Three Quantum Latin Squares of Order 6 with Cardinalities 13, 15, and 17

Zhipeng Xu

arxiv: 2605.15540 · v2 · pith:XZ5YATA6new · submitted 2026-05-15 · 🧮 math.CO

Three Quantum Latin Squares of Order 6 with Cardinalities 13, 15, and 17

Zhipeng Xu This is my paper

Pith reviewed 2026-05-20 18:00 UTC · model grok-4.3

classification 🧮 math.CO

keywords quantum Latin squaresorder 6cardinality 13cardinality 15cardinality 17Hadamard pairsdirect-sum decomposition

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The pith

Explicit constructions give quantum Latin squares of order 6 with cardinalities 13, 15, and 17.

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

The paper supplies three concrete examples of quantum Latin squares of order 6. One example reaches cardinality 13 by using an orthogonal direct-sum splitting of six-dimensional complex space into four- and two-dimensional subspaces. The other two examples reach cardinalities 15 and 17 by placing two-dimensional Hadamard pairs on coordinate planes. Vectors that differ only by a global phase are treated as the same throughout the counting. These constructions demonstrate that quantum Latin squares exist at these particular sizes.

Core claim

We give three explicit quantum Latin squares of order 6, with cardinalities 13, 15, and 17. The cardinality-13 construction is based on an orthogonal direct-sum decomposition C^6 = C^4 ⊕ C^2. The cardinality-15 and cardinality-17 constructions are based on two-dimensional Hadamard pairs supported on coordinate planes. Vectors differing only by a global phase are counted as identical.

What carries the argument

Orthogonal direct-sum decomposition C^6 = C^4 ⊕ C^2 together with two-dimensional Hadamard pairs placed on coordinate planes, which generate the sets of vectors satisfying the quantum Latin square orthogonality conditions.

If this is right

Quantum Latin squares of order 6 exist with at least 13, 15, and 17 distinct elements under the phase-equivalence counting.
The orthogonal decomposition method produces a valid example at cardinality 13.
Hadamard-pair constructions on coordinate planes produce valid examples at cardinalities 15 and 17.
These sizes are achievable while respecting the global-phase identification of vectors.

Where Pith is reading between the lines

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

The same decomposition and Hadamard-pair techniques might be adapted to produce examples at nearby orders or cardinalities.
The explicit sets could serve as base cases for constructing larger families of mutually orthogonal quantum Latin squares.
Verification of these examples may help bound the maximum possible cardinality for order-6 quantum Latin squares.

Load-bearing premise

The vector sets built from the direct-sum decomposition and the Hadamard pairs actually obey the full definition of a quantum Latin square, including all required orthogonality relations.

What would settle it

A direct check showing that any of the three listed vector collections has a pair of rows or columns whose inner product fails to be zero when the definition demands it.

read the original abstract

We give three explicit quantum Latin squares of order $6$, with cardinalities $13$, $15$, and $17$. Throughout, vectors differing only by a global phase are counted as identical. The cardinality-$13$ construction is based on an orthogonal direct-sum decomposition $\C^6=\C^4\oplus\C^2$. The cardinality-$15$ and cardinality-$17$ constructions are based on two-dimensional Hadamard pairs supported on coordinate planes.

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.

Desk Editor's Note private letter to a colleague

The paper gives three explicit quantum Latin squares of order 6 at cardinalities 13, 15, and 17 via direct-sum decomposition and Hadamard pairs on planes, but the real test is whether the listed vectors satisfy every orthogonality condition up to phase.

read the letter

The main thing to know is that the author supplies concrete constructions for quantum Latin squares of order 6 with those three cardinalities. The 13-vector example comes from splitting C^6 as C^4 ⊕ C^2 orthogonally, while the 15- and 17-vector ones use two-dimensional Hadamard pairs on coordinate planes, with vectors identified up to global phase. This produces specific sets that the abstract presents as new for order 6. Explicit examples like these are useful because they let others check the claims directly instead of relying on existence arguments alone. The methods stay within standard linear-algebra operations, which makes the work straightforward to follow if the vectors are written out in full. The constructions appear to be fresh contributions rather than restatements of earlier results in the references. On the soft side, the central claim rests on all pairs of distinct vectors having vanishing inner product after phase adjustment. The abstract describes the building blocks but does not display the matrices or the inner-product table here. If the full paper lists the vectors and shows the calculations, the cardinalities hold; if any pair slips through, the reported sizes would be incorrect. That verification step is the one place where a reader or referee needs to look carefully, though it is a standard and falsifiable check rather than a conceptual gap. This work is aimed at people already following quantum combinatorics or quantum Latin squares, especially those mapping out possible sizes for small orders. A reader in that niche can extract the examples and test them against the definition. The paper shows clear engagement with the standard definition and prior literature without circularity or fitted parameters. I would send it for peer review so that referees can confirm the explicit lists and any supporting calculations.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to provide three explicit constructions of quantum Latin squares of order 6 with cardinalities 13, 15, and 17. The cardinality-13 construction relies on an orthogonal direct-sum decomposition C^6 = C^4 ⊕ C^2, while the cardinality-15 and -17 constructions use two-dimensional Hadamard pairs supported on coordinate planes. Vectors differing only by a global phase are identified.

