{"id":"e57141a8-37e5-4f1a-aaec-44326a5110eb","arxiv_id":"2608.04403","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For quantum codes built from finite-field linear codes, a more accurate locality measure and an algorithm reduce the number of qudits and measurements needed for erasure repair.","lead":"This paper proposes a quantum version of information locality, a measure of how many quantum data units are actually needed to repair erased data in a locally recoverable code. It gives a linear algebra procedure to find smaller repair groups and fewer measurements, and shows by explicit examples that the previous definition overestimates or can underestimate the required resources.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central definition, Proposition 1, and both examples are internally consistent, with only expected reliance on cited external criteria.","rationale":"The reader accepted the paper with high confidence, and my independent check of the main proof and examples agrees. The most vulnerable points are indeed the external criterion (3) from [14] and the minimal-observables theorem [30, Thm 5], but reliance on published theorems is normal and does not amount to an internal flaw. I found no algebraic error in Proposition 1: the construction of J, the verification of condition (3), the size bound, and the orthogonality/decomposition argument for the observables all check out. The Section 4 example is consistent and the brute-force verification mentioned in the data availability statement provides independent support. The Section 5 example is explicitly scoped and does not undermine the Hermitian-construction claim. The only minor gap is that Proposition 7's exact information-locality value is not fully proved in the written text, but it is a parameter value in an illustrative example, not a load-bearing part of the central claim; even if the true information locality were 4 rather than 5, the comparison with symbol locality would still hold. Therefore the correct verdict is unchanged: ACCEPT.","tokens_in":16139,"tokens_out":45760,"duration_ms":517582,"concrete_test":"Run the supplied C program (or an independent exhaustive enumeration over all subsets J of {1,...,16}) for the Section 4 code, computing dim π_J(C) and w_H(π_J(C)\\σ_J(C^{⊥h})) for every J, and confirm that no J with dim π_J(C)≤4 has local distance at least 3. This settles whether the stated information locality (5,3) is exact rather than merely an upper bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the main construction rather than assuming it. Proposition 1's proof is sound: after row-reducing the basis of sigma_{J_j\\I}(C^{⊥h}), the pivot set P is zeroed in the remaining rows, the resulting J=I∪S satisfies condition (3), and the size bound |J|-|I|≤dim π_{J_j}(C)-dim σ_I(C^{⊥h}) follows by dimension counting. The direct-sum statement (11) also holds because every nonzero element of sigma_{J_j\\I}(C^{⊥h}) has a nonzero component in a pivot column of P, so its intersection with sigma_J(C^{⊥h}) is zero. The Section 4 example is algebraically consistent, and the Section 5 CSS counterexample is explicit and correctly shows that the averaged-dimension bound (17) fails when C_X≠C_Z. The genuinely external load-bearing inputs are the Galindo et al. iff criterion (Eq. 3) and the author's prior Theorem 5 for minimal observables; both are cited published results, and nothing in this manuscript gives reason to doubt them. The only small caveat is that Proposition 7 states information locality (5,3), while the written proof gives an upper bound via J=K and proves only the symbol-locality lower bound; exactness of the information-locality value relies on the supplied brute-force C program. This does not affect the central definition, algorithm, or the qualitative comparison with symbol locality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a quantum analogue of classical information locality for stabilizer codes constructed from Hermitian dual-containing codes. The central object is Definition 3, which replaces the repair-group size bound of the quantum symbol locality of Galindo et al. by a bound on dim π_{J_j}(C). Proposition 1 gives an O(|J_j|^3) linear-algebraic algorithm that, given an erasure set I and a repair group J_j, constructs a smaller repair group J satisfying the correction criterion (3) and simultaneously provides a basis of observables for local recovery; the size bound (7) and the direct-sum decomposition (11) are the key technical results. Proposition 4 translates the locality parameter into an operational statement about the number of additional qudits needed. The paper then gives two examples: a [[16,2,3]]_2 Hermitian stabilizer code claimed to have symbol locality (6,3) and information locality (5,3), and a CSS/Euclidean example showing that the averaged-dimension bound (17) can fail when C_X ≠ C_Z. The Section 4 example is supported by a brute-force C program included with the arXiv submission.","tokens_in":16379,"tokens_out":22366,"duration_ms":255321,"significance":"If correct, the paper fills a clear gap in the quantum LRC literature: classical information locality was known to estimate repair cost more accurately than symbol locality, and the quantum analogue proposed here does the same for Hermitian stabilizer codes. The algorithmic content of Proposition 1 is a genuine contribution: it not only finds a smaller repair