chru´ sci ´ nski, B. C. Hiesmayr, and D. McNulty (2022), J. Phys. A: Math. Theor. 15, 505303

    work page 2022

  26. [26]

    Bae, J., B. C. Hiesmayr, and D. McNulty (2019), New J. Phys. 21, 013012

    work page 2019

  27. [27]

    Sohbi, J

    Baek, K., A. Sohbi, J. Lee, J. Kim, and H. Nha (2020), New J. Phys. 22, 093019

    work page 2020

  28. [28]

    Balan, R., B. G. Bodmann, P . G. Casazza, and D. Edidin (2009), J. Fourier Anal. Appl. 15, 488

    work page 2009

  29. [29]

    A., and S

    Ballester, M. A., and S. Wehner (2007), Phys. Rev. A 75, 022319

    work page 2007

  30. [30]

    Bandeira, A. S., N. Doppelbauer, and D. Kunisky (2022), Sampl. Theory Signal Process. Data Anal. 20, 18

    work page 2022

  31. [31]

    Bandyopadhyay, S., P . O. Boykin, V . Roychow- dhury, and F. Vatan (2002), Algorithmica34, 512

    work page 2002

  32. [32]

    Mandayam (2013), Phys

    Bandyopadhyay, S., and P . Mandayam (2013), Phys. Rev. A 87, 042120

    work page 2013

  33. [33]

    (2019), arXiv:1910.06911

    Banica, T. (2019), arXiv:1910.06911

    work page arXiv 2019

  34. [34]

    Bichon, and J.-M

    Banica, T., J. Bichon, and J.-M. Schlenker (2009), J. Funct. Anal. 257, 2864

    work page 2009

  35. [35]

    Nicoara (2007), Panamer

    Banica, T., and R. Nicoara (2007), Panamer. Math. J. 17, 1

    work page 2007

  36. [36]

    Dietrich, and C

    Barrera Acevedo, S., H. Dietrich, and C. Lionis (2024), Des. Codes Cryptogr. , 1

    work page 2024

  37. [37]

    ˇCernoch, K

    Bartkiewicz, K., A. ˇCernoch, K. Lemr, A. Mir- anowicz, and F. Nori (2016), Phys. Rev. A 93, 062345

    work page 2016

  38. [38]

    Cramer, and M

    Baumgratz, T., M. Cramer, and M. B. Plenio (2014), Phys. Rev. Lett. 113, 140401

    work page 2014

  39. [39]

    Bavaresco, J., M. T. Quintino, L. Guerini, T. O. Maciel, D. Cavalcanti, and M. T. Cunha (2017), Phys. Rev. A 96, 022110

    work page 2017

  40. [40]

    Mutually unbiased bases in multi-partite continuous- variable systems,

    Beales, A., and S. Weigert (2024), “Mutually unbiased bases in multi-partite continuous- variable systems,” Unpublished

    work page 2024

  41. [41]

    Nicoara (2008), Lin

    Beauchamp, K., and R. Nicoara (2008), Lin. Alg. Appl. 428, 1833

    work page 2008

  42. [42]

    Peres (2000), Phys

    Bechmann-Pasquinucci, H., and A. Peres (2000), Phys. Rev. Lett. 85, 3313

    work page 2000

  43. [43]

    Smotrovs (2008), in Math- ematical Methods in Computer Science , edited by J

    Belovs, A., and J. Smotrovs (2008), in Math- ematical Methods in Computer Science , edited by J. Calmet, W. Geiselmann, and J. Müller-Quade (Springer) pp. 50–69

    work page 2008

  44. [44]

    Pastawski, and J

    Bendersky, A., F. Pastawski, and J. P . Paz (2008), Phys. Rev. Lett. 100, 190403

    work page 2008

  45. [45]

    Beneduci, R., T. J. Bullock, P . Busch, C. Carmeli, T. Heinosaari, and A. Toigo (2013), Phys. Rev. A 88, 032312

    work page 2013

  46. [46]

    (2005), AIP Conf

    Bengtsson, I. (2005), AIP Conf. Proc. 750, 63

    work page 2005

  47. [47]

    (2007), AIP Conf

    Bengtsson, I. (2007), AIP Conf. Proc. 889, 40

    work page 2007

  48. [48]

    (2010), J

    Bengtsson, I. (2010), J. Phys. Conf. Ser. 254, 012007

    work page 2010

  49. [49]

    Blanchfield, and A

    Bengtsson, I., K. Blanchfield, and A. Cabello (2012), Phys. Lett. A 376, 374. 69

    work page 2012

  50. [50]

    Bruzda, Å

    Bengtsson, I., W. Bruzda, Å. Ericsson, J.-Å. Larsson, W. Tadej, and K. ˙Zyczkowski (2007), J. Math. Phys. 48, 052106

    work page 2007

  51. [51]

    Ericsson (2005), Open Syst

    Bengtsson, I., and Å. Ericsson (2005), Open Syst. Inf. Dyn. 12, 107

    work page 2005

  52. [52]

