{"id":"bf45f57d-37b3-40e9-854a-b5cf9fe5eac2","arxiv_id":"2607.27091","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"First explicit optimal pure CSS-like EAQLRC families are constructed from ℓ-intersection MDS pairs and block parity-check matrices, with Singleton-like optimality and nontrivial locality.","lead":"The paper defines entanglement-assisted quantum locally recoverable codes and builds the first explicit optimal families of them. It removes dual-containing constraints that block many classical LRCs from becoming quantum codes, by using pre-shared entanglement whose receiver halves stay noiseless.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"Full read of Defs 1–6, Thms 1–7, framework §5.1, and both examples. The strongest claim (two explicit optimal pure CSS-like EAQLRC families with flexible parameters and nontrivial locality) is internally supported: CSS-like construction without dual-containing (Thm 2), Singleton-like bound (Thm 4), pure equality criterion (Thm 5), and two applications that check the three Thm-5 conditions and force purity by matching the bound. The sufficient-only locality criterion is a genuine caveat on interpreting “minimum” locality, but the paper’s claims are carefully scoped to constructive r and to nontriviality versus the Thm-3 upper bound; they do not assert a necessary characterization. Cited existence (Lemma 6 / [20]; block GRS Lemma 7 / [28]) is used within stated exceptions. No load-bearing algebraic error surfaced. Reader’s ACCEPT / HIGH / low correctness risk stands; no verdict adjustment.","tokens_in":25789,"tokens_out":606,"duration_ms":54232,"concrete_test":"Recompute Example 1 explicitly: with the given H_X, H_Z over F_5, verify rank(H_X H_Z^T)=1, dim(C_X ∩ C'_Z)=1, both codes are [5,3,3]_5 MDS, paired weight-4 dual supports exist for every coordinate, and the EA-CSS parameters are exactly J5,2,3;1K_5 with constructive r=3 < Thm-3 bound 4. If any of these fail, the Thm-6 claim weakens; if all hold, the central existence claim is confirmed on a concrete instance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The softest point remains the one the reader flagged: locality is certified only by sufficient support conditions (Thm 1 / paired dual supports in Thm 2), so the constructive r is an upper bound on true minimum locality. This does not undermine the central claim. The families in Thms 6–7 are still EAQLRCs with the stated r (Def. 6 only requires recovery sets of size ≤ r), they meet equality in the Singleton-like bound (13) for that r via the Thm 5 criterion, and nontriviality is witnessed by r strictly below the Thm 3 upper bound (Remarks 4, Examples 1–2). Purity is forced cleanly: construction gives δ ≥ d while (13) gives 2δ ≤ 2d. No hidden dual-containing assumption, gap in the rank/D-matrix calculations, or break in the ℓ-intersection or block-matrix existence conditions was found that would falsify the parameter claims.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces entanglement-assisted quantum locally recoverable codes (EAQLRCs), defining locality for EAQECCs so that a single transmitted-qudit erasure is recovered from at most r other transmitted qudits plus Bob’s noiseless entangled halves. It gives sufficient support conditions on extended stabilizers for an EASC to have locality r (Theorem 1), a CSS-like construction from two classical codes without dual-containing (Theorem 2 / Corollary 1), an upper bound on locality (Theorem 3), and a Singleton-like bound 2δ ≤ n+c−κ−2⌈κ/r⌉+4 for CSS-like EAQLRCs (Theorem 4), with necessary and sufficient conditions for pure codes to meet equality (Theorem 5). A three-step framework using a nonsingular diagonal matrix D to tune entanglement while preserving paired dual supports then yields two explicit infinite families of optimal pure CSS-like EAQLRCs: one from ℓ-intersection MDS pairs (Theorem 6) and one from block parity-check matrices with diagonal twists (Theorem 7), both with flexible parameters and nontrivial localities, claimed as the first explicit EAQLRC families.","tokens_in":26021,"tokens_out":988,"duration_ms":28566,"significance":"The work cleanly removes the dual-containing bottleneck that has limited qLRC constructions from classical LRCs, and supplies the first explicit optimal pure EAQLRC families with nontrivial locality. The technical core is standard and carefully executed: Pauli commutation for local recovery, EA-CSS parameters via rank(H_X H_Z^T), locality upper bounds by column-basis counting, and Singleton-like optimality reduced to classical LRC Singleton plus pure-code ceiling conditions. The diagonal-D framework is a useful constructive device for adjusting c without breaking paired supports. If the claims hold—as the derivations indicate—they open a systematic route from the large literature on optimal cLRCs to quantum codes with local recovery under entanglement assistance, and they set concrete benchmarks for subsequent EAQLRC work.","major_comments":[],"minor_comments":[{"comment":"After Theorem 3 the paper correctly notes that Theorems 2–1 give only sufficient conditions, so the constructive r is an upper bound on true minimum