{"id":"75407d07-c1e9-4dd6-ae4a-5ee6dc9fd061","arxiv_id":"2607.16110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Bulk regions in AdS3/CFT2 get a holographic secret-sharing distance d and thresholds (r,s), with r = n - d + 1; pure states satisfy s = d - 1 while mixed states can satisfy s >= d.","lead":"The paper introduces a combinatorial secret-sharing description for how well a bulk region in AdS3/CFT2 survives erasures of boundary intervals. It derives how the distance, reconstruction threshold, and secrecy threshold are related, and shows that mixed-state holographic schemes can be \"superadditive,\" giving more secrecy but less robustness than pure-state schemes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's appendix asserts n−s_b=r_b without justification; the additive/superadditive classification is not rigorously established as written.","rationale":"The paper's main theorems are largely combinatorial and appear correct; in particular, Theorem 4.1 is essentially the definition of distance, and the distance/secret-threshold relations in Theorem 4.2 can be derived from monotonicity plus no-cloning. The reader's identified weakest assumption, geometric complementarity (Eq. 4.7), is standard for a pure state on the full boundary including purifier O, and so is not where I would place the stress. The actual soft spot is the proof of Theorem 4.2 in Appendix A.2: the equality n−s_b=r_b is asserted without the missing no-cloning step. This is load-bearing because Theorem 4.2 is the paper's central classification. The reader's CONDITIONAL verdict is appropriate; I recommend no change, but the authors should expand the proof of Theorem 4.2 (or supply the combinatorial lemma) before the classification is taken as established.","tokens_in":28160,"tokens_out":29402,"duration_ms":259601,"concrete_test":"Implement an exhaustive search over all up-closed access structures A⊆2^[n] (n≤6) with [n]∈A and no two disjoint authorized sets. For each structure compute m=min authorized size, M=max unauthorized size, s=m−1, d=n−M, r=M+1, and check: (i) all complements of s-sets are authorized iff s=d−1; (ii) otherwise some s-set and its complement are both unauthorized iff s≥d. If a counterexample appears, Theorem 4.2 is false; if none appears, the missing no-cloning argument can be inserted and the proof repaired. For a holographic check, recompute R_min^b for the n=5 mixed phase in Fig. 27 and verify the same relations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.2 is the classification claim underlying the paper's central message, but the proof in Appendix A.2 contains an unsupported equality. After choosing E with |E|=s_b, the additive-case branch asserts 'we have n−s_b=r_b' merely from b⊆EW([n]\\E). This does not follow from Definition 4.2: r_b is the smallest size such that every boundary set of that size (and hence every larger one) is authorized, while the additive hypothesis only says that the specific complements of all s_b-sets—i.e., the (n−s_b)-sets—are authorized. It leaves open the possibility that some set of size n−s_b−1 is unauthorized, which would give r_b<n−s_b. To obtain n−s_b=r_b one must additionally use the maximality of s_b (there exists an authorized set A of size s_b+1) and the no-cloning/pairwise-intersection lemma (Lemma 3.2) to force A's complement, of size n−s_b−1, to be unauthorized. That forces the maximum unauthorized size M to be exactly n−s_b−1. The appendix does not supply this step, nor does it cite where it is proved. Since Theorem 4.2 underpins the additive/superadditive dichotomy and Corollary 4.1, the central claim is not fully supported as written. This is a proof-support gap, not a demonstrated counterexample; the statement may be true, but the provided justification is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces combinatorial holographic quantum secret sharing (CHQSS), a coarse-grained framework in which a bulk subregion b in AdS3/CFT2 is characterized by the family R_b of boundary subsets whose entanglement wedges contain b. It defines a distance d_b (Definition 3.3), a uniform reconstruction threshold r_b (Definition 4.2), and a uniform secret threshold s_b (Definition 4.3). The main formal claims are Theorem 3.1 (d_max = ceil(n/2)), Theorem 4.1 (r_b = n - d_b + 1), and Theorem 4.2, which gives an additive/superadditive dichotomy: if every s_b-subset has an authorized complement then s_b = d_b - 1, while otherwise s_b >= d_b. Corollary 4.1 states that pure-state CHQSS are additive and mixed-state schemes may be superadditive. The paper also presents numerical phase-transition data for symmetric n = 3,4,5 setups and constructs families of CHQSS schemes, including a perfect threshold scheme at the graph center and perfect non-threshold schemes away from it. Proofs are collected in Appendix A; additional examples appear in Appendix B.","tokens_in":28511,"tokens_out":24179,"duration_ms":199956,"significance":"If the central theorems are correct, CHQSS provides a clean, parameter-free combinatorial characterization of how bulk logical information is protected against boundary erasures, and it draws a sharp