{"id":"c57741e9-e185-4bac-81c9-384eda22127b","arxiv_id":"2608.10972","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Adding carefully chosen auxiliary local sources lets a single Bell-like inequality certify that all original sources in any quantum network are nonlocal.","lead":"This paper proposes a way to certify that every source in a quantum network is nonlocal using one Bell-like inequality, by adding extra local sources and parties to the network. If correct, it would make full network nonlocality tests much cheaper for complex network topologies.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (2.2) in Theorem 1 factorizes expectations across the 'independent' party A_t although A_t may share local sources with Γ̄; without this step the bound (2.1) and hence the FQNN criterion in Corollary 5 are unproven.","rationale":"I agree with the reader's weakest-assumption diagnosis. The factorization in Eq. (2.2) is load-bearing: it is the only step that connects the h-independent-party expression to the known (h-1)-party bound, and every corollary uses it. My explicit local-hidden-variable example shows the step is not a harmless inequality but a false statistical-independence assumption. The paper's numerical SLSQP results are also not reproducible (no code or coefficient values are given), and Eq. (2.6) for w is asserted without derivation, but the invalid factorization alone is enough to make the central claim unproven. This does not show the auxiliary-source idea is unworkable; it shows the manuscript does not establish it. Since the reader's verdict is already REJECT for the same core reason, I recommend no change.","tokens_in":15702,"tokens_out":12310,"duration_ms":114507,"concrete_test":"Independent analytic check: instantiate the triangular network with h=1, Γ={1}, Γ̄={2,3}, and a local hidden variable λ∈{±1} shared by A_1 and A_2. Set A_{x1=0}=A_{x1=1}=λ, A_{x2=0}=λ, A_{x2=1}=A_{x3=0}=A_{x3=1}=1. Then I=⟨A^+_{x1}A_{x2=0}A_{x3=0}⟩=1, J=0, while the factorized upper bound in Eq. (2.2) gives |⟨A^+_{x1}⟩||⟨A_{x2=0}A_{x3=0}⟩|=0. Hence the inequality step fails. If the authors supply a corrected proof of (2.1) that does not use this factorization, then rerun Examples 1-5 with reproducible code; otherwise the central claim remains unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 breaks at the first inequality, Eq. (2.2). For h=1 it asserts |E[A_t^+ B0]| ≤ |E[A_t^+]| |E[B0]| (and an analogous A_t^- term), i.e., that A_t's outcome is statistically independent of the operator B0 = Π_{i∈Γ\\{t}} A_{xi}^+ Π_{j∈Γ̄} A_{xj=0}. The hypotheses do not give this. 'A_t is independent' only means A_t shares no source with the other members of Γ; it may share sources with Γ̄. 'A_t receives only local sources' does not make those sources private; a local source can be a classical hidden variable shared between A_t and a party in Γ̄, which correlates A_t with B0. Explicit counterexample to the step: in a triangular network with h=1, let a classical bit λ=±1 be shared by A_t and a Γ̄ party. Choose A_{xt=0}=A_{xt=1}=λ and A_{xj=0}=λ for that Γ̄ party, all other observables equal to 1. Then A_t^+ = λ and B0=λ, so |E[A_t^+ B0]|=1 while |E[A_t^+]||E[B0]|=0. Eq. (2.2) is therefore false as a general step. Since Eq. (2.2) is the only place where the h-party problem is reduced to the (h-1)-party bound from Ref. [24], the w-QNL inequality (2.1) is not established; Corollary 5 and all numerical demonstrations inherit this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a method to certify full quantum network nonlocality (FQNN) in arbitrary quantum networks by adding auxiliary local sources and parties, and then testing a single nonlinear Bell-like inequality on the enlarged network. The central theoretical result is Theorem 1, which asserts an l-QNL inequality for arbitrary networks, and Corollary 5, which uses this inequality to certify FQNN. The paper illustrates the method on triangular, chain, cyclic, and tree-shaped networks, using Werner states and SLSQP optimization to exhibit violations.","tokens_in":16086,"tokens_out":10201,"duration_ms":94130,"significance":"If Theorem 1 were correct, the paper would provide a significant advance: a single-inequality test for FQNN in arbitrary network topologies, improving on existing star-specific or decomposition-based methods. The constructive idea of adding auxiliary local sources to increase the independence number is appealing, and the paper contains explicit constructions for several network families. However, the central proof contains a load-bearing error in Eq. (2.2), and