locality, and optimality is relative to that r. A short explicit sentence in the abstract or the statement of Theorems 6–7 (e.g., “locality at most r”) would prevent readers from over-reading “locality r” as a proven minimum.","section":"§3–§5, Theorems 6–7"},{"comment":"A compact parameter table comparing the new EAQLRC families (n, κ, δ, c, r) against the best known pure CSS/Hermitian qLRCs of similar length and distance would make the gain from entanglement assistance more visible; the text currently relies on narrative comparison.","section":"§5 / Conclusion"},{"comment":"In Example 1 the matrices are small enough that dim(C_X ∩ C'_Z)=1 and the Thm 3 bound equal to 4 can be stated as a one-line verification; likewise for Example 2. Adding these one-line checks in the examples would help readers confirm nontriviality without recomputing.","section":"Examples 1–2"},{"comment":"Notation: lab(G) and supp_Q(G) are introduced in §2 and used heavily in Theorem 1; a brief reminder at the start of §3.2 would ease reading. Also, the phrase “CSS-like EAQLRC” is used for both pure and impure codes from Theorem 2—consistent, but worth one clarifying sentence when the Singleton bound is stated for both.","section":"§2–§4"},{"comment":"Minor typographical/consistency items: AMS classification line is fine; ensure “EAQLRC” vs “EA-CSS code Q(C_X,C_Z)” is used consistently when purity is discussed; in Theorem 7 the condition r>2d−4≥2 is slightly dense—splitting “r≥3 and d≥3 with r>2d−4” may help.","section":"Throughout / Theorem 7"}],"recommendation":"accept","confidential_remarks":"Solid first paper on the topic; the reader’s and skeptic’s assessments align with mine. No load-bearing gaps found. Accept is appropriate; the minor points are presentation only. Fit for a coding-theory / IEEE TIT-style venue is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper does the clean thing the subfield needed: it defines locality for entanglement-assisted codes, removes the dual-containing bottleneck that blocks most good classical LRCs, and actually builds the first explicit optimal families.\n\nWhat is new is concrete. They model single-qudit erasure with Bob’s halves noiseless, give sufficient support conditions on extended stabilizers (Thm 1), and get a CSS-like construction from any two classical codes via rank(H_X H_Z^T) with no dual-containing requirement (Thm 2). The locality upper bound (Thm 3) and Singleton-like bound 2δ ≤ n+c−κ−2⌈κ/r⌉+4 (Thm 4) are standard reductions to classical LRC Singleton plus EA-CSS dimension formulas; the pure-code equality criterion (Thm 5) is useful. The framework with a nonsingular diagonal D that tunes c while preserving paired supports is a practical device. Thms 6–7 then deliver two infinite families—ℓ-intersection MDS pairs and block parity-check matrices—attaining the bound with nontrivial locality (r strictly below the Thm 3 ceiling), with clean examples.\n\nThe soft spot the reader flagged is real but proportionate: locality is certified by sufficient support conditions, so the stated r is an upper bound on true minimum locality. That does not break the claims. Definition 6 only requires recovery sets of size ≤ r; the families meet equality in (13) for that r via Thm 5; purity is forced (construction δ ≥ d, bound 2δ ≤ 2d); and nontriviality is witnessed explicitly. No hidden dual-containing assumption or gap in the rank/D calculations turned up on a full read. Existence leans on cited MDS-intersection and block-matrix lemmas, which is normal for this literature.\n\nThis is for people who build qLRCs or EAQECCs and want usable parameters without dual-containing. Math is checkable from the text. I would send it to referees and would cite the families and the framework.","headline":"Solid first definition and constructions of EAQLRCs: dual-containing-free CSS route, Singleton-like bound with purity criterion, and two explicit optimal families with nontrivial locality.","tokens_in":26678,"tokens_out":528,"would_cite":true,"duration_ms":9244,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B35","81P45"],"pacs":[],"model":"grok-4.5","headline":"Entanglement assistance removes dual-containing barriers and yields the first explicit optimal quantum locally recoverable codes.","keywords":["locally recoverable code","entanglement-assisted quantum code","CSS construction","ℓ-intersection pair","block parity-check matrix","Singleton-like bound","quantum erasure correction"],"falsifier":"For the concrete parameters of Theorems 6 and 7 (or the worked Examples 1–2), compute the actual minimum recovery-set sizes of the resulting EA-CSS code and check whether 2δ equals n+c−κ−2⌈κ/r⌉+4 with that true r; any strict improvement in locality or failure of purity would break the optimality claim.","tokens_in":26660,"feed_emoji":"🔗","tokens_out":1066,"duration_ms":19355,"temperature":0.7,"pith_summary":"Standard quantum locally recoverable codes force classical ingredients to be dual-containing or self-orthogonal, so many well-designed classical LRCs cannot be used. This paper defines entanglement-assisted quantum LRCs (EAQLRCs), in which pre-shared noiseless entanglement halves let