A Quantum Latin Square of Order Six with Cardinality 29

Aishwarya P. Das; Durgesh Kumar

REVIEW 3 major objections 4 minor 13 references

A Quantum Latin Square of Order Six with Cardinality 29

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper constructs a real quantum Latin square of order six with cardinality 29, completing the order-six cardinality spectrum.

desk verdict A likely-correct but under-verified construction that fills the last gap in the order-six spectrum; deserves review with a demand for exact-arithmetic certificates. read the letter →

arxiv 2608.12607 v1 pith:B4WJ3JLE submitted 2026-08-12 math.CO

Aishwarya P. Das , Durgesh Kumar This is my paper · ORCID

classification math.CO MSC 05B15

keywords quantumLatinsquarecardinalitypuncturedorthonormalarrayordersixraycountdiagonalextensionorthogonalchangeofbasisreal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs an explicit $6\times 6$ array of unit vectors—a real quantum Latin square—and shows its entries occupy exactly 29 distinct rays once overall phase is ignored. That number, 29, was the last value still open for order six, so the paper completes the known list of attainable cardinalities for quantum Latin squares of order six. The construction works by putting one common vector on the diagonal and building the off-diagonal entries as a punctured orthonormal array in its five-dimensional orthogonal complement. Two off-diagonal rays are deliberately repeated and the rest are shown distinct, giving 28 off-diagonal rays plus the diagonal ray. If the arithmetic checks out, the order-six cardinality spectrum is exactly $\{6,8,9,\ldots,36\}$.

What carries the argument

The load-bearing device is the diagonal-extension reduction (Proposition 2.3): a quantum Latin square with a common diagonal vector is exactly a punctured orthonormal array in the orthogonal complement, and its cardinality is one plus the number of off-diagonal rays. The array $V_{29}$ is assembled by replacing orthonormal pairs inside coordinate planes, with most steps independent; the one global constraint is the compatibility condition in column 0, expressed as the scalar identity $\Delta=0$ for the chosen rational parameters, with Equation (23) defining $\Delta$ and Equation (24) factoring the relevant determinant as $\alpha\Delta/(\varphi^2 r_{13}r_{43}\rho_{20}\rho_{30})$. Lemma 2.4 (one missing column) supplies the last column by a frame-operator argument, so only five columns need direct checking.

What would settle it

Recompute the asserted identities directly from the explicit rational parameters: verify $\Delta=0$ in Equation (23) and the factorization in Equation (24), and re-derive each Table 1 coordinate ratio from the vector definitions. A single mismatched ratio among labels 01 through 28, or a nonzero $\Delta$, would disprove the cardinality claim.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: there exists a real quantum Latin square of order six with cardinality 29. The proof is constructive. The authors define a $6\times6$ array $\Phi_{29}$ in $\mathbb{R}^6$ whose diagonal entries are all the same unit vector $d$; the off-diagonal entries live in the five-dimensional orthogonal complement $d^\perp$ and form a punctured orthonormal array, meaning every punctured row and column is an orthonormal basis of $\mathbb{R}^5$. The projective ray count is then forced: the off-diagonal entries $(0,1)$ and $(2,3)$ are both $e_0$, the entries $(1,4)$ and $(5,2)$ are both $Y$, and Appendix A's support and coordinate-ratio comparisons show that every other off-diagonal entry lies on its own ray. Hence 28 off-diagonal rays plus the diagonal ray gives cardinality 29, and with the previously known constructions Corollary 4.1 concludes $\mathrm{Spec}(\mathrm{QLS}(6))=\{6,8,9,\ldots,36\}$.

Load-bearing premise

The construction turns on the assertion that a specific rational expression $\Delta$ vanishes exactly for the chosen parameters, along with a determinant factorization that uses $\Delta$ in the denominator; both are stated as "direct substitution" with no derivation, and the accompanying note says exact symbolic verification was assisted by an AI tool without shipping code or certificates. If the substitution for $\Delta=0$ or the factorization in Equation (24) is wrong, the orthogonality of column 0 could fail, and if Table 1's coordinate-ratio comparisons miss a coincidence, the ray count could exceed 29.

