{"id":"1b56df41-f600-4b30-b11d-9439ff42b58a","arxiv_id":"2608.06854","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Entanglement-assisted quantum locally recoverable codes can be constructed from arbitrary classical LRC pairs, and this paper proves bounds, optimality conditions, and explicit constructions for them.","lead":"This paper shows that pre-shared entanglement lets quantum codes with local repair be built from pairs of ordinary classical storage codes, without the self-orthogonality that earlier constructions required. It derives new limits on such codes, identifies when they can be optimal, and gives explicit families that reach or miss those limits.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed purity of the Tamo–Barg and general cyclic-pair constructions (Theorems 9–12) is asserted without proof: Theorem 1 only guarantees δ = min{wt(C2\\(C1⊥∩C2)), wt(C1\\(C1∩C2⊥))}, and equality with min{d1,d2} requires minimum-weight codewords to avoid the hull intersection, which is never…","rationale":"The reader's weakest assumption precisely identifies the same load-bearing concern: the purity of Theorems 9–12 is asserted without proof. My independent reading of the full text confirms that Theorem 11's proof stops after computing dimension and entanglement, omitting any analysis of whether the minimum-weight codewords of the constituent cyclic codes avoid the relevant hull intersections. For the Tamo–Barg family, Proposition 2 determines only |S∩S⊥|, i.e. the hull dimension, not the weights of codewords outside the hull; likewise Theorems 9–10 simply state 'pure' without deriving δ=wt(C\\(C∩C⊥)). Since Theorem 1's distance expression is a minimum of weights of set differences that can exceed the individual minimum distances, the claimed parameter triples and the optimality boundary in Section VI rest on an unverified assumption. The concern does not affect the correctness of the bound derivations in Section IV or the LCD-based constructions in Section VII, where purity is automatic; those parts of the paper have independent support. The conditional verdict remains appropriate: the paper should either prove purity for the Tamo–Barg and cyclic-pair constructions, or weaken the claims to the actual distances from Theorem 1.","tokens_in":39664,"tokens_out":9383,"duration_ms":97667,"concrete_test":"For the Tamo–Barg code of Theorem 10 with q=13, r=3, ℓ=7 (n=12), compute the hull H=C_TB∩C_TB⊥ and the minimum weight of the nontrivial coset C_TB\\H by enumerating c+H for a single c∈C_TB\\H; if minwt(C_TB\\H)>q−ℓ=6, then the claimed pure parameter δ=q−ℓ is false and the purity assumption in Theorems 9–10 fails. Independently, for Example 2, check whether a minimum-weight codeword of C1 lies in C1∩C2⊥; if so, Theorem 12's distance min{d1,d2}=14 is understated and the purity claim is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is the unproven purity of the non-LCD constructions. Theorem 11 states parameters [[n, n−|(−Z1)∪Z2|, min{d1,d2}; |(−Z1)∩Z2|]]_q, but its proof only computes dimension and entanglement; it never shows wt(C2\\(C1⊥∩C2)) = d2 or wt(C1\\(C1∩C2⊥)) = d1. Theorems 9–10 similarly rest on Proposition 2, which computes only the hull size s=|S∩S⊥|, not whether the hull contains minimum-weight codewords. Theorem 1's distance formula can be strictly larger than min{d1,d2}; hence the claimed 'pure' parameters and the optimality boundary in Section VI are unsupported. The issue is load-bearing because the Tamo–Barg non-optimality claim (Remark 11) and the cyclic-pair optimality discussion (Remark 12) both presuppose δ=min{d1,d2}; if the actual distance is larger, equality in (12) could occur outside the claimed k≤r regime, and the parameters from Theorem 12 would be misreported. The LCD-based maximally entangled constructions (Theorem 13/14) and the GV achievability bounds are unaffected, since LCD gives C∩C⊥={0} and purity is automatic.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a framework for entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built from pairs of classical LRCs via a CSS-like construction that does not require dual-containment. It defines EA-qLRCs through recovery channels, proves a sufficient stabilizer-based locality criterion (Theorem 2), derives four explicit converse bounds (Singleton-, Griesmer-, Plotkin-, and sphere-packing-like) together with a Cadambe-Mazumdar-like bound, characterizes equality in the Singleton-like bound for pure codes, constructs families from Tamo-Barg and cyclic codes, provides optimal maximally entangled constructions from LCD cyclic codes, and establishes two Gilbert-Varshamov-like achievability bounds for q>3. It closes with a unified comparison of all bounds in the maximally entangled