the quantum code be built from two classical codes without that constraint. It gives sufficient support conditions on extended stabilizers for locality r, a CSS-like construction, an upper bound on locality, and a Singleton-like bound, plus exact conditions under which pure codes meet the bound. A general framework then turns pairs of classical LRCs into optimal pure EAQLRCs; applying it to MDS intersection pairs and block parity-check matrices produces two infinite families with flexible parameters and nontrivial localities—the first explicit EAQLRC families claimed in the literature.","feed_headline":"First explicit optimal quantum LRCs via entanglement aid","feed_subtitle":"Shared noiseless pairs drop dual-containing rules and produce two infinite optimal families","key_machinery":"The CSS-like EA construction (Theorem 2) together with the three-step framework of Subsection 5.1: start from parity-check matrices of optimal classical LRCs that already satisfy paired dual supports, rescale one matrix by a nonsingular diagonal D to tune the entanglement count c = rank(HX D HZ⊤) while preserving supports and locality, then prove purity so the resulting pure EAQLRC meets the Singleton-like bound.","core_discovery":"Entanglement-assisted stabilizer codes admit locality r whenever, for each transmitted position, two extended stabilizers exist whose joint support outside that position has size at most r and whose local Pauli actions distinguish X and Z errors; from this the authors obtain a CSS-like EAQLRC construction from any two classical codes whose duals satisfy a paired-support condition, without dual-containing. Pure codes meeting the derived Singleton-like bound 2δ ≤ n+c−κ−2⌈κ/r⌉+4 are completely characterized, and two explicit infinite families attain that bound with nontrivial locality.","pith_inferences":["Because the locality certificate is only sufficient, a follow-up necessary-and-sufficient stabilizer criterion could shrink reported localities and tighten the Singleton-like bound for the same codes.","The same diagonal-tuning idea may lift other classical LRC families (Tamo–Barg, good polynomials, design-supported codes) into EAQLRCs once paired-support dual checks are arranged.","If impure EAQLRCs can be shown to beat the pure Singleton-like equality cases, the paper’s optimality framework would need an impure counterpart.","The entanglement-assisted setting suggests a natural next question: hierarchical or (r,δ)-locality for EAQLRCs, paralleling recent ordinary qLRC extensions."],"forward_implications":["Any pair of optimal classical LRCs whose duals share small joint supports can be turned into an optimal pure EAQLRC without forcing dual-containing.","The diagonal rescaling D gives a systematic way to vary entanglement consumption c and logical dimension while keeping length, distance, and locality fixed.","ℓ-intersection MDS pairs yield optimal pure Jn, n+c−2d+2, d; cKq EAQLRCs of locality n−d+1 for a wide admissible range of c.","Block parity-check matrices yield optimal pure Ju(r+1), ur−2d+4+s, d; u+sKq EAQLRCs of locality r for 0≤s≤d−2 under the stated arithmetic conditions.","These constructions supply the first concrete benchmark families against which future EAQLRC bounds and constructions can be compared."],"fun_headline_variants":["Entanglement aid drops dual-containing for optimal quantum LRCs","First explicit pure EAQLRC families from classical LRC pairs","CSS-like EAQLRCs meet Singleton-like bound with nontrivial locality","Shared noiseless pairs yield two infinite optimal EAQLRC families","Support conditions give locality-r entanglement-assisted stabilizer codes"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Locality is only guaranteed by sufficient support conditions on stabilizers or dual codewords; the true minimum locality of a constructed code could be smaller than the r used to claim optimality.","fun_headline_variants_meta":{"raw":{"variants":["Entanglement aid drops dual-containing for optimal quantum LRCs","First explicit pure EAQLRC families from classical LRC pairs","CSS-like EAQLRCs meet Singleton-like bound with nontrivial locality","Shared noiseless pairs yield two infinite optimal EAQLRC families","Support conditions give locality-r entanglement-assisted stabilizer codes"]},"model":"grok-4.5","effort":"low","cost_usd":0.004065,"raw_usage":{"total_tokens":1308,"prompt_tokens":837,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":40648000,"prompt_tokens_details":{"text_tokens":837,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":400,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":837,"tokens_out":71,"duration_ms":7200,"temperature":1.0,"reasoning_tokens":400,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T11:25:15.275172+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For the concrete parameters of Theorems 6 and 7 (or the worked Examples 1–2), compute the actual minimum recovery-set sizes of the resulting EA-CSS code and check whether 2δ equals n+c−κ−2⌈κ/r⌉+4 with that true r; any strict improvement in locality or failure of purity would break the optimality claim.","supporting_citations":[],"review_version":1}