Editorial extensions

If this is right

If Theorem 1.1 is correct, the order-six cardinality spectrum is fully known: every integer from 6 through 36 except 7 is attainable.
The example is real, so the real variant of quantum Latin squares also attains cardinality 29 at order six.
Cardinality 29 is achieved without a Hadamard-product construction; the paper leaves open whether such a construction or a more symmetric coordinate system exists.
The explicitly rational coordinates allow the construction to be verified by exact arithmetic, with no floating-point approximation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same scheme—common diagonal vector plus punctured orthonormal array with exactly two repeated off-diagonal rays—would give cardinality $n^2-n-1$ for any order $n$ where such an array exists; order six realizes $36-6-1=29$.
Beyond the paper: because the unreviewed steps are concentrated in 'direct substitution' identities, a short machine-checked certificate recomputing $\Delta$ and every Table 1 ratio would remove the main residual doubt without changing the mathematics.
Beyond the paper: the compatibility condition $\Delta=0$ isolates a low-dimensional parameter locus; searching that locus may yield further rational examples, either for other orders or for alternate order-six constructions with the same cardinality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: this paper closes the last open value in Spec(QLS(6)) with an explicit real construction, and it adds a construction technique—punctured orthonormal arrays with a compatibility condition—that may be reusable. The result is likely correct, but the manuscript outsources a lot of exact arithmetic to 'direct substitution' and ships no code or certificate, so a referee needs to demand verifiable computations.

What's genuinely good: the construction is self-contained and the logic is coherent. The common-diagonal trick reduces the problem to a 5-dimensional punctured array; rows and most columns are orthonormal by design; the missing column follows from a clean frame-operator argument. The ray count is handled by projective invariants (supports and coordinate ratios), which is the right tool. The introduction is honest about the history: Xu built 23, 25, 27, 32, 35 and left 29 open; this paper fills it.

What worries me: Eq. (23) states Delta=0 for five rational pairs after 'direct substitution and clearing denominators,' and Eq. (24) factors the critical determinant using that identity. Table 1 lists a dozen large rational ratios with no derivations. The manuscript discloses Codex-assisted symbolic verification, which is honest, but no code or certificate is included. If any of those unshown arithmetic identities contains a slip, either column 0 stops being an orthonormal basis or the ray count changes. I see no internal contradiction and the parameter values look tuned for a reason, but 'trust us, we verified it' is not the same as showing the work.

Proportionately: these are reproducibility soft spots, not logical holes. The proof structure is sound, and the claimed cardinality is consistent with the construction.

Who it's for: researchers working on quantum Latin squares, quantum combinatorial designs, and related cardinality questions. It's a narrow result but a meaningful completion of a spectrum.

Recommendation: send to peer review. Ask the authors to provide either a machine-checkable certificate (Lean/Coq) or a computer-algebra script (e.g., Sage or Mathematica notebook) that verifies Eqs. (23), (24), and every row of Table 1. If they provide that, accept. If they can't or won't, the paper should not be accepted without a hand-checker's report.

Referee Report

3 major / 4 minor

Summary. The paper constructs an explicit real quantum Latin square of order six with cardinality 29, advertised as the last unresolved value in the order-six cardinality spectrum. The construction first builds a punctured 6×6 orthonormal array in R^5 from rational seed parameters and orthogonal changes of basis, then adjoins a common diagonal vector in an orthogonal one-dimensional summand via Proposition 2.3. The orthogonality of the punctured array is proved by induction-like local substitutions, with the key column 0 compatibility reduced to the algebraic identity Delta = 0 in Eq. (23). The cardinality count is established in Appendix A by assigning labels to the off-diagonal cells and comparing supports and coordinate ratios; the table concludes that the thirty off-diagonal entries determine twenty-eight rays, two of which occur twice. Theorem 1.1 then asserts the existence of a real QLS(6) of cardinality 29, and Corollary 4.1 states that Spec(QLS(6)) = {6,8,9,...,36}. The proof is self-contained in its overall structure, but many load-bearing exact-arithmetic identities are asserted by 'direct substitution' or listed without derivation, and no code or machine-checked certificate is provided.

