Large sets of mutually orthogonal quantum Latin squares

Robin Simoens; Simeon Ball

REVIEW 2 major objections 2 minor 14 references

Large sets of mutually orthogonal quantum Latin squares

T0 review · 2 major / 2 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves that any set of n−2 mutually orthogonal quantum Latin squares of order n is classical, lowering the upper bound for non-classical sets to n−3, and it constructs sets of d−1 MOQLS(q) for prime powers q when d>1 divides q−1.

desk verdict A solid, checkable advance on MOQLS bounds; the central upper-bound proof leans on an unpublished lemma with a terse sketch, but the argument survives scrutiny. read the letter →

arxiv 2607.12933 v2 pith:DJMSL3CJ submitted 2026-07-14 math.CO quant-ph

Simeon Ball , Robin Simoens This is my paper · ORCID

classification math.COquant-ph MSC 05B1581P70

keywords quantumLatinsquaremutuallyorthogonalsquaresnon-classicalFrobeniusringfinitefielddirectionsdeterminedbyafunctioncompletemappingsupperbounds

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

Quantum Latin squares are the quantum-mechanical generalization of Latin squares: each row and column is an orthonormal basis of C^n. This paper asks how many such squares can be mutually orthogonal in a single set while remaining genuinely quantum, and it improves both sides of the answer. It proves that a set of n−2 MOQLS of order n is necessarily classical—so no non-classical set can reach that size, lowering the upper bound for M(n) from n−2 to n−3. On the constructive side, it shows that for every prime power q whose order minus one has a proper divisor d>1, there is a set of d−1 MOQLS(q) with exactly one non-classical member, built from permutations of the field that determine few directions. These improved bounds matter because MOQLS are used in quantum teleportation, dense coding, and quantum combinatorial designs.

What carries the argument

Two pieces of machinery carry the argument. First, the pattern of a quantum Latin square—the binary matrix recording which basis states appear in each entry—plus the 'quantum sudoku' rules: standard form, unitary patterns, and zero overlap. These force severe restrictions on supports in any large mutually orthogonal set: at an off-first-row cell in a set of n−2 squares, rule #2 rules out coordinate j and rule #3 makes the n−2 supports disjoint, so every entry has weight at most 2. The second piece is Lemma 10, a replacement lemma converting squares of weight ≤2 into classical squares while preserving orthogonality and only shrinking supports. For the construction, the central object is the s

What would settle it

Look for a non-classical pair of mutually orthogonal quantum Latin squares of order 4; since n−2 = 2, the theorem says no such pair can exist, so producing one would disprove the upper-bound claim.

Watch

Extended reading notes

Core claim

The paper's central result is a rigidity theorem: for every n≥3, any set of n−2 mutually orthogonal quantum Latin squares of order n is classical, so M(n)≤n−3. The proof starts with a hypothetical non-classical set in standard form; the mutual-orthogonality rules force every off-first-row entry to have support of size at most 2, and Lemma 10 (proved only in outline here, with details in the authors' earlier preprint) replaces each square by a classical one whose supports only shrink while preserving all orthogonality relations. The n−2 classical squares can then be extended to n−1 classical MOLS by a known theorem, and the extension forces a collision that contradicts orthogonality. On the c

Load-bearing premise

The load-bearing premise is Lemma 10 (Section 3, proof sketched, full version deferred to the authors' preprint [2]): a quantum Latin square whose entries never use more than two basis states can be replaced by a classical square that remains orthogonal to every square it was originally orthogonal to; if that lemma fails, the reduction in Theorem 5 collapses and only the weaker previous upper bound n−2 survives.

Editorial extensions

If this is right

No non-classical set of MOQLS can reach size n−2: any set that large is classical, so the largest non-classical set has at most n−3 squares.
For every prime power q with q−1 having a proper divisor d>1, there exist d−1 MOQLS(q), exactly one non-classical, giving explicit lower bounds.
The numerical bounds in Table 2 replace Table 1 for orders up to 17; for instance M(16) lies between 4 and 13 instead of 3 and 14.
The Frobenius-ring construction works for any finite ring with a generating character, not only fields, and reduces to classical MOLS when all functions are affine.
Any set of n−2 MOQLS(n), even if it appears quantum, is isotopic to a classical set—a rigidity statement about the boundary of the problem.