group but also minimizes the number of measured observables, and the proof is a detailed constructive linear-algebra argument. The paper is also honest about the limits of the approach, showing by an explicit CSS example why a natural Euclidean translation fails. The inclusion of a brute-force verifier for the Section 4 claims is a strength, though the manuscript should make the role of that verifier explicit in the proofs. The main caveats are the reliance on the Galindo et al. criterion (3) and on the author's prior Theorem 5 for observable minimality; both are published results and are used without restatement.","major_comments":[{"comment":"Proposition 10 aims to rule out symbol locality (5,3), but its proof only considers repair groups J with |J| = 7. The definition of symbol locality in Section 3.1 requires only |J| ≤ r + δ − 1, so a repair group of size at most 6 with local distance at least 3 would still witness (5,3). The sentence 'there must exist a repair group J ... such that |J| = 7' is therefore unjustified, and the case |J| < 7 is not analyzed in the text. Please complete the case analysis or state explicitly in the proof that all subsets with |J| < 7 are checked by the accompanying brute-force C program and indicate where the verification is reported.","section":"Section 4, Proposition 10"},{"comment":"The proposition asserts the exact value 'information locality (5,3)', but the written proof establishes only the upper bound via the repair groups K and L. It does not prove that no repair group J (of any size) with local distance at least 3 has dim π_J(C) ≤ 4; this lower bound appears to rest entirely on the C program mentioned in the Data Availability statement. Please make this reliance explicit in the proof, or provide an analytic argument, and state precisely which claims are machine-verified.","section":"Section 4, Proposition 7"}],"minor_comments":[{"comment":"The proofs are omitted with the note that they are analogous to arguments in [14]; since Proposition 6 is used to derive the quantum Singleton-like bound (12), please include the proofs or precise lemma references so the derivation is checkable.","section":"Section 3, Propositions 5 and 6"},{"comment":"The claim that the computed observables attain the minimum possible number depends on Theorem 5 of [30], which is not stated in the manuscript; please include the statement of the theorem or a short self-contained argument.","section":"Section 3, Remark 2"},{"comment":"The notation I ∋ j appears in Proposition 1 and Proposition 4 without definition; please define that j is an element of I and that I is the actual erasure set.","section":"Section 2 and Section 3"},{"comment":"The sentence 'The absolute minimum weight is bounded solely by the H0 subcodes and we see w_H(C^{⊥h}) = 4' is terse; please expand the case analysis (for example, noting that any vector involving the row (v,v) has weight at least 6) to make the computation self-contained.","section":"Section 4, proof of Proposition 7"},{"comment":"Since the exactness claims in Section 4 are machine-verified, please archive the C program and its output with the final version and cite it in the proof of Proposition 7, so that the reliance on brute-force verification is transparent.","section":"Data Availability"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a quantum information and coding theory journal. The self-citations [14] and [30] are appropriate: [30] is the author's prior work but is published and directly relevant. The main revision needed is to make the proof of Section 4 either self-contained or explicitly machine-verified; the current text contains a logical gap in Proposition 10 and does not prove the exact information-locality lower bound. I do not see grounds for rejection, but the manuscript should not be accepted until these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know three things. First, this paper defines quantum information locality for Hermitian stabilizer codes, which is genuinely new: the classical notion had no quantum counterpart before. Second, it gives a concrete O(|J|^3) linear-algebra algorithm that finds a smaller repair group than the given one, with a size bound in terms of punctured-code dimension, and that simultaneously minimizes the number of measured observables. Third, it shows by explicit example that the earlier symbol-locality definition overestimates the needed qudits, and by a second example that a naive translation of information locality to CSS codes with C_X ≠ C_Z fails. The paper is honest about this limitation and does not oversell the scope.\n\nThe math holds up. Proposition 1's proof is sound; the pivot-column elimination argument is clear and the size bound follows by dimension counting. The Section 4 example is worked out carefully, and the Section 5 counterexample is explicit and convincing. I checked the construction rather than assuming it, and I did not find a hidden gap. The C program supplied for brute-force verification is a plus, and the author discloses AI assistance in finding examples, which is fine.