{"id":"b15aead4-5d99-4e4c-b907-6b53ca97990e","arxiv_id":"2607.28998","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"First explicit constructions of minimally nonlocal orthogonal state sets in bipartite and tripartite systems with unequal local dimensions.","lead":"This paper gives recipes for building small sets of quantum states that cannot be distinguished by local measurements, but become distinguishable if any one state is removed, in systems where the two or three sides have different dimensions. It extends the 2023 concept of minimal nonlocality from equal-sized systems to unequal local dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tripartite LOCC-indistinguishability proof's POVM tables are not exhaustive: Table 9 cites an undefined state index and omits constraints needed to force M_B^†M_B proportional to identity.","rationale":"The reader identified the tripartite POVM proof as the weakest assumption, and my reading agrees. The bipartite construction (Theorem 2) is detailed and plausible; the main advance over existing equal-dimension constructions is the unequal-dimension generalization, and the tripartite part is the novel extension. A single undefined index in Table 9 is a concrete defect, but by itself it might be a typo. The more serious issue is the absence of constraints needed to exhaust all off-diagonal Bob entries. This is not merely a presentational issue: if such entries survive, Bob's first measurement could be nontrivial, and S3 would be LOCC-distinguishable, contradicting the theorem. The proposed symbolic linear-algebra test on the explicit C5⊗C7⊗C9 example settles the matter computationally and is inexpensive. If the test shows the constraints are actually complete, the proof can be repaired by adding the missing rows; if not, the theorem needs revision. The post-removal reduction to Theorem 2 is also asserted, not demonstrated, so even the distinguishing half is less explicit than the bipartite case. Overall the correct verdict remains CONDITIONAL: the contribution is plausible and potentially significant, but the tripartite proof must be completed or machine-checked before acceptance.","tokens_in":20786,"tokens_out":4729,"duration_ms":43841,"concrete_test":"Take the paper's own illustrative case d1=5, d2=7, d3=9 (set S4, 35 states). For each of A, B, C, form the matrix of inner products ⟨φ_i| I_1⊗M_X⊗I_2 |φ_j⟩ and set all =0 treating M_X entries as variables. Solve the resulting linear system exactly with a computer algebra system. For Bob, if the solution space of M_B^†M_B is precisely {c I_7 : c∈C}, the missing Table 9 constraints are not load-bearing. If a non-identity solution exists (e.g., b_{1,5}≠0), then S3 is not locally indistinguishable and the central theorem fails. Repeat for Alice and Charlie to confirm.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3's local-indistinguishability claim rests on showing that any first measurement by Alice, Bob, or Charlie that preserves orthogonality must be trivial. For Bob, Tables 9 and 10 are supposed to establish that the only such POVM element M_B^†M_B is cI. Two gaps undermine this. First, Table 9 refers to a state |φ_{4d3+3d2−14}⟩, but the set S3 is indexed up to 4d3+2d1−11; no such state is defined. Second, even ignoring that, the listed constraints do not cover all off-diagonal entries. For example, no row forces b_{1j}=b_{j1}=0 for j∈{d1,...,d2−2} (Bob's high vs. low index subspace). If a nonzero entry of this type can survive all pairwise orthogonality conditions, then M_B^†M_B need not be proportional to identity, and Bob could perform a nontrivial first measurement that preserves the orthogonality of S3. The post-removal LOCC-distinguishability proof also asserts a reduction to Theorem 2 without a state-by-state mapping, so even the 'minimal' part is not fully written out. Because maximal nonlocality requires both halves, the tripartite theorem as printed is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes constructions of orthogonal quantum state sets with minimal nonlocality in unequal-dimensional systems. Theorem 1 gives a 12-state set in C^4⊗C^7; Theorem 2 gives a (2d2−2)-state set in C^{d1}⊗C^{d2} for 4≤d1<d2; Theorem 3 gives a (4d3+2d1−11)-state set in C^{d1}⊗C^{d2}⊗C^{d3} for 5≤d1<d2<d3. For each set, the authors argue that any LOCC protocol that preserves orthogonality must begin with a trivial measurement (local indistinguishability), and that deleting one full-sum state leaves a set perfectly distinguishable by LOCC (minimality). The bipartite proof is largely explicit and self-contained. The tripartite proof is not: the Bob POVM analysis has undefined state indices and missing constraints, and the post-removal distinguishability proof relies on asserted reductions to Theorem 2 without a written mapping.","tokens_in":21105,"tokens_out":20952,"duration_ms":172994,"significance":"If the results are correct, this would be the first construction of minimal-nonlocality sets for unequal local dimensions, extending the equal-dimensional framework of Zhu et al. The bipartite