{"id":"ddeaa7cc-f045-40ac-b491-caed05eb619b","arxiv_id":"2509.07086","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Local extensions produce PPT bound entangled states of Schmidt number 3, 4, and 5 in record dimensions 4×5, 7×7, and 9×9.","lead":"This paper introduces 'local extensions', the inverse of peel-off projections, as a systematic way to build higher-dimensional quantum states that keep the PPT property, and uses them to construct record-small bound entangled states with Schmidt number 3 in 4×5 and Schmidt numbers 4, 5 in 7×7, 9×9. The framework has clean bounds, giving researchers a reusable toolkit for charting how much hidden entanglement high-dimensional states can carry, provided the algebraic certificat","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's printed polynomial identity is false: the LHS expands to psi_00^4 + psi_02 psi_20 (psi_00^2 + 2 psi_01 psi_10), so the Nullstellensatz step is not established; SM F's k=4,5 certificates are also unreleased.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing point: the algebraic certificate for Theorem 5 is invalid as printed, and the SM F certificates for k=4,5 are not shipped. My independent expansion confirms the reader's calculation. Because the local-extension framework itself is not implicated, and because the claims may be salvageable by a corrected certificate, the appropriate verdict is conditional rather than rejection. The reader already made this call, so no verdict adjustment is needed. The concrete test would settle whether the conditional status can be upgraded or whether the paper's record claims should be withdrawn pending new certificates.","tokens_in":14810,"tokens_out":4343,"duration_ms":42064,"concrete_test":"Recompute the ideal membership in exact arithmetic over Q (e.g. SageMath/Singular): form I = <g1,...,g5> from Eq. (12), compute a Groebner basis, and reduce psi_00^4 (or psi_00 itself). If the normal form is 0, produce explicit polynomial multipliers p_i with sum p_i g_i = psi_00^M; if nonzero, the Theorem 5 certificate is false. Independently run the same exact Groebner-basis check for the SM F states rho^(4) and rho^(5), reducing alpha^4 and alpha^5 against the ideal of k x k minors of the range coordinate matrix, and release the script and certificates for public verification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5 rests on the claimed identity\n\npsi_00(g5 - g3 - g4) - psi_02 g1 + psi_20 g2 = psi_00^4.\n\nDirect expansion contradicts this. Using the paper's own definitions,\n\ng5 - g3 - g4 = psi_00^3 + psi_00 psi_02 psi_20,\n\nso the left-hand side equals\n\npsi_00^4 + psi_02 psi_20 (psi_00^2 + 2 psi_01 psi_10),\n\nnot psi_00^4. Thus the displayed Nullstellensatz certificate is algebraically incorrect. The conclusion that psi_00^4 lies in the ideal generated by {g1,...,g5} does not follow from the printed computation, and without an ideal-membership certificate the lower bound SN(rho_4x5) > 2 is not established. This is the load-bearing step for the paper's headline 'smallest known' claim: a 4x5 PPT state of Schmidt number 3. The same kind of gap affects SM F: the k=4 and k=5 memberships are asserted from SAGE Groebner-basis runs, but no code or certificate is shipped, so the 7x7 and 9x9 record claims are not independently verifiable. The framework results (Theorem 2, the extension constraints) appear sound, and the failure may be repairable by a different multiplier choice, but as printed the central record claims lack a valid algebraic proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'local extensions' as the inverse of peel-off projections and uses them to construct bipartite PPT states with high Schmidt number. The main framework result (Theorem 2) states that a local projection drops the Schmidt number by at most one; the authors also derive counts of local extensions from birank data and prove some low-dimensional unextendibility results. The headline application is a claimed PPT state of Schmidt number 3 in 4×5 dimensions, called the smallest system known, followed by 7×7 (Schmidt number 4) and 9×9 (Schmidt number 5) examples from a recursive family of generalized grid states. The lower bounds on Schmidt number are proved via the range criterion, with the algebraic