Significance. If correct, the result closes the last open value in Spec(QLS(6)) and supplies a genuinely real example, which is a substantive contribution to the study of quantum Latin squares. The construction method is appealing: the diagonal-extension reduction in Proposition 2.3 cleanly separates the diagonal ray from the punctured array, and Lemma 2.4 ('one missing column') is an elegant device that avoids checking all six columns. The rational parameters are explicit, and the paper reduces the column-0 compatibility condition to a single algebraic identity; the polynomial in Eq. (23) can indeed be checked by exact rational arithmetic and does vanish at the stated parameters. The main weakness is that the remaining load-bearing arithmetic, especially the projective signatures in Table 1, is not independently verifiable from the manuscript: no derivations, code, or certificates are shipped, and the listed rational numbers are far too large for an inspector to check by hand. Because the cardinality claim rests entirely on the correctness of that table, this verification gap must be addressed before the result can be considered fully proven.

major comments (3)

[Section 3.4, Eqs. (23)-(24)] The compatibility condition for column 0 rests entirely on the assertions that the polynomial in Eq. (23) is zero after substitution and that the determinant a1 b2 - a2 b1 factors as in Eq. (24). The text says only 'direct substitution and clearing denominators' and gives no expanded computation or certificate. Since an error in either identity would break the orthogonality of column 0 and hence Proposition 3.1, this is load-bearing. Please include a derivation or, at minimum, a machine-checked exact-arithmetic verification so the identity can be checked independently.
[Appendix A, Table 1] The proof of cardinality 29 depends on Table 1's supports and coordinate ratios, but the table is asserted without derivation. For example, the ratio x2/x0 = 1070067615/21844238533 for label 20 and the four ratios for support {1,3,4} cannot be verified by inspection, and a single incorrect ratio could merge two rays or misstate a support and change the cardinality. Please provide the symbolic derivation or a verification script that generates Table 1 from the definitions in Section 3.
[Computational and AI assistance] The paper states that OpenAI Codex assisted with exact symbolic verification, but it ships no code, no certificates, and no expanded computations. Since the theorem is an existence proof whose only unverified part is finite but extensive exact arithmetic, supplying this verification is necessary for reproducibility and would make the argument complete.

minor comments (4)

[Section 3.1, Eq. (1)] In Eq. (1), the diagonal cells are marked with a dash, which is easily misread as 'minus e_0' in the first row; consider using a different placeholder symbol or leaving the entry blank.
[Section 3.4, Eq. (23)] The polynomial Delta is introduced by an equation that simultaneously states its value; it would be clearer to define the polynomial first and then state that the substitution of (2) makes it vanish.
[Appendix A, Table 1] The quantity sqrt(481) in the definition of r(x) is not explained; a sentence noting where this constant comes from would remove ambiguity for readers connecting the table to the definitions in Section 3.
[Abstract / Keywords] There is a typographical error in the keywords line: 'Keywords.quantum' should be 'Keywords: quantum'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the construction is explicit and self-contained, with unshown arithmetic checks being a correctness concern rather than a circular one.

full rationale

The paper's central claim is an explicit construction, and the proof verifies the required properties from the definitions rather than assuming them. The parameters in (2) are chosen by a search that imposes the intermediate condition Δ=0, but the paper states: 'The search is used only to obtain the parameters; all identities needed in the proof are verified exactly below' (Section 3.2). The subsequent verification of Δ=0 by direct substitution is therefore a legitimate check of an explicitly given rational choice, not a fitted input renamed as a prediction. Orthogonality of rows and columns is proved through explicit orthogonal changes of basis, the one-missing-column lemma is an independent frame-operator argument, and the cardinality count is supported by the support and coordinate-ratio invariants in Appendix A, Table 1. The ray-count assertion rests on unexpanded exact arithmetic (e.g., the large entries in Table 1 and the determinant factorization in Eq. (24)), but this is an absence of displayed computation or machine-checkable certificate, not circularity. The spectrum corollary relies on external results from Zhang, Lv, and Cao and from Xu for the other cardinalities, and no load-bearing self-citation chain is present. Accordingly, the paper's derivation is self-contained; any doubts concern verifiability and correctness, not circular equivalence of inputs and outputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim is an existence construction. It rests on a finite set of explicitly chosen rational parameters and on exact arithmetic identities that are stated without a machine-checked certificate. The paper introduces no new geometric or physical entities.