the paper does not provide a valid derivation of the main inequality. Consequently, the claimed FQNN certification criterion and all numerical demonstrations that rely on it are not established. The significance is therefore contingent on a substantial repair of the proof.","major_comments":[{"comment":"The proof also states, immediately after Eq. (2.2), that removing the independent party A_t leaves a network that is (h−1)-independent. This is not true in general. For example, in the triangular network with h=1, removing the independent party A1 leaves parties A2 and A3 connected by a source, so the remaining network has independence number 1, not 0. In the 4-cycle with Γ={1,3}, removing vertex 1 leaves a path on vertices 2,3,4, which has independence number 2, not h−1=1. Therefore the appeal to Ref. [24] for an (h−1)-independent network is not valid when the remaining network has a larger independence number.","section":"Section II, Eq. (2.2), proof of Theorem 1"}],"minor_comments":[{"comment":"The closed-form expression for w is asserted without proof. Since w appears in the right-hand side of Ineq. (2.1) and in the number of auxiliary sources to be added, this formula is load-bearing for the examples and Corollary 5. A derivation or an explicit reference is needed.","section":"Section II, Eq. (2.6)"},{"comment":"The numerical results report SLSQP maxima (e.g., |I1|^{1/3}+|J1|^{1/3}=1.357 in Example 1) without providing the optimized measurement coefficients or a global optimality certificate. While a single explicit violating assignment would be sufficient to demonstrate a violation, as written the reader cannot verify the reported values. The statement in Example 1 that SLSQP 'can always' find suitable measurements is an unsupported heuristic claim.","section":"Examples 1–5"},{"comment":"The paragraph beginning 'Based on the preceding analysis, we first introduce an l-QNL inequality...' appears twice verbatim in the text; one occurrence should be removed.","section":"Section II, repeated paragraph"},{"comment":"There is a typographical error in Eq. (2.9): 'α11_{r1,r2}=0' should likely be 'α11_{r1,r2}', and the equation is missing the summation structure present in Eq. (2.8).","section":"Eq. (2.9)"}],"recommendation":"reject","confidential_remarks":"The main proof gap in Eq. (2.2) is severe and affects every subsequent claim in the manuscript. This is not a local presentation issue; the central inequality is unsupported. The paper's reliance on Ref. [24], authored by a coauthor, is not circular per se, but it means the validity of the new result inherits the correctness of that prior work, and the new proof step is invalid. Given the scope of the flaw, I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of Yang et al. The idea is genuinely appealing: instead of decomposing a network into stars to certify full quantum network nonlocality, add auxiliary local sources and parties, then certify l-QNN with a single inequality. If it worked, it would remove a real experimental obstacle, and the construction is concrete—explicit N(n,m), explicit inequalities for chains, cycles, trees, and general networks. The paper is worth reading for that.\n\nThe problem is the proof of Theorem 1. Eq. (2.2) factorizes the independent party A_t from the rest of the network: |⟨A_t^+ B0⟩| ≤ |⟨A_t^+⟩| |⟨B0⟩| (and similarly with A_t^-). That step is false. 'Independent' here only means A_t shares no source with the other parties in Γ; it can share a local hidden variable with parties in Γ̄, which correlates A_t with B0. The stress-test counterexample works: in a triangle with h=1, a classical bit λ shared between A_t and a Γ̄ party, with A_xt = B0 = λ, gives |⟨A_t^+ B0⟩|=1 while |⟨A_t^+⟩||⟨B0⟩|=0. So Eq. (2.2) is not justified, and since the (h-1)-party reduction from Ref. [24] enters through that step, the w-QNL inequality (2.1) is not established. Corollary 5 and the examples inherit the gap.\n\nTwo more issues. Eq. (2.6) for w is asserted without proof; it may be true, but it's load-bearing because w sets the number of auxiliary sources and the inequality threshold. And the numerical violations are SLSQP maxima without global optimality certificates, with no code or explicit coefficients, so they are suggestive but not reproducible. That said, the failure is in the proof, not in the concept. The auxiliary-source reduction is new, and the examples show the intended logic clearly.