core being an ideal-membership statement certified by a displayed polynomial identity in Theorem 5 and, for larger k, by SAGE Groebner-basis computations reported in SM F.","tokens_in":15122,"tokens_out":6519,"duration_ms":68451,"significance":"If the central construction is valid, the paper would supply a genuinely useful constructive toolkit for high-dimensional bound entanglement: Theorem 2 and its Schur-complement reformulation (Theorem 6) are clean, explicit, and appear sound, and the grid-state family is elegant. The claimed 4×5 Schmidt-number-3 PPT state would be a record threshold, and the (2k−1)×(2k−1) family with Schmidt number k would improve on earlier constructions and suggest an interesting (d+1)/2 scaling conjecture. The paper also has the virtue of framing the lower-bound certification as an exact algebraic problem (integer-coefficient polynomial ideal membership), which is the right approach for reproducible proofs. However, the record claims stand or fall on the correctness of those algebraic certificates, and as printed the main certificate is not correct.","major_comments":[{"comment":"The proof that SN(ρ_4x5)>2 rests entirely on the claimed identity ψ00(g5 − g3 − g4) − ψ02 g1 + ψ20 g2 = ψ00^4. Expanding the left-hand side using the definitions in Eq. (12) gives ψ00^4 + ψ02 ψ20 (ψ00^2 + 2 ψ01 ψ10), not ψ00^4. The ψ02 ψ20 term does not vanish identically, so the displayed Nullstellensatz certificate is algebraically false. Consequently, the membership of ψ00^4 in the ideal generated by {g1,...,g5} is not established, and the conclusion that every Schmidt-rank-2 state in the range is orthogonal to |e*> is unsupported. Unless a correct polynomial identity (or another valid membership certificate) is supplied, the 4×5 'smallest known' claim is not proven.","section":"Proof of Theorem 5, Eq. (12) and following displayed identity"},{"comment":"The statements that α^k lies in the ideal of k×k minors for k=4,5 are asserted from SAGE Groebner-basis runs, but no code, computation logs, or certificates are provided. Because the ideal-membership certificates are the only evidence for SN(ρ^(4))=4 and SN(ρ^(5))=5, these record claims are not independently verifiable. This is especially concerning given the false identity in Theorem 5: the reader has no way to check whether the SAGE computations suffer from a similar issue. Please include reproducible scripts (or a human-checkable algebraic certificate) for each claimed membership, or state precisely which polynomial reductions were performed.","section":"SM F, ideal-membership claims for k=4 and k=5"},{"comment":"The proof parametrizes a vector |ψ> in R(ρ_4x5) by the matrix in Eq. (11) without deriving why every vector in the range has this coordinate pattern. If the parametrization intentionally defines the range (e.g., via the grid-state edges), this should be stated explicitly and justified; if it is only a subset, then the range-criterion argument does not cover all Schmidt-rank-2 vectors in R(ρ_4x5), and the contradiction would fail even with a correct ideal-membership certificate.","section":"Theorem 5, range parametrization"}],"minor_comments":[{"comment":"The notation 'g:={...}=0' is confusing: it suggests a set of equations rather than a set of polynomials. Please write g_i explicitly and then state the polynomial equations g_i=0.","section":"Eq. (12)"},{"comment":"The definition of d_i uses d_i ≡ min(i, 2k−1−i), but the index range of δ_i is not stated precisely; the text later uses 1≤i≤2k−2. Please make the index ranges of all edge variables uniform.","section":"SM F, Eq. (F2)"},{"comment":"The partial transpose is applied to subsystem A in Eq. (7) but to subsystem B in SM D (where (ρ_3×3)^TB is computed). The asymmetry is harmless but should be flagged so readers do not misread the formulas as inconsistent.