    Grassl, and G

    Bengtsson, I., M. Grassl, and G. McConnell (2024), arXiv:2403.02872

    work page arXiv 2024

  53. [53]

    On discrete structures in finite Hilbert spaces

    Bengtsson, I., and K. ˙Zyczkowski (2017), arXiv:1701.07902

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  54. [54]

    H., and G

    Bennett, C. H., and G. Brassard (1984), in Pro- ceedings of IEEE International Conference on Com- puters, Systems and Signal Processing (IEEE) pp. 175–179

    work page 1984

  55. [55]

    V ., and J

    Berry, M. V ., and J. P . Keating (1999), SIAM Rev. 41, 236

    work page 1999

  56. [56]

    Kharaghani (2010), Cryptogr

    Best, D., and H. Kharaghani (2010), Cryptogr. Commun. 2, 199

    work page 2010

  57. [57]

    Entov, and L

    Biran, P ., M. Entov, and L. Polterovich (2004), Commun. Contemp. Math. 6, 793

    work page 2004

  58. [58]

    (1985), in Recent Advances in Four- ier Analysis and Its Applications , edited by J

    Björck, G. (1985), in Recent Advances in Four- ier Analysis and Its Applications , edited by J. S. Byrnes and J. L. Byrnes (Springer) pp. 131–140

    work page 1985

  59. [59]

    Fröberg (1991), J

    Björck, G., and R. Fröberg (1991), J. Symbolic Comput. 12, 329

    work page 1991

  60. [60]

    Saffari (1995), C

    Björck, G., and B. Saffari (1995), C. R. Acad. Sci. 320, 319

    work page 1995

  61. [61]

    Jungnickel, and B

    Blokhuis, A., D. Jungnickel, and B. Schmidt (2002), Proc. Amer. Math. Soc. 130, 1473

    work page 2002

  62. [62]

    MUBs in infinite di- mensions: The problematic analogy between L2(R) and Cd,

    Blume-Kohout, R. (2008), “MUBs in infinite di- mensions: The problematic analogy between L2(R) and Cd,” Talk presented at “Seeking SICs: A Workshop on Quantum Frames and Designs” (Ontario, Canada)

    work page 2008

  63. [63]

    Haas (2018), Proc

    Bodmann, B., and J. Haas (2018), Proc. Amer. Math. Soc. 146, 2601

    work page 2018

  64. [64]

    (1928), Nature 121, 580

    Bohr, N. (1928), Nature 121, 580

    work page 1928

  65. [65]

    Zhdanovskiy (2016), J

    Bondal, A., and I. Zhdanovskiy (2016), J. Math. Sci. 216, 23

    work page 2016

  66. [66]

    Borkała, J. J., C. Jebarathinam, S. Sarkar, and R. Augusiak (2022), Entropy 24, 350

    work page 2022

  67. [67]

    (1926), Z

    Born, M. (1926), Z. Phys. 37, 863

    work page 1926

  68. [68]

    The "mean king's problem" with continuous variables

    Botero, A., and Y. Aharonov (2007), arXiv:0710.2952

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  69. [69]

    (2018), Quantum 2, 111

    Bouchard, F., et al. (2018), Quantum 2, 111

    work page 2018

  70. [70]

    Boykin, P . O., M. Sitharam, M. Tarifi, and P . Woc- jan (2005), arXiv:quant-ph/0502024

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  71. [71]

    Boykin, P . O., M. Sitharam, P . H. Tiep, and P . Wocjan (2007), Quantum Inf. Comput.7, 371

    work page 2007

  72. [72]

    Dall’Arno, and A

    Brandsen, S., M. Dall’Arno, and A. Szymusiak (2016), Phys. Rev. A 94, 022335

    work page 2016

  73. [73]

    L., and P

    Braunstein, S. L., and P . Van Loock (2005), Rev. Mod. Phys. 77, 513

    work page 2005

  74. [74]

    Bravyi, S., and J. A. Smolin (2011), Phys. Rev. A 84, 042306

    work page 2011

  75. [75]

    Quantum Key Distribution Highly Sensitive to Eavesdropping

    Brierley, S. (2009), arXiv:0910.2578

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  76. [76]

    Weigert (2008), Phys

    Brierley, S., and S. Weigert (2008), Phys. Rev. A 78, 042312

    work page 2008

  77. [77]

    Weigert (2009), Phys

    Brierley, S., and S. Weigert (2009), Phys. Rev. A 79, 052316

    work page 2009

  78. [78]

    Weigert (2010), J

    Brierley, S., and S. Weigert (2010), J. Phys. Conf. Ser. 254, 012008

    work page 2010

  79. [79]

    Weigert, and I

    Brierley, S., S. Weigert, and I. Bengtsson (2010), Quantum Inf. Comput. 10, 803

    work page 2010

  80. [80]

    H., and H

    Bruck, R. H., and H. J. Ryser (1949), Canad. J. Math. 1, 9

    work page 1949

Showing first 80 references.