regime.","tokens_in":39998,"tokens_out":11364,"duration_ms":115355,"significance":"If the construction claims are fully justified, this is a substantial contribution to quantum locally recoverable coding: it opens classical LRC families to quantum local recovery via entanglement assistance, supplies explicit finite-length and asymptotic bounds, and provides several optimal or near-optimal families. The stabilizer criterion of Theorem 2, the subcode-puncturing reduction in Lemma 5 and Theorem 3, and the LCD-based maximally entangled constructions of Section VII are coherent and appear sound. The Gilbert-Varshamov-type achievability results for q>3, obtained through monomial equivalence to LCD codes, are a useful and non-obvious extension. The principal unresolved point is the unproved purity of the non-LCD Tamo-Barg and general cyclic-pair constructions, which affects the reported parameters and the optimality boundary claimed in Sections VI and VII.","major_comments":[{"comment":"The purity of the Tamo-Barg and general cyclic-pair constructions is asserted but not proved. Theorem 1 defines the EA distance as δ = min{wt(C1\\(C1∩C2⊥)), wt(C2\\(C1⊥∩C2))}; equality δ = min{d1,d2} requires that all minimum-weight codewords of each constituent code avoid the corresponding hull intersection. Proposition 2 computes only the hull dimension s=|S∩S⊥|, and the proof of Theorem 11 computes only κ and c from Lemma 8; neither verifies the required weight condition. Consequently, the parameters in Theorem 9 ([[q−1, 1, ≥q−ℓ, r; ...]]), the distance min{d1,d2} in Theorems 11-12, the reported [[36,11,14;12]] code in Example 2, and the non-optimality conclusions of Remarks 11-12 all rest on an unverified hypothesis. If the actual δ is strictly larger than min{d1,d2}, equality in (12) could occur outside the claimed k≤r regime, and the optimality boundary would change. The authors should either prove that minimum-weight codewords of the constituent codes avoid the relevant hull intersections or restate the theorems with δ in place of min{d1,d2} and adjust the optimality discussion accordingly.","section":"Section VI, Theorems 9-12 and Remarks 11-12"},{"comment":"The asymptotic form of the sphere-packing-like bound is derived by fixing τ while n→∞, but the maximum in (16) is taken over τ ranging up to Θ(n). The true leading-order behavior requires optimizing over α = τ/n, and the resulting expression depends on r. Using only fixed τ yields a valid but strictly weaker upper bound; the claimed r-independence in Remark 7 and the tightness comparison in Remark 8 and Figures 7-8 are therefore not established for the actual bound (16). The authors should provide the optimized asymptotic form or explicitly label (23) as a non-optimized relaxation and qualify the hierarchy and tightness claims accordingly.","section":"Section IV, Eq. (16) and Eq. (23)"}],"minor_comments":[{"comment":"The notation [m]† = {0,...,m} and [m] = {1,...,m} is introduced in Section II, but later sections use [ℓ−1], [n−1], and similar sets inconsistently; for example, Proposition 2 writes S∩S⊥ using [ℓ−1] where [ℓ−1]† appears intended. Please standardize the interval notation.","section":"Notation, throughout"},{"comment":"Theorem 13 states parameters [[n,k,d,n−k]]_q; the standard notation is [[n,k,d;n−k]]_q. Also, since the theorem is in the LRC section, it should explicitly state that the underlying classical code C is assumed to have locality r.","section":"Theorem 13"},{"comment":"The abstract says the converse bounds are derived for 'pure CSS-like EA-qLRCs,' but Theorem 3 and Corollaries 2-4 do not require purity and are stated for any CSS-like EA-qLRC from Proposition 1. The wording should be adjusted to avoid suggesting that purity is needed for the bounds.","section":"Abstract and Section IV"},{"comment":"Theorem 3 says the code is 'constructed from Proposition 1 using two classical [n,k_i,d_i]_q codes C_i of locality r.' Proposition 1 requires, for each coordinate, a pair of checks whose union support has size at most r+1, which is stronger than each C_i having classical locality r. Please state the exact hypothesis needed or clarify that the classical locality of C_i is a consequence of the Proposition 1 condition.","section":"Section VI, Theorem 3 statement"},{"comment":"In Example 2, the minimum distances d1 and d2 are initially obtained as lower bounds from Lemma 7; they become exact only after applying the classical LRC Singleton bound. Please make that two-step justification explicit at the point where d1=14 and d2=20 are stated as exact.","section":"Example 2 and Remark 12"}],"recommendation":"major_revision","confidential_remarks":"The paper is substantial and likely publishable after the purity gap is resolved. The most important issue is the unproved purity of the non-LCD constructions in Section VI; if the authors can prove the required weight conditions or revise the parameters and optimality claims, the paper would be a solid contribution. The asymptotic sphere-packing issue is also worth fixing, though it is secondary to the main finite-length results. The paper appropriately acknowledges concurrent work by Li et al., which reduces novelty concerns, and the LCD-based families in Section VII appear to be the cleanest part of the construction story."