\n\nMy take: this is a promising start that needs a real proof fix. I'd send it to a referee rather than desk reject—the construction deserves scrutiny, and a referee might see a way to salvage the inequality or at least document the gap for the authors. But as it stands, the central claim is unsupported.","headline":"Clever auxiliary-source construction, but Theorem 1 hinges on a false factorization step, so the central claim is unproven as written.","tokens_in":710,"tokens_out":670,"would_cite":false,"duration_ms":50758,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40"],"pacs":["03.65.Ud","03.67.-a"],"model":"deepseek-v4-flash","headline":"This paper claims that full quantum network nonlocality can be certified in any network topology by violating a single nonlinear Bell-like inequality after enlarging the network with auxiliary local sources and parties.","keywords":["full quantum network nonlocality","network nonlocality","Bell-like inequalities","quantum networks","hierarchical network nonlocality","independent parties","Werner states","SLSQP optimization"],"falsifier":"Take the triangular-network construction of Example 1 and set all five sources in the enlarged network to local Werner states with $p_i=1/3$, then apply the same SLSQP optimization to inequality (2.7): if the left-hand side exceeds $2^{1/3}$, the claim that violation forces at most two local sources is false, and the central criterion collapses.","tokens_in":15475,"feed_emoji":"🔗","tokens_out":6908,"duration_ms":63268,"temperature":0.7,"pith_summary":"The paper claims that full quantum network nonlocality—the property that every source in a network must be nonlocal—can be certified with a single Bell-like inequality, regardless of network topology. The method enlarges the original network with a carefully counted number of auxiliary local sources and parties, then tests the enlarged network with one nonlinear inequality. If that inequality is violated, the enlarged network can contain at most a known number of local sources; because exactly that many local sources were intentionally inserted, all original sources must be nonlocal. This would replace the current practice of decomposing an arbitrary network into many star subnetworks and testing each separately.","feed_headline":"One Bell-like inequality certifies all-source nonlocality","feed_subtitle":"Adding auxiliary local sources lets a single violation prove every original source in any network was nonlocal.","key_machinery":"The central objects are independent parties (spatially separated parties sharing no common source), the independence number $h$ of the network, and the topological parameter $w$: the minimum number of local sources that forces some independent party to receive only local sources. The paper defines $N(n,m)=w-1$ from the source-sharing sets, adds that many auxiliary local sources and parties, and builds a nonlinear inequality whose left-hand side is an $h$-th-root sum of two correlation products $I$ and $J$ built from binary observables $A^\\pm_{x_i}$. The bound $|I|^{1/h}+|J|^{1/h}\\le 2^{(h-1)/(2h)}$ is tight enough to certify that a violating network has at most $w-1$ local sources.","core_discovery":"The central claim is that full quantum network nonlocality of an arbitrary network $\\Xi(n,m)$ can be certified by a single nonlinear Bell-like inequality, provided the network is first expanded to $\\Xi_{\\mathrm{new}}(n,m)$ by adding $N(n,m)$ auxiliary local sources and corresponding parties to a chosen non-independent party. With $h$ the independence number of the original network and $\\Gamma$ a maximum independent set, the enlarged network has $h+N(n,m)$ independent parties and the test reads $|I_{\\mathrm{gen}}|^{1/(h+N(n,m))}+|J_{\\mathrm{gen}}|^{1/(h+N(n,m))}\\le 2^{(h+N(n,m)-1)/(2(h+N(n,m)))}$. A violation means the enlarged network contains at most $w-1=N(n,m)$ local sources; since $N(n,m)$ local sources were deliberately inserted, every original source must be nonlocal. The authors present this as a constructive, topology-independent replacement for decomposing networks into star-shaped pieces.","pith_inferences":["Beyond the paper's examples, the same expansion strategy could likely certify higher-level hierarchical network nonlocality by inserting $w-l$ auxiliary local sources and testing the corresponding $l$-level inequality; the authors only treat the $l=1$ full-nonlocality case.","The auxiliary-source count $N(n,m)$ depends on the chosen maximum independent set, so minimizing the overhead over valid choices is an open optimization problem the paper does not address.","Because the auxiliary sources are local classical resources, the added experimental cost is classical rather than quantum; whether the