distinction between pure-state (additive) and mixed-state (superadditive) holographic encodings. The connection to superadditivity and the failure of exact holographic QEC is a useful conceptual contribution. The paper is explicit and self-contained: the definitions are operational, the claims are concrete, and the examples for n = 3,4,5 are easy to verify once the proofs are repaired. The paper does not include code or machine-checked proofs, and the numerical phase-transition data are only summarized graphically, but the framework is concrete enough for the proofs to be checked by hand.","major_comments":[{"comment":"The proof of the additive branch contains an unsupported equality. After choosing E with |E|=s_b, the text asserts that b⊆EW([n]\\E) implies n−s_b=r_b. This does not follow from the definitions: r_b is the smallest m such that every m-subset is authorized, while the assumption concerns only a single complement. To obtain r_b=n−s_b one must use the maximality of s_b (there exists an authorized set A of size s_b+1) and Lemma 3.2 to show that the complement of A, of size n−s_b−1, is unauthorized; alternatively, one can prove s_b=d_b−1 directly from the distance definition. As written, Theorem 4.2 and Corollary 4.1 are not fully supported. The converse direction also contains an undefined expression 'R\\E', which should be '[n]\\E'.","section":"Appendix A.2, Theorem 4.2"},{"comment":"The proof identifies the maximum of |E| in X_E_b with d_b−1, but this is not immediate. X_E_b is a uniform condition over all erasures of size at most |E|, whereas d_b is the least size of a single erasure that leaves no authorized set. These notions differ; for example, in the §4.2 case b1, d_b1=1 and the erasure {A2} is correctable, yet {A2} is not in X_E_b1 because the erasure {A1} is not correctable. The proof needs to show: (i) every erasure of size ≤d_b−1 is correctable and, by monotonicity and no-cloning, its complement is authorized; and (ii) any uniform threshold t≥d_b would contradict the definition of d_b. Without this argument, Theorem 4.1 is not rigorously established.","section":"Section 4.1, Theorem 4.1, Eqs. (4.13)–(4.14)"},{"comment":"The upper bound d_b ≤ min(u_b, n−u_b+1) is fine, but the realizability part of the proof only shows that d_b ≤ ceil(n/2) for the constructed b. To prove the lower bound d_b ≥ ceil(n/2), the authors must also show that no erasure of size ceil(n/2)−1 destroys all minimal authorized sets. In the constructed case where R_min_b consists of all ceil((n+1)/2)-subsets, this follows because the complement of such an erasure has size at least ceil((n+1)/2) and is itself a minimal authorized set. The counting in Eqs. (A.11)–(A.12) does not by itself establish equality. The theorem statement may be true, but the proof as written is incomplete.","section":"Appendix A.1, Theorem 3.1"}],"minor_comments":[{"comment":"In item 2, the second occurrence of 'R' should be '\\bar R'; as written the condition reads 'b⊄EW(R) and b⊄EW(R)', which loses the complement. This is likely a typographical error but should be corrected.","section":"Theorem 4.2 statement"},{"comment":"The phase-transition points are reported only graphically and to three significant figures, and no code or tabulated data are provided. The claim that the transition points extracted from Markov gaps match the geometric ones to four significant figures cannot be verified from the manuscript. Please provide numerical data or code, or an analytic determination of the transition points.","section":"Section 2.1"},{"comment":"The sentence 'In a general QSS, the second equation implies a ramp CHQSS' is misleading under the paper's uniform definitions of r_b and s_b. The example b1 in §4.2 has r_b−s_b>1 but is a perfect non-threshold scheme with no intermediate sets. The subsequent qualification about empty intermediate sets is helpful, but the wording should be revised to avoid equating r−s>1 with ramp behavior.","section":"After Eq. (4.27)"},{"comment":"The geometric complementarity assumption EW(R^c)=EW(R)^c, including the purifier O, is load-bearing for Lemma 3.2 and Theorem 4.1. The paper assumes it without discussion of its domain of validity for mixed states. A brief statement of when this complementarity is expected to hold (e.g., classical RT surfaces in a pure global state) would strengthen the presentation.","section":"Definition 4.1, item 3"}],"recommendation":"major_revision","confidential_remarks":"The paper's main ideas are sound and the framework is interesting, but the proof of Theorem 4.2—the central classification claim—contains a genuine gap, and Theorem 4.1 and Theorem 3.1 also need small but real missing arguments. None of these appear to be fatal; they are localized to Appendix A and can be repaired within the manuscript's scope. I would be willing to review a revised version. The numerical section would also benefit from code or data so that the claimed phase-transition points and their consistency check are verifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a useful paper, and the core construction is new. The CHQSS distance, reconstruction/secrecy thresholds, d_max = ceil(n/2), and the additive/superadditive dichotomy are not in the prior holographic QEC or QSS literature. The paper does a good job connecting entanglement wedge nesting and complementarity to QSS access structures, and the phase-transition scan for n=3,4,5 with multiple distinct transition points is a concrete addition. Your conditional verdict is roughly right.