{"id":"97bd0c61-158f-4082-9b47-52b37fa70978","arxiv_id":"2507.20154","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Non-classical idempotent, mutually orthogonal, and self-orthogonal quantum Latin squares exist for all sufficiently large dimensions, with explicit finite exception lists.","lead":"This paper constructs new kinds of quantum Latin squares, quantum versions of Latin squares whose rows and columns are orthonormal bases, and proves they exist in nearly all dimensions. It gives an existence map for three special families, leaving only a short list of small possible exceptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-classicality of the constructed squares is asserted, not proved; the gap is real but readily closable.","rationale":"The reader's weakest_assumption is exactly the load-bearing gap I find: the paper asserts non-classicality of every constructed square without proving that no global unitary can align all block-local non-computational bases to a single computational basis. The concern is real and it underlies all four existence theorems in the strongest_claim. However, the concern is also easily resolved by a short linear-algebra argument, so it does not warrant rejecting the paper; it does warrant keeping the verdict conditional until the lemma is written down. I considered whether a more serious flaw exists in Construction 2.3's mutual-orthogonality proof, where the text claims two non-zero overlaps force two different intersection basis vectors. That claim is indeed unproved and the written argument is wrong as stated, but the intended conclusion can be repaired using the idempotent-MOQLS property that (|0>,|0>) cannot occur off-diagonal because it already occurs at the diagonal cell. Thus the central constructions appear sound, but they are not yet fully rigorous as written. The reader's CONDITIONAL verdict is therefore the right recommendation, with no change needed from this stress-test pass.","tokens_in":23016,"tokens_out":21680,"duration_ms":218349,"concrete_test":"Re-derive the missing non-classicality lemma for Construction 2.2 in the smallest nontrivial case: v=7 with a PBD block B0={0,1,2} whose block basis is |0>, |+>=(|1>+|2>)/sqrt(2), |->=(|1>-|2>)/sqrt(2). Assume a unitary U maps every entry of the constructed square to a computational-basis vector. Since the square contains entries |1> and |2> (they appear in rows/columns outside B0 through ordinary PBD blocks), U|1>=|a> and U|2>=|b> with |a>,|b> distinct computational basis vectors. Then U|+>=(|a>+|b>)/sqrt(2), which is not a computational basis vector, contradiction. If this one-line argument is formalized and shown to extend unchanged to Constructions 2.3, 2.6, 2.8, and 2.11, the non-classicality gap is closed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every existence theorem in the paper is a statement about non-classical squares, and every construction (Constructions 2.2, 2.3, 2.6, 2.8, 2.11) concludes non-classicality with a single sentence: the square is non-classical because some entries are computational while others come from a non-computational basis of a block subspace. That is not a proof. The equivalence relation for QLS allows an arbitrary global unitary U, so one must rule out the possibility that a single U sends all mixed bases to one computational basis. The claim is very likely true: if U maps the computational basis vectors |i> and |j> to distinct basis vectors |a> and |b>, then it cannot also map (|i>+|j>)/sqrt(2) to a basis vector, since U(|i>+|j>)/sqrt(2) = (|a>+|b>)/sqrt(2). Because the constructions contain both the superposed block vectors and the constituent computational vectors as entries (e.g., rows outside the holes contain the full computational basis, and every non-zero PBD element appears in some block not containing 0), this argument blocks the needed global unitary. But the manuscript never supplies it. The reader identified exactly this as the weakest assumption, and it is load-bearing: if the assertion failed for any construction, the corresponding existence theorem would lose its 'non-classical' content. A secondary issue is that Construction 2.3's proof of mutual orthogonality across different blocks contains an unjustified assertion that two non-zero overlaps would force two