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The framework results (Theorems 2 and 6) appear sound and are likely publishable; the issue is specifically the validity of the algebraic certificates behind the record Schmidt-number claims. If the authors can supply a corrected identity for Theorem 5 and reproducible SAGE code/certificates for SM F, the paper may be suitable. I do not recommend rejection at this stage, but the current proofs of the headline claims are not correct as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing: the framework is worth taking seriously. Theorem 2—projections drop the Schmidt number by at most one, extensions raise it by at most one—is clean, and I verified the Schur-complement argument by expanding it. The birank counting for existence of extensions and the unextendibility results for 3×3 rank-4 states give a genuinely new dual perspective on peel-off theorems. The paper is also honest about what it has not proved: the (d+1)/2 scaling is explicitly a conjecture, not an overclaim.\n\nThe soft spot is in the headline constructions. The identity in the proof of Theorem 5, ψ00(g5−g3−g4)−ψ02g1+ψ20g2 = ψ00^4, is false as printed. Expanding the left side gives ψ00^4 + ψ02ψ20(ψ00^2 + 2ψ01ψ10). I checked this with the definitions in Eq. (12). So the Nullstellensatz argument does not go through, and the claimed Schmidt number 3 for the 4×5 state is not established by the printed proof. SM F makes the same kind of claim for k=4 and 5, but the SAGE computations are asserted with no code or certificates shipped; the 7×7 and 9×9 records are not independently verifiable.\n\nThese are mechanical corrections, not a failure of the whole approach. The framework might still work, and a different polynomial multiplier could repair the identity. But as it stands, the central record claims lack a valid proof. I don't see a circularity problem; the self-citations are prior record benchmarks, not load-bearing inputs.\n\nWho is this for? People working on Schmidt numbers and PPT bound entanglement will care about the extension framework and the explicit states. The record claims should be treated as conditional until the algebraic certificates are released and corrected. I'd send it to a serious referee—the framework is interesting enough, and the flaw is concrete and fixable—but I'd require a corrected certificate and released SAGE code before trusting the dimensional records.","headline":"The local-extension framework is a real contribution, but the headline 4×5 record rests on a false polynomial identity and the 7×7/9×9 records on unreleased computations.","tokens_in":15758,"tokens_out":3779,"would_cite":true,"duration_ms":31292,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.67.Mn"],"model":"deepseek-v4-flash","headline":"Local extensions, the inverse of peel-off projections, are introduced as a constructive scaffold: a PPT state with Schmidt number 3 in 4×5, rising to Schmidt number 5 in 9×9.","keywords":["Schmidt number","PPT states","bound entanglement","local extensions","generalized grid states","peel-off theorems","range criterion","high-dimensional entanglement"],"falsifier":"Expand the polynomial identity printed in the proof of Theorem 5: substituting g1 through g5 gives ψ00(g5−g3−g4)−ψ02 g1 + ψ20 g2 = ψ00^4 + ψ02ψ20(ψ00^2 + 2ψ01ψ10), not ψ00^4. So the printed Nullstellensatz step is not a valid certificate. A corrected identity, or a certified Gröbner-basis membership test for the same ideal, would settle whether the 4×5 state really has Schmidt number 3; without one, that record claim is unsupported.","tokens_in":14556,"feed_emoji":"⚛️","tokens_out":10393,"duration_ms":99631,"temperature":0.7,"pith_summary":"The paper introduces local extensions—the inverse operation of peel-off projections—as a systematic way to grow the local dimension of a bipartite state while controlling its Schmidt number, the number of entangled degrees of freedom. Its central theorem bounds how fast the Schmidt number can grow: extending one local dimension can raise it by at most one, and projecting can lower it by at most one. Using this, the authors construct a positive-partial-transpose (PPT) state with Schmidt number three in local dimensions 4×5, which they claim is the smallest system known to host such entanglement, and a family of generalized grid states reaching Schmidt number four in 7×7 and five in 9×9. If the algebraic certificates hold, these are the lowest-dimensional hosts known for bound entangled states of high Schmidt number, and the construction gives a concrete route toward probing how this number scales.","feed_headline":"4×5 state hosts Schmidt number 3 bound entanglement","feed_subtitle":"The