Reading between the lines

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

The upper-bound method likely leaves room: the same pattern-counting argument does not rule out non-classical sets of size n−3, and the authors leave open whether n−3 sets are forced classical; if the technique extends, the true bound may fall further.
Because the construction produces exactly one non-classical square, large sets are for the most part classical; this suggests that non-classicality is a rare, fragile property and that the interesting quantum behavior may live only in small sets or at specific orders.
The direction-counting connection suggests a testable analogy: extremal permutations of finite fields that determine few directions might yield maximal MOQLS sets, and the problem of 3 MOQLS(10) could be approached by looking for a permutation of Z/10Z with no direction m in the required set.
The paper's construction depends on a non-affine permutation with many undetermined directions; such functions over composite orders may be harder to find, so the lower bound for non-prime-power orders could be much poorer—a gap worth investigating.

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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

Short version: this is a solid advance on a narrow problem. Theorem 5 drops the upper bound for large non-classical MOQLS from n−2 to n−3, and Corollary 7 gives the first general lower bounds for prime powers. I read through Theorem 5 with the stress-test worry in mind: the proof leans on Lemma 10, which is taken from the authors' unpublished preprint and proved only as a sketch. The worry was that applying the local unitary to all 110...0 entries might break orthogonality with a partner that has partially overlapping support. That worry does not survive contact with the text. For weight-≤2 vectors, orthogonality forces supports to be either equal or disjoint; partial overlap is impossible. So the transformation preserves each pair's inner product, and rows/columns stay orthonormal because any entry not transformed has support disjoint from {1,2} and is untouched. The lemma is correct, but the paper is too terse: it should state the equal-or-disjoint support fact explicitly and give the full proof or a clear pointer to [2].

Other strengths: the Frobenius-ring approach in Section 4 is a genuine generalization of the Huang–Li construction, and the connection to direction sets in finite fields is a nice, checkable idea. Example 15 works, and the numbers in Table 2 (4 MOQLS(11), 7 MOQLS(17)) follow from the construction. The counting in Theorem 5 is carefully done; I did not find gaps after the support fact is supplied.

The soft spots are minor: the dependence on an unpublished companion paper for a lemma that is central to the upper bound, and a few places where the text says "we can" without showing the intermediate step (e.g., the statement that at position (2,j) the entries cover exactly the stated set). Both are fixable in revision. The self-citation is not a red flag; the lemma is auxiliary and its logic is transparent once filled in.

Who is this for? People who work on quantum Latin squares, MOQLS, or related combinatorial designs. It will not change the world, but it settles a clean problem and poses a natural open one. I would be happy to referee it myself. Recommendation: send it to peer review. Ask the authors to expand Lemma 10 or to make the companion preprint available and cite the lemma number, and to spell out the support argument. That is a revision, not a reject.

Referee Report

2 major / 2 minor

Summary. The paper studies the maximum size M(n) of a non-classical set of mutually orthogonal quantum Latin squares of order n. Its main upper-bound result (Theorem 5) asserts that any set of n−2 MOQLS(n) is classical, giving M(n) ≤ n−3; the proof combines the quantum sudoku rules with a reduction (Lemma 10) of quantum squares without high-weight entries to classical squares, followed by an application of Shrikhande's MOLS extension theorem. The lower-bound part (Theorem 6, Corollary 7) constructs, for finite Frobenius rings and for prime powers q whose q−1 has a proper divisor d>1, a set of d−1 MOQLS(q) with exactly one non-classical member, using functions determining few directions. The paper improves the previously known bounds in Table 2.

Significance. Assuming the two flagged issues are fixed, the paper gives a genuine improvement on both sides of the problem: it closes the possibility of n−2 non-classical MOQLS and provides new explicit non-classical families of order q, with sizes that grow with the largest proper divisor of q−1. The constructions are concrete and checkable, and the use of Frobenius rings is a natural framework. The main caveats are that the upper bound depends on Lemma 10, whose proof is only sketched and deferred to an unpublished preprint, and on a correct statement of Shrikhande's theorem. These are fixable but load-bearing; the paper is not fully self-contained as it stands.

major comments (2)