distinct intersection vectors; the correct repair uses the fact that any off-diagonal cell with (|0>,|0>) in both idempotent MOQLS entries would duplicate the diagonal pair, which is impossible. This, too, is absent from the text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces idempotent, self-orthogonal, and holey quantum Latin squares and applies combinatorial design tools—pairwise balanced designs, incomplete Latin squares, and conjugate-orthogonal incomplete Latin squares—to construct non-classical mutually orthogonal QLSs. The main claimed results are: a non-classical idempotent QLS(v) for every v≥6 (Theorem 3.3); a non-classical 2-idempotent MOQLS(v) for v≥6 except possibly {6,...,12,14,15,18,19,23} (Theorem 3.4); a non-classical 2-MOQLS(v) for v≥4 except possibly {4,5,6,7} (Theorem 3.7); a non-classical 3-MOQLS(v) for v≥16 (Theorem 3.8); and a non-classical SOQLS(v) for v≥13 (Theorem 3.9). Theorem 3.2 and the appendix assert that no non-classical idempotent QLS exists for v=3,4,5.","tokens_in":23322,"tokens_out":35394,"duration_ms":335218,"significance":"If the proofs are completed, the paper would extend the known existence ranges for non-classical orthogonal quantum Latin squares and add structural properties (idempotence, self-orthogonality) that are new in this context. The PBD and filling-in-holes constructions are natural and potentially reusable, and the explicit exception sets are falsifiable predictions. The paper does not ship machine-checked proofs or code, and it does not fit any parameters; the central constructions are concrete. The significance is moderate: the area has active connections to unitary error bases, mutually unbiased bases, and k-uniform states, but the value of the paper depends on closing the proof gaps below.","major_comments":[{"comment":"The non-classicality of every constructed (and explicitly displayed) square is asserted rather than proved. For example, the last line of Construction 2.2 says the square is non-classical 'since the elements are from computational basis or non-computational basis', and Construction 2.6 says the same because the holes are filled with squares based on non-computational bases. No argument rules out a global unitary U that sends all the mixed block bases to one computational basis. This is load-bearing: every existence theorem in the paper is about non-classical squares, and if the assertion failed for any construction, the corresponding theorem would still produce QLSs but lose the advertised property. A short proof is available: in each construction the classical parts contain the full computational basis {|0>,...,|v−1>} as entries, while at least one entry is a superposition such as (|i>+|j>)/√2; a unitary that maps every computational-basis entry to a basis vector must permute the computational basis, and then the superposition is not sent to a basis vector. The manuscript should state and prove this lemma once and apply it explicitly to each construction and to the displayed squares in Theorems 3.3 and 3.7.","section":"§2, Constructions 2.2, 2.3, 2.6, 2.8, 2.11; Theorems 3.3 and 3.7"},{"comment":"The proof of mutual orthogonality for two different blocks B1≠B2 is not valid as written. The text claims that if ⟨Φfi(x1,y1)|Φfi(x2,y2)⟩≠0 and ⟨Φfj(x1,y1)|Φfj(x2,y2)⟩≠0, then the first overlap is witnessed by an element xm and the second by an element xn with xm≠xn, so span{|xm>,|xn>}⊆LB1∩LB2, contradicting |B1∩B2|≤1. But LB1∩LB2 is a fixed subspace; if it is one-dimensional, both non-zero overlaps can be caused by the same intersection element, and the claimed contradiction does not follow. A correct argument is needed, for example based on the fact that B0-off-diagonal entries are supported on span(B\\{0}) while B'-off-diagonal entries avoid the intersection element. As written, the mutual orthogonality of the t squares is unproved, and Theorem 3.4 depends on it.","section":"Construction 2.3, proof of Eq. (3)"},{"comment":"The v=5 non-existence proof is a long handwritten case analysis with many branches represented by small diagrams. Several branches are dismissed with 'obviously this case is contradictory to the definition of the QLS', and Cases 2–4 are handled only by saying that they can be discussed 'in the way of Case 1' without presenting the