same extension recipe gives Schmidt number 4 in 7×7 and 5 in 9×9, hinting at linear scaling.","key_machinery":"The central object is the local extension: ρ is a local extension of a core state ρ_c when ρ_c is obtained by projecting ρ onto a local block, and the new rows and columns are chosen so that both ρ and its partial transpose stay positive. A dimension-counting inequality (p+q−mn)n−m bounds how many nontrivial extensions any core admits. The explicit examples are generalized grid states, graphs whose vertices are basis states and whose hyperedges are vectors; the family ρ^(k) has one large diagonal hyperedge plus low-rank edges. Its Schmidt number is certified by the range criterion: every vector of Schmidt rank k−1 in the range must be orthogonal to that large hyperedge. This orthogonality is","core_discovery":"The paper's central claim is that every high-Schmidt-number state sits on top of a chain of lower-Schmidt-number cores, and that building upward—local extension rather than peel-off projection—is a constructive tool. Theorem 2 states that deleting one local dimension lowers the Schmidt number by at most one, so a one-step extension raises it by at most one. The authors then exhibit explicit grid states: ρ_4×5 is a (1,2)-extension of a 3×3 Schmidt-number-2 grid state and is certified by the range criterion to have Schmidt number 3; the recursive family ρ^(k) reaches Schmidt number k in (2k−1)×(2k−1) for k=2,3,4,5, improving earlier constructions in dimensions 2k×2k and (2k−1)×k(k+1)/2. The pa","pith_inferences":["Editorial: Supplying a repaired identity for the 4×5 case, or a certified Gröbner-basis computation for k=4 and 5, is the immediate testable next step; the same software pipeline would settle whether k=6 continues the linear scaling.","Editorial: The 4×5 construction extends the two subsystems asymmetrically, suggesting that non-square dimensions tailored to the core state may beat the homogeneous (2k−1)×(2k−1) family in future searches.","Editorial: The unextendibility results for low-dimensional PPT states of fixed rank hint that the presence of product vectors in the range is the dividing line between extendible and unextendible cores; mapping that boundary would sharpen guesses about where high-Schmidt-number states can exist.","Editorial: Because the Schmidt-number certificate is independent of the state's weights, the lower bound in the grid-state family is robust to renormalization; only the PPT property depends on choosing the weights correctly."],"forward_implications":["A 4×5 PPT state with Schmidt number 3 would be the smallest known host of such bound entanglement, living in a 20-dimensional bipartite space.","The recursive grid-state family would place Schmidt number 4 in 7×7 and Schmidt number 5 in 9×9, improving the best previous dimensional thresholds.","The (d+1)/2 scaling conjecture, if true, implies Schmidt number grows only linearly with local dimension for odd d×d PPT states.","The framework converts the search for high-Schmidt-number PPT states into an algebraic certification task: check whether a monomial lies in the ideal of minors.","The general projection bound means no local dimension reduction can remove more than one Schmidt-number level, a structural constraint on both constructions and witnesses."],"supporting_citations":[{"why":"Supplies the decomposition and peel-off theorems that motivate local extensions and ground Theorem 2.","marker":"[40]"},{"why":"Provides the previous Schmidt-number-k PPT family in (2k−1)×k(k+1)/2 and the generalized grid state methods this paper extends.","marker":"[35]"},{"why":"Establishes the quantum grid state formalism and the algebraic range-criterion technique used to compute Schmidt numbers.","marker":"[49]"},{"why":"Analyzes entanglement properties of grid states via minors of the range-coordinate matrix, the method used in Theorem 5.","marker":"[50]"},{"why":"Shows 2×n PPT states with a product vector in the kernel are separable, used to bound Schmidt number in Observation 3.","marker":"[45]"},{"why":"Classifies rank-four PPT entangled states of two qutrits, used to prove unextendibility in Theorem 4.","marker":"[47]"},{"why":"Classifies 4×4 PPT states of rank 6, used to extend the unextendibility