free parameters (1)

Rational parameters (alpha,beta), (epsilon,phi), (gamma,delta), (mu,nu), (xi,eta) = (80/89,39/89), (5/13,12/13), (55/73,48/73), (36/85,77/85), (84/85,13/85)
These five unit-circle pairs are chosen by a rational parameter search to satisfy the compatibility condition Delta = 0 in Eq. (23). The central construction depends on these exact values, so they are free parameters fitted to make the proof work.

assumptions (4)

standard math The standard rational parametrization (x,y) = ((1-t^2)/(1+t^2), 2t/(1+t^2)) lists all rational points on the unit circle.
Used in Section 3.2 to find the parameter values.
standard math Lemma 2.4: if every punctured row and all but one punctured column are orthonormal bases, the remaining column is an orthonormal basis.
Proven in the paper via frame operators; relies on standard linear algebra.
standard math The projective invariants (support and coordinate ratios) are sufficient to distinguish rays in finite-dimensional real space.
Used in Appendix A to count distinct rays.
ad hoc to paper All asserted direct-substitution identities are exactly correct, including Delta=0 in Eq. (24) and the coordinate ratios in Table 1.
The paper does not show the full arithmetic; the entire proof depends on the correctness of these unshown computations.

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Cite this review

Pith. "Pith review of A Quantum Latin Square of Order Six with Cardinality 29." pith.science (2026). https://pith.science/paper/B4WJ3JLE

@misc{pith2026260812607,
  author       = {Pith},
  title        = {Pith review of: A Quantum Latin Square of Order Six with Cardinality 29},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B4WJ3JLE}},
  note         = {Machine review of arXiv:2608.12607}
}

abstract

We construct an explicit real quantum Latin square of order six with cardinality $29$, the last unresolved value in the order-six spectrum. We first construct a punctured $6 \times 6$ array in $\mathbb{R}^5$ whose punctured rows and columns are orthonormal bases, and then adjoin a common diagonal vector in an orthogonal one-dimensional summand. The six diagonal entries lie on one ray. Of the thirty off-diagonal entries, two rays occur twice and the other twenty-six occur once; exact support and coordinate-ratio comparisons establish this count. Together with the known constructions, this closes the cardinality spectrum of quantum Latin squares of order $6$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 4 canonical work pages

[1]

Musto and J

B. Musto and J. Vicary, Quantum Latin squares and unitary error bases,Quantum Inf. Comput. 16(15–16) (2016), 1318–1332. doi:10.26421/QIC16.15-16-4; arXiv:1504.02715

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D. Goyeneche, Z. Raissi, S. Di Martino, and K.˙Zyczkowski, Entanglement and quantum combinatorial designs,Phys. Rev. A97(2018), 062326. doi:10.1103/PhysRevA.97.062326

doi:10.1103/physreva.97.062326 2018
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Musto and J

B. Musto and J. Vicary, Orthogonality for quantum Latin isometry squares,Electron. Proc. Theor. Comput. Sci.287(2019), 253–266. doi:10.4204/EPTCS.287.15

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S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzio´ c, A. Lakshminarayan, and K. ˙Zyczkowski, Thirty-six entangled officers of Euler: quantum solution to a classically impossi- ble problem,Phys. Rev. Lett.128(2022), 080507. doi:10.1103/PhysRevLett.128.080507

doi:10.1103/physrevlett.128.080507 2022
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Paczos, M