details. Since Theorem 3.2(3) is used to assert that no non-classical 2-idempotent MOQLS(5) exists (Theorem 3.4), the appendix needs to be either completed with all subcases or replaced by a machine-verified exhaustive enumeration. If a missing subcase contains a non-classical idempotent QLS(5), the claimed exception set changes.","section":"Appendix A (proof of Theorem 3.2(3))"},{"comment":"Construction 2.5 is stated as: if there exists an HLS(v;v1,...,vn) and an LS(vs) for 1≤s≤n, then there exists a non-classical QLS(v). Taken literally, filling the holes with Latin squares on the computational basis produces a classical QLS, not a non-classical one. The intended statement must be that the hole squares are quantized with non-computational bases, and the non-classicality then needs the same global-unitary proof as in the first major comment. Because Theorem 3.6 and part of Theorem 3.3 invoke Construction 2.5, this imprecision is load-bearing and should be corrected.","section":"Construction 2.5"}],"minor_comments":[{"comment":"The notation '{6−12,14,15,18,19,23}' should be written unambiguously as '{6,7,8,9,10,11,12,14,15,18,19,23}' or with an en-dash; the minus sign is confusing.","section":"Theorem 3.4"},{"comment":"The displayed array in the proof of Lemma 3.1 is garbled; please redraw it so that the positions of |x⟩, |m⟩, |y⟩, |z⟩, |u⟩, and |v⟩ are clear.","section":"Lemma 3.1"},{"comment":"The citation to Lemma 1.1(4) is appropriate here, since that part gives N(4)≥3; no correction is needed for this citation.","section":"Theorem 3.8"},{"comment":"The proof states that a 2-idempotent MOQLS based on the computational basis exists for k∈{4,5,7,9,10,11}, but for blocks in B0 of Construction 2.3 one must relabel the non-zero part of each block to a non-computational basis. This is a harmless relabeling, but it should be stated explicitly so that the hypothesis of Construction 2.3 is visibly satisfied.","section":"Theorem 3.4, first paragraph"},{"comment":"The phrase 'with contradiction' in Cases 2 and 5 is too terse. The contradiction is that the original holey squares have no outside cell whose tensor product lies in Vs⊗Vs; spelling this out would improve the readability.","section":"Construction 2.6, Cases 2 and 5"}],"recommendation":"major_revision","confidential_remarks":"The main claims are likely salvageable: the missing non-classicality lemma is short and the orthogonality gap in Construction 2.3 has a natural fix. Please also verify the source [26] for Construction 2.5, since the statement as written appears too strong; the authors should rephrase it in their own terms."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the existence theorems are likely correct and are a genuine advance: they push non-classical 2-MOQLS, 3-MOQLS, idempotent MOQLS, and SOQLS to 'all sufficiently large orders with small exception lists,' using PBD and hole-filling constructions that are natural and mostly checkable. Second, the paper never actually proves non-classicality. Every construction ends with a sentence saying the square is non-classical because some entries are computational and others are not. That is not a proof: equivalence allows an arbitrary global unitary, and a mixed set of bases could in principle be rotated into a single basis. The stress-test note is right that the gap is real but easy to close. If a square contains two orthogonal computational basis vectors |a> and |b>, plus a third vector with nonzero coefficients on both, no single orthonormal basis can contain all three; the constructions contain such triples (for instance, any block of size at least 3 provides the diagonal basis vectors and a vector from the block's non-computational basis). Adding a short lemma and checking each construction would fix every non-classicality claim. What is good: the new definitions (idempotent, self-orthogonal, holey QLS) are useful; the PBD construction and the filling-in-holes constructions are clean; the results substantially improve Lemma 1.2, especially for composite orders that were previously open. The explicit 2-MOQLS(8), (10), (11) examples are helpful. The use of COILS to handle orders 26, 27, and 30 is a nice touch. There is no circular reasoning: the constructions rely on standard MOLS, PBD, and COILS existence