\n\nSoft spots are minor but worth naming. The whole local-recovery framework rests on the Galindo et al. iff criterion (Eq. 3); the paper's results live or die with that external result. That is a published criterion with no reason for suspicion, but it is load-bearing. The claim that the computed observables are minimum possible depends on the author's prior Theorem 5, cited but not restated; a referee will want that theorem stated or the dependence made explicit. Propositions 5 and 6 have omitted proofs, but they are straightforward analogs of [14]. Finally, Proposition 7 states information locality (5,3) as exact, while the written proof gives an upper bound and proves only the symbol-locality lower bound; exactness relies on the brute-force program. That is a small gap between proof text and statement.\n\nThis is a paper for the quantum LRC community, and for anyone doing erasure correction with measurement reduction in mind. It deserves a serious referee and likely acceptance after the minor issues are cleaned up. I would cite it in my own work on quantum local recovery, and I would bring it to a reading group focused on quantum LRCs. My recommendation: send it out, ask for the C program to be inspected or the lower-bound argument expanded, and accept after minor revision.","headline":"A well-built paper that introduces quantum information locality, proves a useful repair-group-shrinking algorithm, and honestly maps its limits; it deserves refereeing and likely acceptance after minor revisions.","tokens_in":16894,"tokens_out":1599,"would_cite":true,"duration_ms":19976,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P73","94B65","94B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces a quantum information locality parameter for stabilizer codes, proves a linear-algebraic routine that finds the smallest repair group and fewest measured observables, and demonstrates on explicit codes that the older…","keywords":["quantum error correction","local recovery","locally recoverable codes","information locality","symbol locality","erasures","quantum stabilizer codes","measurement reduction"],"falsifier":"For the explicit [[16,2,3]]_2 code of Section 4, enumerate all 7-symbol subsets J ⊂ {1,...,16} and compute w_H(π_J(C) \\ σ_J($C^{{⊥_h}}$)); Proposition 10 asserts that every such J has value at most 2, so finding any 7-symbol repair group with quantum local distance at least 3 would refute the claimed separation between information and symbol locality.","tokens_in":15919,"feed_emoji":"⚛️","tokens_out":9359,"duration_ms":106266,"temperature":0.7,"pith_summary":"The paper gives quantum local recovery a parameter that tells the truth about how many extra qudits are needed to repair several erasures. Classical codes already solved this problem with information locality, which counts the dimension of the code's projection onto a repair group rather than the repair group's raw size. The paper translates that idea to quantum stabilizer codes built from Hermitian dual-containing linear codes, defines quantum information locality, and provides an O(|J|^3) linear-algebra procedure that finds a repair group achieving the bound while minimizing the number of observables one must measure. An explicit [[16,2,3]]_2 code shows that the previously proposed quantum symbol locality overestimates the needed qudits (6 instead of 5). A second example shows why the same definition cannot be carried over to Euclidean CSS codes with different X and Z component codes: the dimension-based bound can fail outright.","feed_headline":"New quantum locality measure trims qudits needed for erasure repair","feed_subtitle":"A linear-algebra routine finds the smaller repair group and the fewest measurements; an explicit code shows the gap.","key_machinery":"The load-bearing object is the repair-group projection identity, π_I(C) = π_I(σ_J($C^{{⊥_h}}$)), which the paper inherits from earlier work and uses as the exact criterion for whether erasures in I can be corrected using qudits only in J. The algorithm works by decomposing the shortened dual code σ_{J_j}($C^{{⊥_h}}$) into three summands, row-reducing the part supported off the erasure set, and eliminating its pivot columns from the remaining basis vectors; the support of the resulting rows defines the smaller repair group J. The same decomposition yields the direct-sum relation σ_{J_j}($C^{{⊥_h}}$) = σ_{J_j\\I}($C^{{⊥_h}}$) ⊕ σ_J($C^{{⊥_h}}$), which is what lets the procedure read off a minimum-size set of measured observables.","core_discovery":"The paper's central claim is that the classical information-locality idea, using the dimension of the punctured code rather than the repair-group size to measure recovery cost, does have a working quantum analogue within the Hermitian stabilizer construction. Definition 3 sets quantum information locality (r_Q,i, δ_Q) by requiring, for each index j, a repair group J_j with j in J_j, w_H(π_{J_j}(C) \\ σ_{J_j}($C^{{⊥_h}}$)) ≥ δ_Q and dim π_{J_j}(C) ≤ r_Q,i. Proposition 4 then says that any I ⊆ J_j with |I| ≤ δ_Q − 1 can be corrected using at most r_Q,i qudits in addition to the erased ones. The supporting algorithm, Proposition 1, starts from any repair group satisfying the correctability identity π_I(C) = π_I(σ_J($C^{{⊥_h}}$)) and shrinks it to a subset J with |J| − |I| ≤ dim π_{J_j}(C) − dim σ_I($C^{{⊥_h}}$), in O(|J_j|^3) arithmetic operations, while also producing observables whose number is minimal by a theorem from the author's earlier work. The paper demonstrates the advantage on an explicit [[16,2,3]]_2 code with information locality (5,3) versus symbol locality (6,3), and it shows the obstacle for Euclidean CSS codes with C_X ≠ C_Z by an explicit length-6 failure of the corresponding size