[§3, Lemma 10] The central upper bound rests on Lemma 10, quoted from the authors' unpublished preprint [2, Lemma 13]. The proof in §3 is a sketch. It asserts that applying U to every entry with pattern 110...0 preserves orthogonality because any orthogonal entry has either the same support {1,2} or a disjoint support. This dichotomy is true (partial overlap would give a nonzero inner product), but the proof does not state the argument, and more importantly it does not verify explicitly that rows and columns remain orthonormal bases after transforming several entries simultaneously, including the one-changed/one-unchanged case. Since Theorem 5 collapses if Lemma 10 fails, the full proof should be included in this paper (or [2] should be available and the lemma proved in detail).
[§3, Theorem 9] The statement as printed is logically inconsistent: from 'a set of n−3 MOLS can be extended to n−1' it does not follow that 'in particular a set of n−2 MOLS can be extended'—a set of n−2 contains an n−3 subset, but the extension of the subset need not contain the remaining square. The proof of Theorem 5 uses the n−2 extension statement explicitly. Please correct the statement (the intended Shrikhande result appears to be that a set of n−2 MOLS extends to n−1 for n≠4; the n−3 statement is then the consequence). The wording of Problem 17 and reference [4] should also be aligned with the corrected theorem.

minor comments (2)

[§5, Problem 17] The sentence 'It is not known that 7 MOLS(10) do not exist' should be 'It is not known whether 7 MOLS(10) exist'. Also, Bruck–Ryser [4] does not establish the non-existence of a projective plane of order 10; if that fact is intended, the computer proof (e.g., Lam, Thiel, Swiercz, 1989) should be cited.
[§3, Lemma 10 proof] In the displayed patterns, the notation '0*1*...*' is hard to parse; clarify that * denotes an arbitrary bit and specify the coordinate indexing used.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity; minor self-citation in Lemma 10, whose proof sketch is incomplete but not a reduction of the main claim to its inputs.

full rationale

The derivation chain was walked. The upper bound (Theorem 5) depends on Lemma 10, which is quoted from the authors' own preprint [2, Lemma 13]; the proof in §3 is a sketch and does not fully justify the orthogonality-preservation step. This is a load-bearing self-citation, but it is not a circular reduction: Lemma 10 is a distinct technical claim, not a restatement of Theorem 5, and no fitted parameter or data is renamed as a prediction. The lower-bound construction (Theorem 6, Corollary 7) is an explicit construction from standard finite-field direction results (Rédei, Lovász–Schrijver, Blokhuis et al.) and is self-contained. No equation reduces to its own input by construction. Hence no significant circularity; the score of 2 reflects the weak support for Lemma 10, not a circular dependency.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted parameters and no new entities. The construction uses finite-field characters and known direction-counting functions. The only external inputs are Shrikhande's MOLS extension theorem and the authors' prior lemma on classicalizing low-weight quantum Latin squares.

assumptions (3)

standard math Finite fields and modular rings are Frobenius rings with a generating character (Wood [13]).
Used in §2.5 and §4 to define Ψ(f) and to prove Lemmas 8, 12, 13.
standard math Shrikhande's theorem: a set of n−3 (hence n−2) MOLS(n) can be extended to n−1 MOLS(n) for n≠4.
Used in the proof of Theorem 5 (§3) to add a classical square Φ to the n−2 classical MOQLS.
domain assumption Lemma 10 (from the authors' prior work [2]): any QLS with no entry of weight ≥3 can be transformed to a classical QLS preserving orthogonality to any orthogonal QLS.
Central to Theorem 5; proof sketched in §3 but full proof deferred to the self-cited unpublished preprint.

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Pith. "Pith review of Large sets of mutually orthogonal quantum Latin squares." pith.science (2026). https://pith.science/paper/DJMSL3CJ

@misc{pith2026260712933,
  author       = {Pith},
  title        = {Pith review of: Large sets of mutually orthogonal quantum Latin squares},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJMSL3CJ}},
  note         = {Machine review of arXiv:2607.12933}
}

read the original abstract

How large can a set of mutually orthogonal quantum Latin squares (MOQLS) get? We show that a set of n - 2 MOQLS of order n is necessarily classical and construct large non-classical sets of MOQLS of orders that are prime powers, improving both the previously known lower and upper bounds.