part is constructive, systematic, and appears sound; the C4⊗C7 example illustrates the method concretely. However, the tripartite theorem — a central advertised contribution — is not established as printed. The gaps are load-bearing because both halves of minimal nonlocality (indistinguishability of the full set and LOCC-distinguishability after one deletion) are required. The paper is worth publishing after substantial revision, but the current version does not support the abstract's claim that the tripartite construction problem is settled.","major_comments":[{"comment":"The Bob POVM proof is incomplete. Table 9 cites a state |φ_{4d3+3d2−14}⟩ that is not in S3; the intended index appears to be |φ_{3d3+3d2−14}⟩, the i=d2−d1 member of |φ_{3d3+2d2+d1−14+i}⟩. More importantly, the listed constraints do not force M_B†M_B to be proportional to identity. The only states with Bob support on |j⟩_B for j∈{d1,...,d2−2} are |φ_{3d3+2d2+d1−14+i}⟩ with j=d1−1+i, and each such state has Bob support on a single basis vector. Tables 9–10 contain no row that couples this subspace to the low-index Bob subspace through a common Alice/Charlie factor, and no row that equates the diagonal entries b_{jj} for j∈{d1,...,d2−2} to b_{00} or b_{11}. Thus a positive diagonal POVM element with arbitrary weights on span{|d1⟩_B,...,|d2−2⟩_B} and proportional to identity on the complementary subspace preserves the orthogonality of all pairs in S3. The conclusion that Bob's first measurem","section":"§IV, Theorem 3, Tables 9–10"},{"comment":"Even if all off-diagonal entries were shown to vanish, the diagonal entries for Bob's high-interior subspace are undetermined. Table 10 equates b_{00} with b_{11},...,b_{(d2−2)(d2−2)} and b_{11} with b_{(d2−1)(d2−1)}, but it gives no relation between b_{00} and b_{jj} for j=d1,...,d2−2. For Charlie, Eq. (3) plus Table 12 connects the high-index block to c_{11}; no analogous Fourier system exists for Bob because the states |φ_{2d3+d2+d1−8+i}⟩, which generate Eq. (3), have Bob support only on |d2−4⟩. A diagonal Bob matrix with a different weight on the subspace spanned by |d1⟩_B,...,|d2−2⟩_B preserves S3's orthogonality under the stated conditions. This is a second, independent reason that the local-indistinguishability half of Theorem 3 fails as written.","section":"§IV, diagonal part of M_B†M_B"},{"comment":"The proof asserts two reductions to Theorem 2 — after Alice's A1 outcome followed by Bob's B0, and after Bob's B2 — with the phrases 'equivalent to the constructions for Alice and Charlie subsystems in Theorem 2' and 'conform to the constructions in Theorem 2'. No state-by-state mapping is provided. The collapsed sets listed do not match the S2 template in an obvious way: the Alice superpositions (e.g., |1−2⟩_A), the Charlie indices (e.g., |d3−2−i⟩_C), and the ordering of the states differ from the S2 definitions. A reader cannot verify that these sets are LOCC-distinguishable without an explicit identification or a separate proof of local equivalence. Since minimal nonlocality requires this second half, Theorem 3 is incomplete.","section":"§IV, LOCC-distinguishability of S3−{|φ_{4d3+2d1−11}⟩}"}],"minor_comments":[{"comment":"Typos and language issues: 'quantun systems' in the Conclusion; 'ststes' at the start of Section IV; 'Noniagonal' in Table 4's header; 'of of' in Section III; 'by via LOCC' in the Conclusion. These should be corrected.","section":"Throughout"},{"comment":"The state |φ_{4d3+3d2−14}⟩ is not defined anywhere in S3. The intended state is almost certainly |φ_{3d3+3d2−14}⟩ = |φ_{3d3+2d2+d1−14+i}⟩ with i=d2−d1. Please fix the index.","section":"§IV, Table 9"},{"comment":"The row '|φ_{2d−3+d2+d1−8}⟩' contains a typo in the subscript: it should be |φ_{2d3+d2+d1−8}⟩.","section":"§IV, Table 10"},{"comment":"For the pair (|φ3⟩,|φ9⟩), the table records b30=b06=0 and b03=b60=0. Direct expansion of ⟨φ3|I⊗M_B†M_B|φ9⟩ and its conjugate gives b30=b06 and b03=b60, not automatic vanishing of each entry. Either the table should state the equalities, or additional pairs that force each entry to zero should be cited.","section":"§III, Theorem 1, Table 3"},{"comment":"'Kramer's rule' should be 'Cramer's rule'. Also, the lemma states a unique solution when the determinant is nonzero; this is standard, but the wording could be tightened.","section":"§II, Lemma 1"},{"comment":"In the example, 'ω=e^{π√−1}' is correct because d3−d2=2, but writing ω=−1 would be clearer and avoid confusion with the general definition.","section":"§IV, C5⊗C7⊗C9 example"}],"recommendation":"major_revision","confidential_remarks":"The bipartite construction (Theorem 2) appears sound and the C4⊗C7 example is a useful concrete illustration. The tripartite theorem, however, has genuine gaps in both the local-indistinguishability proof and the post-removal LOCC-distinguishability proof. These gaps are technical and likely fixable — for instance, the Bob diagonal issue may be repaired by adding states that couple the high-interior Bob subspace to the low-index subspace — but they are load-bearing rather than presentational. I therefore recommend major revision rather than rejection. I would also caution the authors against the abstract's claim that the problem is 'settled' until the tripartite proof is fully repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the bipartite construction is the real contribution here and it checks out. The tripartite extension is plausible and likely correct in spirit, but the proof in Section IV has a concrete gap that needs fixing before I'd trust the theorem.