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about quantum LRCs: it answers Luo et al.'s question about whether entanglement assistance can give local recoverability, and it does so with a systematic framework. The core structural point is good. Locality needs only two local checks per coordinate, one X-type and one Z-type, and entanglement assistance removes the global dual-containment requirement, so classical LRC families that were previously inaccessible to CSS constructions become usable. Theorem 2 and Proposition 1 are clean. The bounds in Section IV are derived from external classical LRC bounds through a subcode-and-puncturing lemma; no fitted parameters, no circularity. Theorem 7's hull-dimension optimality criterion is genuinely useful, and the LCD cyclic families give real optimal codes. The GV-like achievability for q>3 via monomial equivalence also looks sound on its face.\n\nNow the soft spot, and it is load-bearing for part of the paper. Theorems 9–12 claim pure codes with distance equal to min{d1,d2}, but the proofs never show that minimum-weight codewords avoid the hull intersections. Theorem 1 only gives delta = min{wt(C2\\(C1^perp ∩ C2)), wt(C1\\(C1 ∩ C2^perp))}, which can be strictly larger than min{d1,d2}. The Tamo–Barg non-optimality claim in Remark 11 and the cyclic-pair optimality discussion in Remark 12 both presuppose equality with min{d1,d2}. If the actual distance is larger, those optimality conclusions, and the stated parameters in Theorem 12, are unsupported. This is not fatal for the LCD-based constructions: there C∩C^perp={0}, purity is automatic, and Theorems 13–14 hold. It is also not a problem for the bounds or the optimality criterion as such; the gap is in the construction theorems.\n\nMinor point: the bound comparison is careful, but the claimed sphere-packing crossing window is narrow and the paper does not oversell it. Citation practice is honest, including the concurrent work [19]; the self-citation to [33] is a construction ingredient, not the source of the main bounds.\n\nWho should read this: coding theorists working on quantum LRCs and EAQECCs. It deserves a serious referee, not a desk reject, but the referee should insist on either a proof of purity for the Tamo–Barg and cyclic-pair constructions or a restatement with the actual distance formula, and a re-check of the optimality claims that depend on it. My verdict: send to peer review with conditions.","headline":"Solid EA-qLRC framework with honest bounds and useful hull-dimension criterion, but the purity of the Tamo–Barg and cyclic-pair constructions is asserted, not proved, and the optimality claims lean on it.","tokens_in":40469,"tokens_out":1985,"would_cite":true,"duration_ms":22488,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B15","94B65","81P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that shared entanglement removes the dual-containment barrier that has kept most classical locally recoverable codes out of CSS quantum constructions.","keywords":["Entanglement-assisted quantum locally recoverable codes","quantum locally recoverable codes","CSS construction","hull dimension","LCD codes","cyclic codes","Tamo-Barg codes","Singleton-like bound"],"falsifier":"For the Tamo-Barg example with q=7, r=2, ℓ=4 (the claimed pure code [[6,2,3,2;2]]_7), compute every weight-3 codeword of C_TB and check membership in C_TB∩C_TB^⊥; a single such codeword in the hull would give δ>d=3 and falsify the purity claim.","tokens_in":39438,"feed_emoji":"⚛️","tokens_out":8681,"duration_ms":88819,"temperature":0.7,"pith_summary":"This paper establishes that entanglement assistance erases the dual-containment requirement that has kept most classical locally recoverable codes out of quantum CSS constructions. The authors define entanglement-assisted quantum locally recoverable codes (EA-qLRCs) through recovery channels and prove that two stabilizer generators per erased coordinate, one with an X-error and one with a Z-error, whose combined support has size at most r+1, are enough to guarantee local recovery of that coordinate using the surviving r qudits and the shared ebits. From this they derive a Singleton-like bound on distance, dimension, locality, and entanglement