results. Soft spots, in order of importance. (1) The non-classicality gap noted above affects every theorem. (2) The mutual orthogonality proof in Construction 2.3 has an unjustified assertion that two nonzero overlaps would force two distinct intersection vectors; the repair uses the fact that an off-diagonal pair duplicating the diagonal pair is impossible, but it needs to be written. (3) Theorem 3.8 cites Lemma 1.1(4), which excludes v=4; the needed bound is N(4)=3 from Lemma 1.1(1). (4) The v=5 no-idempotent appendix is a long case analysis with several 'similar to' steps; it may be correct, but it is not machine-checked and should be verified carefully. None of these looks load-bearing with respect to the main theorems. Bottom line: this paper deserves serious refereeing. It is for people working on quantum Latin squares and combinatorial designs, who will want these existence ranges. I would send it out; the fixes are mechanical, and the results are worth having on record. A referee should ask for the non-classicality proof and the Construction 2.3 repair, but should not desk-reject.","headline":"Existence theorems are likely right and are a real advance, but non-classicality is asserted rather than proved; the gap is real, easy to fix, and should not block peer review.","tokens_in":788,"tokens_out":1081,"would_cite":true,"duration_ms":97360,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B15","05B05","81P68"],"pacs":[],"model":"deepseek-v4-flash","headline":"Quantum Latin squares can be genuinely non-classical while also being idempotent, mutually orthogonal, or self-orthogonal, and the paper constructs such squares in almost every dimension using classical combinatorial designs.","keywords":["quantum Latin square","non-classical quantum Latin square","idempotent quantum Latin square","mutually orthogonal quantum Latin squares","self-orthogonal quantum Latin square","pairwise balanced design","holey Latin square","existence constructions"],"falsifier":"A direct test is to check whether all vectors appearing in one of the paper's explicit arrays lie in a single orthonormal basis; if they do, the square is classical. For the $v=6$ idempotent array in Theorem 3.3, the top-left entry $|0\\rangle$ and the entry $|+\\rangle=(|0\\rangle+|3\\rangle)/\\sqrt{2}$ are distinct but not orthogonal, so they cannot both belong to one orthonormal basis; applying this same orthogonality check to every array produced by Constructions 2.2, 2.6, and 2.8 would settle whether the non-classicality premise holds in general.","tokens_in":22844,"feed_emoji":"⚛️","tokens_out":18064,"duration_ms":152362,"temperature":0.7,"pith_summary":"Quantum Latin squares generalize Latin squares by replacing the symbols in each row and column with vectors that form an orthonormal basis of $\\mathbb{C}^v$. The paper aims to show that such squares can be genuinely non-classical—not merely relabeled ordinary Latin squares—while also satisfying idempotency, mutual orthogonality, or self-orthogonality. It establishes that non-classical idempotent QLS($v$) exist for every $v\\geq 6$; that non-classical 2-idempotent MOQLS($v$) exist for every $v\\geq 6$ except possibly $v\\in\\{6,7,8,9,10,11,12,14,15,18,19,23\\}$; that non-classical 2-MOQLS($v$) exist for every $v\\geq 8$ and non-classical 3-MOQLS($v$) exist for every $v\\geq 16$; and that non-classical SOQLS($v$) exist for every $v\\geq 13$. These are existence claims with explicit constructions, and they matter because quantum Latin squares are ingredients for unitary error bases, mutually unbiased bases, $k$-uniform states, and quantum error-correcting codes.","feed_headline":"Non-classical quantum Latin squares exist for almost all orders","feed_subtitle":"The paper proves idempotent, mutually orthogonal, and self-orthogonal variants exist beyond short exception lists.","key_machinery":"The load-bearing machinery is a pair of transfer arguments from classical designs to quantum squares. The pairwise balanced design (PBD) construction treats a PBD($v,K$) as a blueprint: a PBD is a set of blocks in which every pair of elements occurs in exactly one block, and each block is regarded as a subspace spanned by its computational basis vectors. Blocks containing the distinguished element $0$ receive an idempotent QLS