\n\nWhat the paper does well: Theorem 4.1, r_b = n - d_b + 1, follows cleanly from the definitions and is well supported. Theorem 3.1's counting argument plus the realizability condition is solid. The additive/superadditive distinction is conceptually interesting and gives a sharp way to see the mixed-state obstruction to exact holographic QEC. The Appendix B examples are helpful, though the figures are heavy.\n\nSoft spots, in proportion:\n\n1. The proof of Theorem 4.2 in A.2 is incomplete. The line asserting n-s_b = r_b after choosing E with |E|=s_b does not follow from b ⊆ EW([n]\\E) alone. One needs the maximality of s_b and the pairwise-intersection lemma to rule out an authorized set of size n-s_b-1. Since Theorem 4.2 underpins the paper's central classification claim, this needs to be spelled out. I don't see a counterexample; it looks repairable, but as written the proof has a real gap.\n\n2. Geometric complementarity for mixed states is a nontrivial assumption. The paper flags it and leans on it heavily. That is acceptable if stated clearly, but it deserves more scrutiny than it gets.\n\n3. The numerical phase-transition results come with no code or data, and the consistency check is only to four significant figures. Since the computation is deterministic from Eq (2.1), this is minor, but shipping the script would make it independently checkable.\n\n4. Remark 3's no-ramp claim is overbroad. It should be proved under the exact-reconstruction assumption or softened.\n\nWho this is for: people working on holographic QEC/QSS, entanglement wedge reconstruction, and secret-sharing access structures. It deserves a serious referee. I would send it out, but request the A.2 repair and some notation cleanup before acceptance.","headline":"A genuinely new combinatorial translation of holographic QSS with a clean distance/threshold story; the main theorem's appendix proof has a real gap that should be fixed before publication.","tokens_in":28980,"tokens_out":1862,"would_cite":true,"duration_ms":17985,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["11.25.Tq","03.67.Dd","03.67.Pp"],"model":"deepseek-v4-flash","headline":"This paper establishes that in AdS3/CFT2 a single erasure distance d_b controls quantum secret sharing for every bulk cell, forcing the reconstruction threshold r_b = n - d_b + 1 and splitting schemes into additive (pure-state) and superadd","keywords":["holographic quantum secret sharing","entanglement wedge reconstruction","RT-region graph","reconstruction threshold","secret threshold","erasure distance","AdS3/CFT2","entanglement wedge phase transitions"],"falsifier":"Take the symmetric n = 5 mixed-state phase in which only the five-partite entanglement wedge is connected and compute the exact authorized sets for the central cell from the minimal-surface configuration. The paper predicts (d_b, r_b, s_b) = (1, 5, 4); any exact computation that yields a different triple for that cell—or any bulk cell whose minimal authorized sets fail to pairwise intersect—would refute the central classification.","tokens_in":28040,"feed_emoji":"🔐","tokens_out":9444,"duration_ms":81519,"temperature":0.7,"pith_summary":"The paper tries to prove that the quality of holographic encoding in AdS3/CFT2—how easily a bulk region's logical information can be reconstructed from the boundary, how well it stays secret, and how robust it is to erasures—is fixed entirely by combinatorial data: which entanglement wedges contain the bulk cell. It defines a distance d_b for each bulk cell (the smallest erasure that destroys all authorized boundary sets), a reconstruction threshold r_b, and a secret threshold s_b, and establishes the exact relations r_b = n - d_b + 1 and a dichotomy: either s_b = d_b - 1 (additive) or s_b ≥ d_b (superadditive). Pure-state schemes are always additive; mixed-state schemes can be superadditive, which is the paper's explanation for why exact holographic quantum error correction can fail. It also derives the maximum distance d_max = ceil(n/2), consistent with the no-cloning bound, and constructs concrete families of perfect threshold and perfect non-threshold schemes from phase-transition geometries in the symmetric n = 3, 4, 5 settings. If correct, this gives a complete combinatorial characterization of holographic secret sharing from entanglement-wedge inclusion alone.","feed_headline":"One distance law sets all AdS3 secret-sharing thresholds","feed_subtitle":"How hard a bulk region is to reconstruct follows from how many erasures it survives; pure states are always additive.","key_machinery":"The central object is the RT-region graph: a coarse-graining of the bulk time slice by the minimal surfaces of all boundary subsets, where each vertex b carries an access structure R_b = {R ⊆ [n] : b ⊆ EW(R)}. Its defining property is the pairwise-intersection lemma: any two minimal authorized sets intersect, which follows from entanglement wedge nesting plus geometric complementarity. That pairwise-intersection fact is what converts the no-cloning principle into the bound 2r_b > n, and it is the engine behind Theorem 4.1 (r_b = n - d_b + 1) and Theorem 4.2 (the additive/superadditive dichotomy).","core_discovery":"The central discovery is that for any bulk cell b contained in EW([n]), the access structure R_b is monotone (by entanglement wedge nesting) and has pairwise-intersecting minimal elements (by nesting plus geometric complementarity). From these two facts, the distance d_b and reconstruction threshold r_b are locked by r_b = n - d_b + 1. The secret threshold s_b then obeys a dichotomy: s_b = d_b - 1 exactly when the entanglement wedge of the complement of each minimal authorized set also contains b (additive), and s_b ≥ d_b when some complement pair leaves b outside both wedges even though b is inside EW([n]) (superadditive). Pure-state schemes always fall in the additive case, while mixed-sta","pith_inferences":["If the paper is right, the same distance-threshold relation r_b = n - d_b + 1 should hold in any holographic theory satisfying entanglement wedge nesting and geometric complementarity, not just pure AdS3; the maximum-distance formula might change with dimension, but the additive/superadditive dichotomy should persist.","The gap d_b - s_b (or r_b - s_b) could serve as an order parameter for holographic phase transitions: in the symmetric examples it jumps at each transition point, so boundary entropy data may be usable to locate bulk cells with superadditive schemes.","One could turn the RT-region graph into explicit erasure codes by assigning a logical state to each cell and checking whether its authorized sets realize a known quantum secret sharing scheme, making the combinatorial classification testable in tensor-network toy models.","Because mixed states can have s_b ≥ d_b, a quantitative trade-off between secrecy and robustness emerges: maximizing one lowers the other. This suggests treating the holographic phase choice as a resource in a future resource-theoretic account of holographic secret sharing."],"forward_implications":["Each bulk cell's combinatorial holographic quantum secret sharing scheme is fully classified by the triple (r_b, s_b, d_b); from the access structure alone one can read off the erasure threshold and the reconstruction and secret thresholds.","The best-protected bulk region achieves d_max = ceil(n/2) and reconstruction threshold floor(n/2) + 1, saturating the holographic no-cloning bound 2r > n in phases where all relevant entanglement wedges are connected.","Pure-state schemes always satisfy d_b - s_b = 1 and r_b + s_b = n, a clean additive relation that underlies exact holographic quantum error correction.","Mixed-state superadditive schemes satisfy s_b ≥ d_b and r_b + s_b ≥ n + 1, so their existence explains why exact holographic quantum error correction can fail in mixed states and ties that failure to genuine multipartite entanglement.","The symmetric n = 3, 4, 5 constructions produce explicit families of both perfect threshold and perfect non-threshold schemes, showing that holography naturally realizes both classes of quantum secret sharing."],"fun_headline_variants":["Distance dictates thresholds in all AdS3 secret sharing","Pure state secret sharing is always additive in AdS3","One law links reconstruction threshold and distance in AdS3","AdS3 secret sharing: threshold equals n minus distance plus one"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire chain rests on the assumption that the entanglement wedge of the complement of any boundary region—including a purifier—is exactly the geometric complement of its entanglement wedge; if that fails for a bulk cell, the distance-threshold relations for that cell need revision.","fun_headline_variants_meta":{"raw":{"variants":["Distance dictates thresholds in all AdS3 secret sharing","Pure state secret sharing is always additive in AdS3","One law links reconstruction threshold and distance in AdS3","AdS3 secret sharing: threshold equals n minus distance plus one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1391,"prompt_tokens":677,"completion_tokens":714,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":421,"completion_tokens_details":{"reasoning_tokens":646}},"tokens_in":421,"tokens_out":714,"duration_ms":6845,"temperature":1.0,"reasoning_tokens":646,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T21:18:15.161778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the symmetric n = 5 mixed-state phase in which only the five-partite entanglement wedge is connected and compute the exact authorized sets for the central cell from the minimal-surface configuration. The paper predicts (d_b, r_b, s_b) = (1, 5, 4); any exact computation that yields a different triple for that cell—or any bulk cell whose minimal authorized sets fail to pairwise intersect—would refute the central classification.","supporting_citations":[],"review_version":1}