single-inequality search remains practical for much larger networks would require studying the scaling of the SLSQP optimization, which the paper does not do."],"forward_implications":["A single violation of one Bell-like inequality would certify full quantum network nonlocality for arbitrary network topologies, eliminating the need for repeated star-subnetwork tests.","For chain, cyclic, and tree-shaped networks, the construction specifies exactly how many auxiliary local sources and parties to add, so the test is directly implementable.","Numerical examples with Werner states of visibility around 0.9–0.96 and auxiliary local sources at $p=1/3$ violate the relevant inequalities, indicating a concrete experimental route.","If the criterion holds, it provides an efficient fault-detection tool for large quantum networks: one inequality would reveal whether any source has degenerated into a classical source.","The inequality parameter depends only on network topology, so for any given network the test can be precomputed from its source-sharing structure."],"supporting_citations":[{"why":"Supplies the quantum bound for $(h-1)$-independent networks that Theorem 1 invokes to bound the reduced correlation terms.","marker":"[24]"},{"why":"Defines full network nonlocality, the target concept the paper aims to certify.","marker":"[25]"},{"why":"Introduced full quantum network nonlocality and chain/star criteria, and provides the decomposition baseline the paper seeks to outperform.","marker":"[32]"},{"why":"Prior optimized inequalities for general networks; the paper positions its single-inequality method against this background.","marker":"[34]"}],"fun_headline_variants":["Single Bell inequality certifies nonlocality in any network","Expand network, then one Bell test reveals all source nonlocality","Auxiliary sources enable a single Bell inequality to verify all nonlocality","Nonlinear Bell inequality proves nonlocality of all sources in any network","One Bell test, after expansion, reveals all sources nonlocal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on assuming in Eq. (2.2) that one party's measurement statistics can be pulled out as a product from the rest of the network's statistics, and this is not guaranteed when that party shares a local source with the other parties; if that step fails, the derived inequality bound and the full-network-nonlocality certification do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Single Bell inequality certifies nonlocality in any network","Expand network, then one Bell test reveals all source nonlocality","Auxiliary sources enable a single Bell inequality to verify all nonlocality","Nonlinear Bell inequality proves nonlocality of all sources in any network","One Bell test, after expansion, reveals all sources nonlocal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000872,"raw_usage":{"total_tokens":3736,"prompt_tokens":868,"completion_tokens":2868,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":2777}},"tokens_in":484,"tokens_out":2868,"duration_ms":18736,"temperature":1.0,"reasoning_tokens":2777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:06:00.841979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the triangular-network construction of Example 1 and set all five sources in the enlarged network to local Werner states with $p_i=1/3$, then apply the same SLSQP optimization to inequality (2.7): if the left-hand side exceeds $2^{1/3}$, the claim that violation forces at most two local sources is false, and the central criterion collapses.","supporting_citations":[{"cited_title":"Computationally efficient nonlinear bell inequalities for quantum networks.Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum bound for $(h-1)$-independent networks that Theorem 1 invokes to bound the reduced correlation terms."},{"cited_title":"Full network nonlocality.Phys","cited_arxiv_id":null,"evidence_quote":"Defines full network nonlocality, the target concept the paper aims to certify."},{"cited_title":"Hierarchical certification of nonclas- sical network correlations.Phys","cited_arxiv_id":null,"evidence_quote":"Introduced full quantum network nonlocality and chain/star criteria, and provides the decomposition baseline the paper seeks to outperform."},{"cited_title":"Verifying hierarchical network nonlocality in general quantum networks.Chinese Physics B, 33(7):070304, jun 2024","cited_arxiv_id":null,"evidence_quote":"Prior optimized inequalities for general networks; the paper positions its single-inequality method against this background."}],"review_version":1}