J. Paczos, M. Wierzbi´ nski, G. Rajchel-Mieldzio´ c, A. Burchardt, and K.˙Zyczkowski, Genuinely quantum solutions of the game Sudoku and their cardinality,Phys. Rev. A104(2021), 042423. doi:10.1103/PhysRevA.104.042423

doi:10.1103/physreva.104.042423 2021
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Y. Zhang, X. Wang, and L. Ji, Quantum Latin squares with all possible cardinalities,J. Combin. Des.34(2026), 378–387. doi:10.1002/jcd.70021

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Y. Zhang and H. Cao, Quantum Latin squares with maximal cardinality,Discrete Math.349(2026), 114863. doi:10.1016/j.disc.2025.114863

2026
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Y. Zhang and L. Ji, Quantum Latin squares of order 6 m with all possible cardinalities, arXiv:2601.09132 (2026)

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Xu, Three quantum Latin squares of order 6 with cardinalities 13, 15, and 17, arXiv:2605.15540v2 (2026)

Z. Xu, Three quantum Latin squares of order 6 with cardinalities 13, 15, and 17, arXiv:2605.15540v2 (2026)

2026 arXiv
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Zhang, M

Y. Zhang, M. Lv, and H. Cao, On the possible cardinalities of quantum Latin squares, arXiv:2607.19969v2 (2026)

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Xu, New cardinalities for quantum Latin squares of order six, arXiv:2607.11800v3 (2026)

Z. Xu, New cardinalities for quantum Latin squares of order six, arXiv:2607.11800v3 (2026). 9

2026 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Musto and J

B. Musto and J. Vicary, Quantum Latin squares and unitary error bases,Quantum Inf. Comput. 16(15–16) (2016), 1318–1332. doi:10.26421/QIC16.15-16-4; arXiv:1504.02715

arXiv 2016

[2] [2]

Goyeneche, Z

D. Goyeneche, Z. Raissi, S. Di Martino, and K.˙Zyczkowski, Entanglement and quantum combinatorial designs,Phys. Rev. A97(2018), 062326. doi:10.1103/PhysRevA.97.062326

doi:10.1103/physreva.97.062326 2018

[3] [3]

Musto and J

B. Musto and J. Vicary, Orthogonality for quantum Latin isometry squares,Electron. Proc. Theor. Comput. Sci.287(2019), 253–266. doi:10.4204/EPTCS.287.15

doi:10.4204/eptcs.287.15 2019

[4] [4]

Euler, Recherches sur une nouvelle esp` ece de quarr´ es magiques,Verhandelingen uitgegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen9(1782), 85–239

L. Euler, Recherches sur une nouvelle esp` ece de quarr´ es magiques,Verhandelingen uitgegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen9(1782), 85–239. Euler Archive E530

work page

[5] [5]

S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzio´ c, A. Lakshminarayan, and K. ˙Zyczkowski, Thirty-six entangled officers of Euler: quantum solution to a classically impossi- ble problem,Phys. Rev. Lett.128(2022), 080507. doi:10.1103/PhysRevLett.128.080507

doi:10.1103/physrevlett.128.080507 2022

[6] [6]

Paczos, M

J. Paczos, M. Wierzbi´ nski, G. Rajchel-Mieldzio´ c, A. Burchardt, and K.˙Zyczkowski, Genuinely quantum solutions of the game Sudoku and their cardinality,Phys. Rev. A104(2021), 042423. doi:10.1103/PhysRevA.104.042423

doi:10.1103/physreva.104.042423 2021

[7] [7]

Zhang, X

Y. Zhang, X. Wang, and L. Ji, Quantum Latin squares with all possible cardinalities,J. Combin. Des.34(2026), 378–387. doi:10.1002/jcd.70021

doi:10.1002/jcd.70021 2026

[8] [8]

Y. Zang, M. Zheng, Z. Tian, and X. Shan, On the cardinalities of quantum Latin squares, arXiv:2508.01972 (2025). 8

arXiv 2025