argument.","marker":"[48]"},{"why":"Provides the range criterion for Schmidt numbers, the detection tool that certifies SN>2.","marker":"[51]"},{"why":"Gives the previous 2k×2k examples that the new homogeneous family improves upon.","marker":"[34]"},{"why":"Used to compute the Gröbner-basis membership certificates for k=4 and 5 in the supplemental material.","marker":"[53]"}],"fun_headline_variants":["Local extensions build bound entanglement up to 9×9","Smallest system yet for Schmidt number 3 bound state","New grid states hit Schmidt number 5 in 9×9","Scaling of bound entanglement tied to local dimensions","Local extension trick hints at (d+1)/2 Schmidt scaling"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The record dimensional claims rest on exact algebraic certificates proving that every low-Schmidt-rank vector in a grid state's range has a zero first coordinate; if any of those certificates—including the one printed for the 4×5 case—fails, the Schmidt-number lower bounds for the record dimensions are not established, even though the extension framework itself may stand.","fun_headline_variants_meta":{"raw":{"variants":["Local extensions build bound entanglement up to 9×9","Smallest system yet for Schmidt number 3 bound state","New grid states hit Schmidt number 5 in 9×9","Scaling of bound entanglement tied to local dimensions","Local extension trick hints at (d+1)/2 Schmidt scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000694,"raw_usage":{"total_tokens":2999,"prompt_tokens":788,"completion_tokens":2211,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":2129}},"tokens_in":532,"tokens_out":2211,"duration_ms":18402,"temperature":1.0,"reasoning_tokens":2129,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T22:54:41.554350+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand the polynomial identity printed in the proof of Theorem 5: substituting g1 through g5 gives ψ00(g5−g3−g4)−ψ02 g1 + ψ20 g2 = ψ00^4 + ψ02ψ20(ψ00^2 + 2ψ01ψ10), not ψ00^4. So the printed Nullstellensatz step is not a valid certificate. A corrected identity, or a certified Gröbner-basis membership test for the same ideal, would settle whether the 4×5 state really has Schmidt number 3; without one, that record claim is unsupported.","supporting_citations":[{"cited_title":"Cones of positive maps and their duality re- lations,","cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition and peel-off theorems that motivate local extensions and ground Theorem 2."},{"cited_title":"Unextendible product bases and bound entanglement,","cited_arxiv_id":null,"evidence_quote":"Provides the previous Schmidt-number-k PPT family in (2k−1)×k(k+1)/2 and the generalized grid state methods this paper extends."},{"cited_title":"Separability in2×n composite quan- tum systems,","cited_arxiv_id":null,"evidence_quote":"Establishes the quantum grid state formalism and the algebraic range-criterion technique used to compute Schmidt numbers."},{"cited_title":"Extreme points of the set of density matrices with pos- itive partial transpose,","cited_arxiv_id":null,"evidence_quote":"Analyzes entanglement properties of grid states via minors of the range-coordinate matrix, the method used in Theorem 5."},{"cited_title":"Entanglement of pure states for a single copy,","cited_arxiv_id":null,"evidence_quote":"Classifies rank-four PPT entangled states of two qutrits, used to prove unextendibility in Theorem 4."},{"cited_title":"Characterization of distillability of en- tanglement in terms of positive maps,","cited_arxiv_id":null,"evidence_quote":"Classifies 4×4 PPT states of rank 6, used to extend the unextendibility argument."},{"cited_title":"Description of rank four entangled states of two qutrits having posi- tive partial transpose,","cited_arxiv_id":null,"evidence_quote":"Provides the range criterion for Schmidt numbers, the detection tool that certifies SN>2."},{"cited_title":"Secure key from bound en- tanglement,","cited_arxiv_id":null,"evidence_quote":"Gives the previous 2k×2k examples that the new homogeneous family improves upon."},{"cited_title":"Quantum grid states and hy- brid graphs,","cited_arxiv_id":null,"evidence_quote":"Used to compute the Gröbner-basis membership certificates for k=4 and 5 in the supplemental material."}],"review_version":1}