consumption, and they characterize exactly when pure CSS-like EA-qLRCs attain it: the constituent classical codes must themselves be optimal LRCs with equal parameters and compatible flooring of k/r, which in the single-code case reduces to a threshold on the hull dimension s=dim(C∩C^⊥). Explicit optimal maximally entangled codes are then built from LCD cyclic LRCs of length dividing q−1 or q+1, while a Tamo-Barg construction is shown to attain the bound only in the regime k≤r where locality is vacuous. The paper closes with two Gilbert-Varshamov-like achievability bounds that hold unconditionally for q>3.","feed_headline":"Entanglement frees quantum local recovery from dual-containment","feed_subtitle":"Pre-shared ebits supply the commutativity CSS lacks, opening optimal cyclic codes and GV-like achievability.","key_machinery":"The load-bearing mechanism is the two-generator local recovery criterion (Theorem 2): if for every coordinate i there are two stabilizer generators—one acting as X on i, one acting as Z on i—whose combined support sits inside a set Γ_i of size at most r+1, then every single-qudit error on i is correctable from Γ_i\\{i} together with the decoder's ebits, because the whole single-qudit operator algebra of the erased system is reproduced by operators on the survivors. The paper's named objects are the CSS-like construction from a classical pair (C_1,C_2), the hull dimension s=dim(C∩C^⊥), which fixes entanglement c=n−k−s and sets the optimality threshold s≤(k−1) mod r, the LCD condition C∩C^⊥={0} that yields maximally entangled codes, and cyclic defining-set machinery in which the local checks are residue classes modulo n/(r+1). A locality-preserving monomial equivalence argument is what carries the Gilbert-Varshamov achievability statements to all q>3.","core_discovery":"On the paper's own terms, the discovery is that locality in entanglement-assisted quantum codes is a purely local support condition, not a global algebraic one: for every coordinate i, parity-check vectors c_1^(i)∈C_1^⊥ and c_2^(i)∈C_2^⊥ with i in both supports and |supp(c_1^(i))∪supp(c_2^(i))|≤r+1 make the CSS-like EA code locally recoverable with locality r, whether or not C_1^⊥⊆C_2. The main performance statement is the converse bound 2δ≤n−κ+c−2⌈κ/r⌉+4, with necessary and sufficient equality conditions in Theorem 6 and the single-code hull version in Theorem 7: an optimal pure CSS-like EA-qLRC from C_1=C_2=C is optimal exactly when C is a classically optimal LRC with hull dimension s≤(k−1) mod r, and when s=0 the code is maximally entangled with c=n−k. The explicit constructions are claimed optimal maximally entangled CSS-like EA-qLRCs from LCD cyclic LRC families of length dividing q±1, with parameters [[n,k,d;n−k]]_q and locality r, while the Tamo-Barg construction is proved to miss optimality for every k>r.","pith_inferences":["Editorial extension: the same two-generator local-support criterion should transplant to entanglement-assisted codes with (r,ρ)-locality, where ρ local checks per coordinate would need 2ρ generators with combined support at most r+ρ; the paper does not treat this.","Editorial extension: since optimality hinges on the hull threshold s≤(k−1) mod r, any systematic family of classical optimal LRCs with hull bounded by that threshold would yield optimal EA-qLRCs; searching beyond LCD codes (s=0) is a natural next step.","Editorial extension: the q=2,3 gap might be closed by explicit binary LCD-LRC families rather than monomial equivalence, because the obstruction is an equivalence failure, not a locality obstruction; the explicit cyclic families already cover some binary parameter ranges.","Editorial extension: the asymptotic comparison suggests the sphere-packing-like bound dominates only when locality r is large relative to length; testing finite-length crossover points for q=2 with r between 6 and 12 would calibrate the regime boundary."],"forward_implications":["Any classical LRC pair whose duals carry two local checks with joint support at most r+1 per coordinate gives an EA-qLRC with locality r; no dual-containment is required.","The Singleton-like bound 2δ≤n−κ+c−2⌈κ/r⌉+4 is universal for CSS-like EA-qLRCs, and equality is characterized by classically optimal constituent codes with equal parameters and ⌈k_1/r⌉=⌈κ/r⌉.","In the single-code case, optimality reduces to the hull condition s=dim(C∩C^⊥)≤(k−1) mod r, and LCD codes yield maximally entangled optimal EA-qLRCs with c=n−k.","Explicit optimal maximally entangled EA-qLRCs exist from LCD cyclic LRCs of length dividing q−1 or q+1, with parameters [[n,k,d;n−k]]_q and locality r.","The Tamo-Barg-based construction attains the Singleton-like bound only in the vacuous regime k≤r, while the Gilbert-Varshamov-like achievability