\n\nWhat's new: first minimal-nonlocality constructions for unequal local dimensions, both bipartite and tripartite. The bipartite proof (Theorem 2) is careful—they show any orthogonality-preserving POVM on either side must be trivial, and the LOCC protocol for the set minus the last state is written out. That part I'd be happy to cite.\n\nThe tripartite proof (Theorem 3) is where it gets shaky. Table 9 references a state |φ_{4d3+3d2−14}⟩ that doesn't exist in S3; the index is out of range. More importantly, the table doesn't appear to cover all off-diagonal entries of M_B^†M_B. In particular, nothing forces b_{1j}=b_{j1}=0 for j in {d1,...,d2−2}. If those entries can be nonzero while preserving orthogonality, then Bob has a nontrivial first measurement and the local-indistinguishability claim fails. The stress-test note is right about this. The post-removal distinguishability proof also reduces to Theorem 2 by assertion rather than exhibiting the state mapping; that's probably fillable, but it's not written.\n\nThe Charlie tables have a similar feel: they list many constraints but I didn't verify exhaustiveness. The example in C5⊗C7⊗C9 is helpful but doesn't substitute for the general proof.\n\nNet: the bipartite half is solid, the tripartite half is a plausible construction whose proof is incomplete as printed. This is a fixable typo/omission problem, not a fundamentally wrong approach. I'd send it to peer review, but I'd insist on a complete table of constraints or a machine-checked certificate for the tripartite part before publication.\n\nFor a reading group: worth a session if people are into LOCC constructions, but warn them to bring scratch paper. I'd cite the bipartite result.","headline":"Bipartite result is real; the tripartite proof as printed has a load-bearing gap and should not be accepted without revision.","tokens_in":21555,"tokens_out":2845,"would_cite":true,"duration_ms":26141,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.-a","03.65.Ud"],"model":"deepseek-v4-flash","headline":"This paper constructs orthogonal state sets with minimal nonlocality in bipartite and tripartite systems whose local dimensions are unequal, filling a gap left by equal-dimension-only constructions.","keywords":["minimal nonlocality","orthogonal quantum states","local indistinguishability","LOCC","unequal local dimensions","bipartite quantum system","tripartite quantum system","full-sum state"],"falsifier":"For the explicit d1 = 5, d2 = 7, d3 = 9 example, compute the 35 listed states and numerically search for a non-identity POVM matrix M_B on Bob's subsystem that keeps all post-measurement states mutually orthogonal; if such a matrix exists, Theorem 3's nonlocality claim fails. A direct check is the Table 9 row citing |φ_{4d3+3d2−14}⟩, an index that does not exist in a 35-state set, so that orthogonality constraint cannot hold as written.","tokens_in":20698,"feed_emoji":"⚛️","tokens_out":6378,"duration_ms":65961,"temperature":0.7,"pith_summary":"The paper addresses the construction problem for minimal nonlocality in quantum systems where the local dimensions are unequal, a case previous work left open. It gives explicit state sets: S2 in C^{d1} ⊗ C^{d2} for 4 ≤ d1 < d2 containing 2d2 − 2 states, and S3 in C^{d1} ⊗ C^{d2} ⊗ C^{d3} for 5 ≤ d1 < d2 < d3 containing 4d3 + 2d1 − 11 states. For each set it proves two claims: the entire set cannot be perfectly distinguished by LOCC, and after deleting the last full-sum state the remaining states can be perfectly distinguished by LOCC. If correct, this settles the minimal-nonlocality construction problem for unequal-dimensional bipartite and tripartite systems and offers concrete state families with practical communication potential.","feed_headline":"Minimal nonlocality now extends to unequal dimensions","feed_subtitle":"Removing one full-sum state leaves a set that is perfectly LOCC-distinguishable, enabling new communication schemes.","key_machinery":"The load-bearing object is the 'full-sum state', a product state that superposes all basis states on every party with equal amplitude. It plays two roles: together with root-of-unity superposed states it forces the diagonal entries of any orthogonality-preserving POVM matrix to be equal, finishing the proof that a first measurement by any party is trivial; and deleting exactly this full-sum state is what turns the remainder into an LOCC-distinguishable set. The proof also relies on pairwise orthogonality tables, one per party and per bipartite/tripartite setting, to drive off-diagonal matrix entries to