whose off-diagonal vectors lie in a non-computational basis of the block's remaining subspace, while all other blocks receive classical idempotent QLSs. Because any two blocks intersect in at most one element, vectors from different blocks are automatically orthogonal except at the shared index, which is what makes rows, columns, and orthogonality checks work. The filling-in-holes constructions start instead from classical holey Latin squares, such as 2-IMOLS($v;4$), 3-IMOLS($v;4$), ISOLS($v;4$), and idempotent conjugate-orthogonal incomplete Latin squares; each hole is then filled with a quantum Latin square on a non-computational basis of that hole subspace, and self-orthogonality is preserved in the same way. Small explicit arrays for $v=6,8,10,11$ handle the low orders, and classical existence theorems for PBDs, IMOLSs, and ISOLSs provide the remaining orders.","core_discovery":"The central discovery is that classical combinatorial design existence can be lifted to quantum non-classical existence. Concretely, the paper claims: (1) for $v\\geq 6$ there is a non-classical idempotent QLS($v$); (2) for $v\\geq 6$ there is a non-classical 2-idempotent MOQLS($v$), except possibly for $v\\in\\{6,7,8,9,10,11,12,14,15,18,19,23\\}$; (3) for $v\\geq 4$ there is a non-classical 2-MOQLS($v$), except possibly for $v\\in\\{4,5,6,7\\}$; (4) for $v\\geq 16$ there is a non-classical 3-MOQLS($v$); and (5) for $v\\geq 13$ there is a non-classical SOQLS($v$). The constructions are explicit: they take a pairwise balanced design or a holey/incomplete Latin square, index rows and columns by a computational basis, and fill some blocks or holes with quantum Latin squares whose vectors come from non-computational bases of the block subspaces. The paper's operative criterion for non-classicality is that mixing computational-basis entries with non-computational block-basis entries prevents the whole array from being equivalent to a classical square.","pith_inferences":["The exception lists in Theorems 3.4 and 3.7 likely reflect the limits of the classical design inputs, not genuine non-existence; closing them by direct search or by specialized quantum designs would be the natural next test.","The paper's non-classicality criterion suggests a sharper quantifier: the cardinality of a QLS, the number of distinct vectors up to phase, could be used to distinguish constructions, with PBD-built squares probably having larger cardinality than hole-filling ones.","If the mixed-bases premise is false in some setting, the constructions would still produce classical squares, so a classification of when block-basis choices force non-classicality would be a useful follow-up.","The same transfer idea could be applied to other classical structures, such as orthogonal arrays or nets, to produce non-classical quantum Latin squares with additional symmetry or higher mutual orthogonality."],"forward_implications":["For every $v\\geq 8$ there is a non-classical pair of mutually orthogonal quantum Latin squares, and for every $v\\geq 16$ a non-classical triple; the only unresolved orthogonality orders are the short exception lists in the theorems.","For every $v\\geq 13$ there is a non-classical self-orthogonal quantum Latin square, so quantum self-orthogonality is available in essentially all large dimensions.","Idempotency does not block non-classicality: non-classical idempotent QLSs start at $v=6$, and non-classical 2-idempotent MOQLSs cover all $v\\geq 6$ outside a finite set.","The constructions are explicit, so each existence theorem yields actual arrays that can be used as ingredients for unitary error bases, mutually unbiased bases, $k$-uniform states, and quantum error-correcting codes.","Because the proofs use classical design existence as a black box, any future improvement in PBD, IMOLS, or ISOLS existence automatically shrinks the exception lists."],"supporting_citations":[{"why":"Supplies the classical existence theorems for pairwise balanced designs and holey/incomplete Latin squares (Lemmas 2.1 and 2.4) that feed the two construction schemes.","marker":"[7]"},{"why":"Supplies the idempotent (3,2,1)-conjugate-orthogonal incomplete Latin squares used in Construction 2.11 to handle the exceptional orders 26, 27, and 30 in Theorem 3.4.","marker":"[2]"},{"why":"Gives