rates hold unconditionally for q>3, leaving q=2,3 open."],"supporting_citations":[{"why":"Supplies the classical LRC Singleton bound d≤n−k−⌈k/r⌉+2 that the paper's Singleton-like EA-qLRC bound is built on.","marker":"[1]"},{"why":"Provides the classical Cadambe-Mazumdar bound, the parity-check augmentation family, and the concatenated-code family used for the achievability theorems.","marker":"[2]"},{"why":"Gives the cyclic LRC construction with defining-set and local-check structure reused in Lemma 9 and the cyclic code constructions.","marker":"[7]"},{"why":"Supplies the classically optimal cyclic LRC-LCD families of length q+1 that appear in Table II for maximally entangled constructions.","marker":"[9]"},{"why":"Establishes the EAQEC parameters and purity framework for arbitrary classical code pairs used in Proposition 1 and throughout.","marker":"[26]"},{"why":"Provides the entanglement-assisted construction machinery referenced as [27, Theorem 21] when deriving the four explicit converse bounds.","marker":"[27]"},{"why":"Gives the condition for a cyclic code to be LCD (Z=−Z), used in Lemma 11 and the maximally entangled constructions.","marker":"[32]"},{"why":"Gives the monomial-equivalence result that any linear code over F_q with q>3 is equivalent to an LCD code, extended to preserve locality.","marker":"[35]"}],"fun_headline_variants":["Entanglement unlocks optimal cyclic quantum LRCs","No dual-containment needed: EA quantum local codes","Hull dimension fixes optimality in EA-qLRCs","Pre-shared ebits relax CSS condition for LRCs","Quantum LRCs freed from dual-containment via entanglement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claimed optimal cyclic and Tamo-Barg constructions are pure because their minimum-weight classical codewords are assumed to avoid the hull intersection with the other constituent's dual; the paper does not prove this avoidance in general, and the claimed quantum distances and optimality rest on it.","fun_headline_variants_meta":{"raw":{"variants":["Entanglement unlocks optimal cyclic quantum LRCs","No dual-containment needed: EA quantum local codes","Hull dimension fixes optimality in EA-qLRCs","Pre-shared ebits relax CSS condition for LRCs","Quantum LRCs freed from dual-containment via entanglement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000261,"raw_usage":{"total_tokens":1724,"prompt_tokens":1206,"completion_tokens":518,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":822,"completion_tokens_details":{"reasoning_tokens":438}},"tokens_in":822,"tokens_out":518,"duration_ms":5743,"temperature":1.0,"reasoning_tokens":438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:37:54.709299+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the Tamo-Barg example with q=7, r=2, ℓ=4 (the claimed pure code [[6,2,3,2;2]]_7), compute every weight-3 codeword of C_TB and check membership in C_TB∩C_TB^⊥; a single such codeword in the hull would give δ>d=3 and falsify the purity claim.","supporting_citations":[{"cited_title":"On the locality of codeword symbols,","cited_arxiv_id":null,"evidence_quote":"Supplies the classical LRC Singleton bound d≤n−k−⌈k/r⌉+2 that the paper's Singleton-like EA-qLRC bound is built on."},{"cited_title":"Bounds on the size of locally recoverable codes,","cited_arxiv_id":null,"evidence_quote":"Provides the classical Cadambe-Mazumdar bound, the parity-check augmentation family, and the concatenated-code family used for the achievability theorems."},{"cited_title":"Cyclic lrc codes and their subfield subcodes,","cited_arxiv_id":null,"evidence_quote":"Gives the cyclic LRC construction with defining-set and local-check structure reused in Lemma 9 and the cyclic code constructions."},{"cited_title":"Constructions of optimal cyclic(r, δ)locally repairable codes,","cited_arxiv_id":null,"evidence_quote":"Supplies the classically optimal cyclic LRC-LCD families of length q+1 that appear in Table II for maximally entangled constructions."},{"cited_title":"Entanglement-assisted quantum error-correcting codes over arbitrary finite fields: C. galindo et al","cited_arxiv_id":null,"evidence_quote":"Establishes the EAQEC parameters and purity framework for arbitrary classical code pairs used in Proposition 1 and throughout."},{"cited_title":"The condition for a cyclic code to have a complementary dual,","cited_arxiv_id":null,"evidence_quote":"Gives the condition for a cyclic code to be LCD (Z=−Z), used in Lemma 11 and the maximally entangled constructions."},{"cited_title":"Linear codes overF q are equivalent to LCD codes forq >3,","cited_arxiv_id":null,"evidence_quote":"Gives the monomial-equivalence result that any linear code over F_q with q>3 is equivalent to an LCD code, extended to preserve locality."}],"review_version":1}