{"id":"5712eeba-0bc7-4a93-b15a-24857f774937","arxiv_id":"2607.12933","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A set of n−2 mutually orthogonal quantum Latin squares of order n must be classical, and for prime powers q the paper constructs d−1 of them, one non-classical, whenever d>1 divides q−1.","lead":"This paper studies quantum Latin squares, grids of quantum states in which every row and column is a basis. It proves a new upper limit on how many such grids can be mutually compatible and builds large new examples for prime-power sizes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central bound rests on Lemma 10 from an unpublished preprint; proof sketch omits the key orthogonality-preservation case, but no concrete counterexample found.","rationale":"The reader correctly identified Lemma 10 as the weakest assumption. I examined the paper's central proof and could not find a concrete flaw in Lemma 10's operation; the sketch appears fillable, and the case analysis for orthogonal supports is likely correct. However, the proof is not self-contained and relies on an unpublished preprint, so the concern is a verification gap rather than a demonstrated error. The remainder of Theorem 5's proof and the Section 4 constructions are mathematically plausible and consistent. Since no load-bearing flaw was established, the reader's ACCEPT verdict need not change, though a full proof or computational test of Lemma 10 would remove the residual risk.","tokens_in":8148,"tokens_out":32906,"duration_ms":282881,"concrete_test":"Give a complete proof of Lemma 10 by enumerating all possible supports of a weight-≤2 vector that can be orthogonal to a weight-2 vector with support {1,2}; show that the only possibilities are exactly {1,2} or supports disjoint from {1,2}. Then prove that applying U to every pattern-110...0 entry preserves the row/column orthonormal-basis property. As an independent computational check, for n=4,5,6 enumerate all quantum Latin squares without entries of weight ≥3 (or sample random ones) and verify that the described iterative classicalization can be carried out while preserving orthogonality to every orthogonal partner. If any counterexample emerges, Theorem 5 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5's upper bound M(n)≤n−3 hinges entirely on Lemma 10, quoted from the authors' own unpublished preprint [2, Lemma 13]. The proof in §3 is only a sketch. The critical step is the assertion that applying a single unitary U to every entry with pattern 110...0 preserves orthogonality within the square and with all orthogonal partners. The sketch claims the only dangerous case is when one entry has pattern 110...0 and the other has either the same support {1,2} or a disjoint support, but it does not justify why an orthogonal entry cannot have a support that partially overlaps {1,2} (e.g., {1,3} or {2,4}) while still being orthogonal because of coefficient cancellation. It also does not explicitly verify that rows and columns remain orthonormal bases after transforming multiple entries simultaneously. If Lemma 10 is false, the reduction of a set of n−2 non-classical MOQLS to a classical set fails, and Theorem 5 collapses. The rest of the proof of Theorem 5 is logically sound: the weight-≤2 bound from rules #2 and #3 is correct, the argument finding two additional weight-2 entries with the same support works, and the final contradiction via the extended classical MOLS is valid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8472,"tokens_out":33445,"duration_ms":270554,"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":[{"comment":"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).","section":"§3, Lemma 10"},{"comment":"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.","section":"§3, Theorem 9"}],"minor_comments":[{"comment":"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.","section":"§5, Problem 17"},{"comment":"In the displayed patterns, the notation '0*1*...*' is hard to parse; clarify that * denotes an arbitrary bit and specify the coordinate indexing used.","section":"§3, Lemma 10 proof"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the main technical concern is not the validity of the constructions but the self-containedness of the upper-bound proof. Lemma 10 is taken from an unpublished preprint by the same authors; if [2] is not yet accepted or publicly available, the proof of Theorem 5 is not complete in this manuscript. The statement of Shrikhande's theorem also appears to have a typo. I would recommend asking for a full proof of Lemma 10 and a corrected Theorem 9 before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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].\n\nOther 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":8912,"tokens_out":10604,"would_cite":true,"duration_ms":93594,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B15","81P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum Latin square","mutually orthogonal quantum Latin squares","non-classical","Frobenius ring","finite field","directions determined by a function","complete mappings","upper bounds"],"falsifier":"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.","tokens_in":8081,"feed_emoji":"🧩","tokens_out":9541,"duration_ms":88175,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quantum Latin sets of size n−2 are always classical","feed_subtitle":"Largest non-classical set drops to n−3; prime-power constructions yield d−1 squares","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["For quantum Latin squares, size n−2 forces classicality","Max non-classical MOQLS size is n−3","Prime powers give n−3 non-classical MOQLS","Quantum Latin squares: n−2 classical, n−3 possible"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["For quantum Latin squares, size n−2 forces classicality","Max non-classical MOQLS size is n−3","Prime powers give n−3 non-classical MOQLS","Quantum Latin squares: n−2 classical, n−3 possible"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001954,"raw_usage":{"total_tokens":7391,"prompt_tokens":572,"completion_tokens":6819,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":316,"completion_tokens_details":{"reasoning_tokens":6756}},"tokens_in":316,"tokens_out":6819,"duration_ms":57210,"temperature":1.0,"reasoning_tokens":6756,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T06:16:27.185226+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}