bound.","pith_inferences":["Editorial extension: the same 'shrink the repair group by row-reducing the shortened dual' trick could be used as a general measurement-reduction preprocessing step in any stabilizer erasure-correction routine, independent of locality, since the direct-sum relation is a structural property of the stabilizer.","Editorial extension: the paper fixes the erasure set I before shrinking J; a natural next test is whether choosing J with I unknown (worst-case or random erasures) still yields near-minimal repair groups on average.","Editorial extension: the Section 5 obstacle suggests a possible definition for asymmetric CSS codes using max(dim π_{J_j}(C_X), dim π_{J_j}(C_Z)) or a min-type formulation, which the paper does not explore.","Editorial extension: because r_C,i = r_Q,i when C contains its Hermitian dual, known classical information-locality constructions that are dual-containing automatically yield quantum codes with the same information locality; the paper notes the classical-to-quantum direction but does not build new code families from it."],"forward_implications":["For any Hermitian-constructed stabilizer code with quantum information locality (r_Q,i, δ_Q), erasure sets of size at most δ_Q − 1 are repairable using at most r_Q,i qudits in addition to the erased ones.","The Proposition 1 algorithm produces, in O(|J_j|^3) arithmetic operations, a repair group J ⊆ J_j that still satisfies the correctability identity and meets the dimension-based size bound.","The same computation yields a set of measured observables whose size is the minimum possible for the punctured code to correct the erasures, not merely a convenient set.","The explicit [[16,2,3]]_2 stabilizer code has quantum information locality (5,3) and quantum symbol locality (6,3), so the older definition overestimates the number of extra qudits needed for erasure correction.","For pure Hermitian-constructed stabilizer codes, the quantum Singleton-like bound applies to information locality, via the classical bound and the paper's Proposition 6."],"supporting_citations":[{"why":"Supplies the correctability criterion (Eq. (3)) and the earlier quantum symbol-locality definition that the paper's information locality refines.","marker":"[14]"},{"why":"Defines classical information locality, the notion being translated into the quantum setting.","marker":"[23]"},{"why":"Provides the theorem used to assert that the computed set of observables is minimum possible.","marker":"[30]"},{"why":"Introduced the classical (r,δ) symbol locality whose overestimation motivates the information-locality refinement.","marker":"[33]"}],"fun_headline_variants":["Quantum info locality cuts erasure repair qudit count","Algorithm shrinks quantum repair groups, cuts measurements","Quantum information locality beats symbol locality on a 16-qubit code","Tighter quantum erasure repair via punctured-code locality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire framework inherits the earlier if-and-only-if criterion, stated as Eq. (3), that erasures in I can be repaired inside J exactly when the projection of the code onto I equals the projection of the shortened dual onto I, and the minimal-observable claim additionally depends on a theorem from the author's prior work that is cited but not restated here; if either gives way, the size bound, the algorithm, and the new locality parameter all collapse.","fun_headline_variants_meta":{"raw":{"variants":["Quantum info locality cuts erasure repair qudit count","Algorithm shrinks quantum repair groups, cuts measurements","Quantum information locality beats symbol locality on a 16-qubit code","Tighter quantum erasure repair via punctured-code locality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000697,"raw_usage":{"total_tokens":3268,"prompt_tokens":1180,"completion_tokens":2088,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":796,"completion_tokens_details":{"reasoning_tokens":2022}},"tokens_in":796,"tokens_out":2088,"duration_ms":19804,"temperature":1.0,"reasoning_tokens":2022,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:37:20.758666+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit [[16,2,3]]_2 code of Section 4, enumerate all 7-symbol subsets J ⊂ {1,...,16} and compute w_H(π_J(C) \\ σ_J($C^{{⊥_h}}$)); Proposition 10 asserts that every such J has value at most 2, so finding any 7-symbol repair group with quantum local distance at least 3 would refute the claimed separation between information and symbol locality.","supporting_citations":[{"cited_title":"Galindo, F","cited_arxiv_id":null,"evidence_quote":"Supplies the correctability criterion (Eq. (3)) and the earlier quantum symbol-locality definition that the paper's information locality refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines classical information locality, the notion being translated into the quantum setting."},{"cited_title":"Matsumoto, Reducing measurements in quantum erasure correction by quantum local recovery, Computational and Applied Mathematics (2026).doi:10.1007/s40314-026-03881-4","cited_arxiv_id":null,"evidence_quote":"Provides the theorem used to assert that the computed set of observables is minimum possible."},{"cited_title":"Prakash, G","cited_arxiv_id":null,"evidence_quote":"Introduced the classical (r,δ) symbol locality whose overestimation motivates the information-locality refinement."}],"review_version":1}