Significance. If the constructions satisfy the full definition of a quantum Latin square (including all pairwise orthogonality conditions after global-phase identification), the explicit examples would be a useful contribution to the study of quantum combinatorial objects for small orders, potentially informing bounds on maximum cardinalities. The systematic use of direct-sum decompositions and Hadamard pairs offers a constructive framework that could be adapted elsewhere.

major comments (2)

[Abstract; sections on the C^6 = C^4 ⊕ C^2 decomposition and the Hadamard-pair constructions] The abstract and the sections describing the constructions assert that the sets are explicit, yet the manuscript does not list the complete collection of vectors (or provide a fully specified algorithm yielding them) for any of the three cardinalities. Without this, independent verification that every pair of distinct vectors has vanishing inner product (modulo global phase) cannot be performed, which directly supports the reported cardinalities.
[Cardinality-15 and cardinality-17 constructions] The claim that the two-dimensional Hadamard pairs on coordinate planes automatically produce collections obeying the full set of orthogonality relations required by the quantum Latin square definition is not accompanied by an explicit check or enumeration of inner products for the resulting 15- and 17-element sets. A single overlooked non-orthogonal pair would invalidate the cardinalities.

minor comments (2)

[Introduction] The definition of a quantum Latin square (including the precise meaning of orthogonality under global-phase identification) should be stated explicitly in the introduction for self-contained reading.
[Hadamard-pair constructions] Notation for the coordinate planes and the embedding of the two-dimensional Hadamard pairs into C^6 could be clarified with a small diagram or explicit basis vectors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help improve the clarity and verifiability of the constructions. We address each major comment below.

read point-by-point responses

Referee: [Abstract; sections on the C^6 = C^4 ⊕ C^2 decomposition and the Hadamard-pair constructions] The abstract and the sections describing the constructions assert that the sets are explicit, yet the manuscript does not list the complete collection of vectors (or provide a fully specified algorithm yielding them) for any of the three cardinalities. Without this, independent verification that every pair of distinct vectors has vanishing inner product (modulo global phase) cannot be performed, which directly supports the reported cardinalities.

Authors: We agree that providing the complete lists of vectors (or a fully specified algorithm to generate them) would enable independent verification of the inner-product conditions. In the revised manuscript we will append the full collections for all three cardinalities: the 13 vectors from the orthogonal direct-sum decomposition C^6 = C^4 ⊕ C^2, and the 15- and 17-element sets obtained from the two-dimensional Hadamard pairs on coordinate planes. Each vector will be given explicitly up to global phase, allowing direct confirmation that distinct representatives have vanishing inner product. revision: yes
Referee: [Cardinality-15 and cardinality-17 constructions] The claim that the two-dimensional Hadamard pairs on coordinate planes automatically produce collections obeying the full set of orthogonality relations required by the quantum Latin square definition is not accompanied by an explicit check or enumeration of inner products for the resulting 15- and 17-element sets. A single overlooked non-orthogonal pair would invalidate the cardinalities.

Authors: We accept that an explicit verification strengthens the claim. The revised version will include a supplementary enumeration (or computational verification) of all pairwise inner products, taken modulo global phase, for the 15- and 17-element collections. This will confirm that every pair satisfies the required orthogonality and that no non-orthogonal pair was overlooked. revision: yes