the original filling-in-holes construction for quantum Latin squares and the prior non-classical MOQLS/SOQLS existence results used as base cases.","marker":"[26]"},{"why":"Shows there is no non-classical QLS(3) and introduces the cardinality measure; the small-order exclusions in Theorems 3.2 and 3.6 rely on it.","marker":"[18]"},{"why":"Supplies the existence of ISOLS(v;4) for all v ≥ 13, the classical input for the self-orthogonal construction in Theorem 3.9.","marker":"[14]"},{"why":"Supplies the displayed ISOLS(10;3) and ISOLS(11;3) designs used to build non-classical 2-MOQLS(10) and 2-MOQLS(11).","marker":"[8]"},{"why":"Also supplies the ISOLS(10;3) and ISOLS(11;3) base designs used in the low-order 2-MOQLS cases.","marker":"[27]"}],"fun_headline_variants":["Non-classical quantum Latin squares exist for almost all orders","Quantum Latin squares: non-classical variants exist for all but a few orders","Almost every order supports non-classical quantum Latin squares","Existence proven: non-classical quantum Latin squares for almost all orders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the unproved assertion that a square mixing computational-basis vectors with vectors from non-computational bases of block subspaces is automatically non-classical; if some single invertible change of basis could rotate all those vectors into one common orthonormal basis, the constructed squares would be classical and the existence theorems would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Non-classical quantum Latin squares exist for almost all orders","Quantum Latin squares: non-classical variants exist for all but a few orders","Almost every order supports non-classical quantum Latin squares","Existence proven: non-classical quantum Latin squares for almost all orders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0007,"raw_usage":{"total_tokens":3176,"prompt_tokens":973,"completion_tokens":2203,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":2130}},"tokens_in":589,"tokens_out":2203,"duration_ms":15971,"temperature":1.0,"reasoning_tokens":2130,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:02:42.241613+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test is to check whether all vectors appearing in one of the paper's explicit arrays lie in a single orthonormal basis; if they do, the square is classical. For the $v=6$ idempotent array in Theorem 3.3, the top-left entry $|0\\rangle$ and the entry $|+\\rangle=(|0\\rangle+|3\\rangle)/\\sqrt{2}$ are distinct but not orthogonal, so they cannot both belong to one orthonormal basis; applying this same orthogonality check to every array produced by Constructions 2.2, 2.6, and 2.8 would settle whether the non-classicality premise holds in general.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical existence theorems for pairwise balanced designs and holey/incomplete Latin squares (Lemmas 2.1 and 2.4) that feed the two construction schemes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the idempotent (3,2,1)-conjugate-orthogonal incomplete Latin squares used in Construction 2.11 to handle the exceptional orders 26, 27, and 30 in Theorem 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original filling-in-holes construction for quantum Latin squares and the prior non-classical MOQLS/SOQLS existence results used as base cases."},{"cited_title":"Paczos, M Wierzbi´nski, G Rajchel-Mieldzio´c, et al., Genuinely quantum solutions of the game Sudoku and their cardinality","cited_arxiv_id":null,"evidence_quote":"Shows there is no non-classical QLS(3) and introduces the cardinality measure; the small-order exclusions in Theorems 3.2 and 3.6 rely on it."},{"cited_title":"Heinrich, L","cited_arxiv_id":null,"evidence_quote":"Supplies the existence of ISOLS(v;4) for all v ≥ 13, the classical input for the self-orthogonal construction in Theorem 3.9."},{"cited_title":"D´enes, A","cited_arxiv_id":null,"evidence_quote":"Supplies the displayed ISOLS(10;3) and ISOLS(11;3) designs used to build non-classical 2-MOQLS(10) and 2-MOQLS(11)."},{"cited_title":"Zhu, Orthogonal latin squares with subsquares","cited_arxiv_id":null,"evidence_quote":"Also supplies the ISOLS(10;3) and ISOLS(11;3) base designs used in the low-order 2-MOQLS cases."}],"review_version":2}