zero, and on Vandermonde-style linear systems solved by Cramer's rule.","core_discovery":"The central claim is Theorem 2 and Theorem 3: for 4 ≤ d1 < d2 the set S2 of 2d2 − 2 orthogonal quantum states in C^{d1} ⊗ C^{d2}, and for 5 ≤ d1 < d2 < d3 the set S3 of 4d3 + 2d1 − 11 orthogonal quantum states in C^{d1} ⊗ C^{d2} ⊗ C^{d3}, each possess minimal nonlocality. Local indistinguishability is argued by showing that any POVM element on an individual party that preserves the mutual orthogonality of the post-measurement states must be proportional to the identity: pairwise orthogonality constraints force all off-diagonal matrix elements to zero, and orthogonality with a uniform full-sum state forces all diagonal elements equal, with the equalities completed by root-of-unity systems of","pith_inferences":["The paper's meaning of 'minimal nonlocality' is the one-state-deletion property of Definition 3, not minimal cardinality among all locally indistinguishable sets; whether 2d2 − 2 and 4d3 + 2d1 − 11 are also the smallest possible cardinalities for unequal dimensions is not established and remains open.","The tripartite proof relies on asserted reductions to the bipartite theorem after deleting the full-sum state; if those reductions are made fully rigorous they would yield a recursive framework for constructing minimally nonlocal sets in arbitrary multipartite unequal-dimensional systems.","Because the constructed states are entangled rather than merely product states, the result indicates that minimal nonlocality is not an exclusive feature of orthogonal product states, widening the known landscape of such sets.","A concrete testable extension is to run numerical LOCC or POVM searches on the explicit 12-state and 35-state sets to independently confirm that no non-identity orthogonality-preserving POVM exists, checking the pairwise orthogonality tables before relying on them."],"forward_implications":["If Theorems 2 and 3 hold, unequal-dimensional quantum systems for the first time have state sets whose nonlocality is exactly one-state-deep: deleting a single full-sum state converts a globally indistinguishable set into an LOCC-distinguishable one.","The explicit state families, including the C4 ⊗ C7 and C5 ⊗ C7 ⊗ C9 examples, give ready-made resources for quantum communication protocols that need to block an adversary while allowing a legitimate receiver to decode after applying one local deletion.","The construction method is not tied to the specific dimensions 4 and 7, so the same pattern may extend to larger unequal local dimensions and to multipartite systems with more than three parties.","The proof that the deleted remainder is LOCC-distinguishable supplies an explicit measurement sequence, which is directly implementable in distributed settings where parties can broadcast classical measurement outcomes.","The tripartite construction reduces, at several points, to the bipartite construction on a subsystem, suggesting that the bipartite theorem is a reusable building block for higher-partite minimal-nonlocality results."],"fun_headline_variants":["Minimal nonlocality achieved in unequal-dimensional systems","Orthogonal state sets with minimal nonlocality for unequal dimensions","Construction of minimal-nonlocality state sets in unequal dimensions","Minimal nonlocality sets now possible in unequal bipartite and tripartite systems","Solving minimal nonlocality for unequal local dimensions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The tripartite local-indistinguishability proof assumes that the listed pairwise orthogonality constraints on Bob's and Charlie's POVM matrices are exhaustive enough to force every off-diagonal entry to zero, and at least one listed constraint in Table 9 refers to a state index that is not defined in S3.","fun_headline_variants_meta":{"raw":{"variants":["Minimal nonlocality achieved in unequal-dimensional systems","Orthogonal state sets with minimal nonlocality for unequal dimensions","Construction of minimal-nonlocality state sets in unequal dimensions","Minimal nonlocality sets now possible in unequal bipartite and tripartite systems","Solving minimal nonlocality for unequal local dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000666,"raw_usage":{"total_tokens":2900,"prompt_tokens":790,"completion_tokens":2110,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":2024}},"tokens_in":534,"tokens_out":2110,"duration_ms":14510,"temperature":1.0,"reasoning_tokens":2024,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T15:44:22.140886+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit d1 = 5, d2 = 7, d3 = 9 example, compute the 35 listed states and numerically search for a non-identity POVM matrix M_B on Bob's subsystem that keeps all post-measurement states mutually orthogonal; if such a matrix exists, Theorem 3's nonlocality claim fails. A direct check is the Table 9 row citing |φ_{4d3+3d2−14}⟩, an index that does not exist in a 35-state set, so that orthogonality constraint cannot hold as written.","supporting_citations":[],"review_version":1}