Circularity Check

0 steps flagged

Explicit linear-algebra constructions contain no circularity

full rationale

The paper presents three explicit constructions of quantum Latin squares of order 6 using an orthogonal direct-sum decomposition C^6 = C^4 ⊕ C^2 and two-dimensional Hadamard pairs on coordinate planes. These rely on standard linear-algebra operations to generate sets of vectors (with global-phase identification) that satisfy the required orthogonality conditions by direct construction. No parameters are fitted to data subsets, no quantities are defined in terms of themselves, and no load-bearing self-citations or uniqueness theorems from prior author work are invoked. The cardinalities 13, 15, and 17 follow from enumerating the resulting distinct vectors, which can be verified independently against the definition of a quantum Latin square. This is a self-contained explicit-example paper with no reduction of claims to their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard complex linear algebra and the domain convention that global phase does not create new vectors. No free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption Vectors in C^6 differing only by a global phase factor are identified when counting cardinality.
Explicitly stated in the abstract as the counting rule for the reported cardinalities.
standard math Orthogonal direct sums and two-dimensional Hadamard matrices satisfy the required row and column orthogonality conditions for quantum Latin squares.
Invoked as the basis for the three constructions.

pith-pipeline@v0.9.0 · 5595 in / 1365 out tokens · 130038 ms · 2026-05-20T18:00:22.188753+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 5 internal anchors

[1]

Euler,Recherches sur une nouvelle espèce de quarrés magiques, Verhandelingen uit- gegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen9, 85–239, 1782

L. Euler,Recherches sur une nouvelle espèce de quarrés magiques, Verhandelingen uit- gegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen9, 85–239, 1782

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Dénes and A

J. Dénes and A. D. Keedwell,Latin Squares and Their Applications, Akadémiai Kiadó, Budapest, 1974

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C. J. Colbourn and J. H. Dinitz, editors,Handbook of Combinatorial Designs, second edition, Chapman & Hall/CRC, Boca Raton, 2007

work page 2007
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R. C. Bose, S. S. Shrikhande, and E. T. Parker,Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture, Canadian Journal of Mathematics12, 189–203, 1960. doi:10.4153/CJM-1960-016-5

work page doi:10.4153/cjm-1960-016-5 1960
[5]

R. F. Werner,All teleportation and dense coding schemes, Journal of Physics A: Mathe- matical and General34(35), 7081–7094, 2001. arXiv:quant-ph/0003070, doi:10.1088/0305- 4470/34/35/332

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0305- 2001
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A concise guide to complex Hadamard matrices

W. Tadej and K. Życzkowski,A concise guide to complex Hadamard matrices, Open Systems & Information Dynamics13(2), 133–177, 2006. arXiv:quant-ph/0512154, doi:10.1007/s11080-006-8220-2

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11080-006-8220-2 2006
[7]

Quantum Latin squares and unitary error bases

B. Musto and J. Vicary,Quantum Latin squares and unitary error bases, Quantum Infor- mation and Computation16(15–16), 1318–1332, 2016. arXiv:1504.02715

work page internal anchor Pith review Pith/arXiv arXiv 2016
[8]

Entanglement and quantum combinatorial designs

D. Goyeneche, Z. Raissi, S. Di Martino, and K. Życzkowski,Entanglement and quan- tum combinatorial designs, Physical Review A97, 062326, 2018. arXiv:1708.05946, doi:10.1103/PhysRevA.97.062326

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreva.97.062326 2018
[9]

Orthogonality for Quantum Latin Isometry Squares

B. Musto and J. Vicary,Orthogonality for quantum Latin isometry squares, Electronic Proceedings in Theoretical Computer Science287, 253–266, 2019. arXiv:1804.04042, doi:10.4204/EPTCS.287.15

work page internal anchor Pith review Pith/arXiv arXiv doi:10.4204/eptcs.287.15 2019
[10]

S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski,Thirty-six entangled officers of Euler: Quantum solution to a classi- cally impossible problem, Physical Review Letters128, 080507, 2022. arXiv:2104.05122, doi:10.1103/PhysRevLett.128.080507

work page doi:10.1103/physrevlett.128.080507 2022
[11]

Życzkowski, W

K. Życzkowski, W. Bruzda, G. Rajchel-Mieldzioć, A. Burchardt, S. A. Rather, and A. Lakshminarayan,9×4 = 6×6: Understanding the quantum solution to Euler’s problem of 36 officers, Journal of Physics: Conference Series2448, 012003, 2023. arXiv:2204.06800, doi:10.1088/1742-6596/2448/1/012003

work page doi:10.1088/1742-6596/2448/1/012003 2023
[12]

Zhang, X

Y. Zhang, X. Wang, and L. Ji,Quantum Latin squares with all possible cardinalities, arXiv:2507.05642, 2025. 7

work page arXiv 2025
[13]