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Reference graph

Works this paper leans on

14 extracted references · 2 linked inside Pith

[2]

Ball and R

S. Ball and R. Simoens. Thirty-six quantum officers are entangled, 2026. arXiv:2603.02334

arXiv 2026
[4]

R. H. Bruck and H. J. Ryser. The nonexistence of certain finite projective planes.Canadian Journal of Mathematics, 1:88–93, 1949

1949
[1]

A. S. ´Arnad´ ottir and D. E. Roberson. Group Invariant Quantum Latin Squares, 2025. arXiv:2501.00196

arXiv 2025
[3]

Blokhuis, A

A. Blokhuis, A. E. Brouwer, and T. Sz˝ onyi. The number of directions determined by a functionfon a finite field.J. Combin. Theory Ser. A, 70(2):349–353, 1995

1995
[5]

Y. Han, Y. Zang, H. Zhang, and Z. Tian. The existence of non-classical orthogonal quantum Latin squares, 2025. arXiv:2507.20154

arXiv 2025
[6]

Helwig, W

W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo. Absolute maximal entanglement and quantum secret sharing.Phys. Rev. A, 86:052335, 2012

2012
[7]

Huang and Y

L. Huang and Y. Li. The Existence of Diagonal Quantum Latin Squares with Maximum Cardinality,
[8]

Lov´ asz and A

L. Lov´ asz and A. Schrijver. Remarks on a theorem of R´ edei.Studia Sci. Math. Hungar., 16(3-4):449– 454, 1983

1983

Show all 14 references

[9]

B. Musto. Constructing mutually unbiased bases from quantum Latin squares. InProceedings 13th In- ternational Conference on Quantum Physics and Logic, volume 236 ofElectron. Proc. Theor. Comput. Sci. (EPTCS), pages 108–126, 2017. 7

2017
[10]

Musto.Quantum Latin squares and quantum functions: applications in quantum information

B. Musto.Quantum Latin squares and quantum functions: applications in quantum information. PhD thesis, University of Oxford, 2019

2019
[11]

R´ edei.Lacunary polynomials over finite fields

L. R´ edei.Lacunary polynomials over finite fields. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Translated from German by I. F¨ oldes

1973
[12]

S. S. Shrikhande. A Note on Mutually Orthogonal Latin Squares.Sankhy¯ a: The Indian Journal of Statistics, Series A, 23(2):115–116, 1961

1961
[13]

J. A. Wood. Duality for modules over finite rings and applications to coding theory.Amer. J. Math., 121(3):555–575, 1999

1999
[14]

Y. Zang, P. Facchi, and Z. Tian. Quantum combinatorial designs and k-uniform states.J. Phys. A: Math. Theor., 54(50):505204, 2021. Simeon Ball Department of Mathematics Universitat Polit`ecnica de Catalunya C. Pau Gargallo 14, 08028 Barcelona, Spain E-mail address:simeon.micha...

2021

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2]

Ball and R

S. Ball and R. Simoens. Thirty-six quantum officers are entangled, 2026. arXiv:2603.02334

arXiv 2026

[2] [4]

R. H. Bruck and H. J. Ryser. The nonexistence of certain finite projective planes.Canadian Journal of Mathematics, 1:88–93, 1949

1949

[3] [1]

A. S. ´Arnad´ ottir and D. E. Roberson. Group Invariant Quantum Latin Squares, 2025. arXiv:2501.00196

arXiv 2025

[4] [3]

Blokhuis, A

A. Blokhuis, A. E. Brouwer, and T. Sz˝ onyi. The number of directions determined by a functionfon a finite field.J. Combin. Theory Ser. A, 70(2):349–353, 1995

1995

[5] [5]

Y. Han, Y. Zang, H. Zhang, and Z. Tian. The existence of non-classical orthogonal quantum Latin squares, 2025. arXiv:2507.20154

arXiv 2025

[6] [6]

Helwig, W

W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo. Absolute maximal entanglement and quantum secret sharing.Phys. Rev. A, 86:052335, 2012

2012

[7] [7]

Huang and Y

L. Huang and Y. Li. The Existence of Diagonal Quantum Latin Squares with Maximum Cardinality,

[8] [8]

Lov´ asz and A

L. Lov´ asz and A. Schrijver. Remarks on a theorem of R´ edei.Studia Sci. Math. Hungar., 16(3-4):449– 454, 1983

1983