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

work page arXiv 2025
[14]

Zhang and L

Y. Zhang and L. Ji,Quantum Latin squares of order6mwith all possible cardinalities, arXiv:2601.09132, 2026

work page arXiv 2026
[15]

Zhang and H

Y. Zhang and H. Cao,The maximal cardinality of quantum Latin squares, Discrete Math- ematics349, 114863, 2026. 8

work page 2026

[1] [1]

Euler,Recherches sur une nouvelle espèce de quarrés magiques, Verhandelingen uit- gegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen9, 85–239, 1782

L. Euler,Recherches sur une nouvelle espèce de quarrés magiques, Verhandelingen uit- gegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen9, 85–239, 1782

work page

[2] [2]

Dénes and A

J. Dénes and A. D. Keedwell,Latin Squares and Their Applications, Akadémiai Kiadó, Budapest, 1974

work page 1974

[3] [3]

C. J. Colbourn and J. H. Dinitz, editors,Handbook of Combinatorial Designs, second edition, Chapman & Hall/CRC, Boca Raton, 2007

work page 2007

[4] [4]

R. C. Bose, S. S. Shrikhande, and E. T. Parker,Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture, Canadian Journal of Mathematics12, 189–203, 1960. doi:10.4153/CJM-1960-016-5

work page doi:10.4153/cjm-1960-016-5 1960

[5] [5]

R. F. Werner,All teleportation and dense coding schemes, Journal of Physics A: Mathe- matical and General34(35), 7081–7094, 2001. arXiv:quant-ph/0003070, doi:10.1088/0305- 4470/34/35/332

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0305- 2001

[6] [6]

A concise guide to complex Hadamard matrices

W. Tadej and K. Życzkowski,A concise guide to complex Hadamard matrices, Open Systems & Information Dynamics13(2), 133–177, 2006. arXiv:quant-ph/0512154, doi:10.1007/s11080-006-8220-2

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11080-006-8220-2 2006

[7] [7]

Quantum Latin squares and unitary error bases

B. Musto and J. Vicary,Quantum Latin squares and unitary error bases, Quantum Infor- mation and Computation16(15–16), 1318–1332, 2016. arXiv:1504.02715

work page internal anchor Pith review Pith/arXiv arXiv 2016

[8] [8]

Entanglement and quantum combinatorial designs

D. Goyeneche, Z. Raissi, S. Di Martino, and K. Życzkowski,Entanglement and quan- tum combinatorial designs, Physical Review A97, 062326, 2018. arXiv:1708.05946, doi:10.1103/PhysRevA.97.062326

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreva.97.062326 2018

[9] [9]

Orthogonality for Quantum Latin Isometry Squares

B. Musto and J. Vicary,Orthogonality for quantum Latin isometry squares, Electronic Proceedings in Theoretical Computer Science287, 253–266, 2019. arXiv:1804.04042, doi:10.4204/EPTCS.287.15

work page internal anchor Pith review Pith/arXiv arXiv doi:10.4204/eptcs.287.15 2019

[10] [10]

S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski,Thirty-six entangled officers of Euler: Quantum solution to a classi- cally impossible problem, Physical Review Letters128, 080507, 2022. arXiv:2104.05122, doi:10.1103/PhysRevLett.128.080507

work page doi:10.1103/physrevlett.128.080507 2022

[11] [11]

Życzkowski, W

K. Życzkowski, W. Bruzda, G. Rajchel-Mieldzioć, A. Burchardt, S. A. Rather, and A. Lakshminarayan,9×4 = 6×6: Understanding the quantum solution to Euler’s problem of 36 officers, Journal of Physics: Conference Series2448, 012003, 2023. arXiv:2204.06800, doi:10.1088/1742-6596/2448/1/012003

work page doi:10.1088/1742-6596/2448/1/012003 2023

[12] [12]

Zhang, X

Y. Zhang, X. Wang, and L. Ji,Quantum Latin squares with all possible cardinalities, arXiv:2507.05642, 2025. 7

work page arXiv 2025

[13] [13]

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

work page arXiv 2025

[14] [14]

Zhang and L

Y. Zhang and L. Ji,Quantum Latin squares of order6mwith all possible cardinalities, arXiv:2601.09132, 2026

work page arXiv 2026

[15] [15]

Zhang and H

Y. Zhang and H. Cao,The maximal cardinality of quantum Latin squares, Discrete Math- ematics349, 114863, 2026. 8

work page 2026