{"id":"1a36ba31-c6b1-4769-bd1b-43583cfb98bc","arxiv_id":"2412.00795","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Generalizes detection length to entanglement depth, intactness, and nonlocality, exhibits maximal gaps between detection lengths, and gives SDP-constructed witnesses, with a noise analysis claiming shorter witnesses are more robust under bit-flip errors.","lead":"This paper extends the detection length framework, a metric for how many parties must be measured together to verify entanglement or nonlocality, and uses semidefinite programming to construct witnesses of chosen length. It exhibits states where entanglement and genuine multipartite entanglement require very different measurement lengths, and argues shorter witnesses can be more robust to bit-flip measurement errors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3's uniform (1-2epsilon)^k noise model does not apply to the paper's own mixed-weight witnesses, so Eq. 20's shorter-witness robustness window is not established.","rationale":"Good-faith reading: the paper's core is the detection-length framework, a gap example (Prop. 1), and SDP witness constructions, with the advertised quantitative conclusion that shorter witnesses can beat longer ones under bit-flip measurement noise. Prop. 1 appears internally sound: the two-party marginal rho_{1,2} is NPT for p > 1/2, giving l_Ent = 2, while the constructed sigma in C(rho, S_{n-1}) is biseparable for the relevant cuts, forcing l_GME = l_Bipa = n in the claimed cases. The SDP witnesses are plausible numerical outputs that can be checked independently. The load-bearing weak point is Prop. 3 and its use in Eq. (20). The proof's uniform shrinkage by (1-2epsilon)^k assumes every non-identity Pauli term has weight exactly k and that every local factor is equally sensitive to the error; neither is true of W1/W2. This is an internal inconsistency with the paper's own length definition, not a disagreement with outside consensus. The concrete test can settle whether the claimed window survives; if it does not, the paper needs a revised noise analysis or the conclusion must be restricted to witnesses built from uniform full-weight Pauli terms. Since the detection-length framework and Prop. 1 are independent of this flaw, conditional acceptance, as the reader concluded, remains appropriate.","tokens_in":21999,"tokens_out":20211,"duration_ms":195402,"concrete_test":"For N = 11, expand W1 and W2 from Eq. (18) explicitly in the Pauli basis, record each term's weight m (and its X/Z composition), and compute the measured expectation of each witness under the paper's depolarizing channel Eq. (14) plus term-wise bit-flip/outcome-flip noise, using the actual Tr(W). Solve for the depolarizing thresholds p*(W1) and p*(W2). Then check whether p*(W1) > p*(W2) > 0 holds throughout the interval 0.0857 < epsilon < 0.0905 claimed after Eq. (20); if it fails, the advertised robustness window is not implied by the construction. As a secondary check, verify Tr(W1) = d and Tr(W2) = d; if not, renormalize as the paper presumably intended and see whether the p* formulas in Eq. (20) still match.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3 (Appendix A, Eqs. A4-A8) starts from W = I - sum_i P_i, where each non-identity P_i is assumed to be a k-length Pauli operator, and then multiplies every Pauli expectation by (1-2epsilon)^k. For a k-length witness as defined in Sec. IV (maximal support k), non-identity terms may have weight m < k, and an error on a party where P_i acts as identity does not affect that term. Under a classical outcome-flip model the correct shrinkage is (1-2epsilon)^m; under the physical bit-flip channel described in the text, X factors are unchanged while Z/Y factors shrink. The paper's own examples in Eq. (18) violate the premise: W1 contains S1 = X1Z2 (weight 2) alongside S2 = Z1X2Z3 (weight 3), and W2 is a product of factors with mixed weights, so Eqs. (15)-(16) do not follow for the witnesses used in the comparison. There is also a normalization inconsistency: as printed Tr(W1) = 2, not d, while the p* formulas in Eq. (20) use values corresponding to Tr(W rho) = -1 for W1 and -2K for W2, so the exact window (0.0857, 0.0905) for N = 11 is not reproducible from the displayed constructions. Because the advertised shorter-witness robustness claim rests on this calculation, it is currently unsupported; the qualitative statement may survive a corrected term-by-term analysis, but Eq. 20 as written cannot be accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the \"detection length\" framework introduced by Shi et al. to bipartite entanglement, genuine multipartite entanglement (GME), nonlocality, entanglement depth, and entanglement intactness. It defines detection lengths for these categories, derives relationships among them, and proves two analytic results: Proposition 1 gives a family of states with a maximal gap between the entanglement detection length (2) and the GME detection length (n), and Proposition 2 identifies a two-parameter family with GME detection length n. The paper also formulates semidefinite programs (SDPs) to construct entanglement witnesses and Bell inequalities of prescribed length, and studies the noise robustness of these witnesses under global depolarizing and bit-flip noise. The central advertised quantitative claim is that, under bit-flip measurement errors, witnesses with shorter detection length can be more noise-tolerant than longer ones (Eq. 20).","tokens_in":22352,"tokens_out":19239,"duration_ms":162204,"significance":"If the results hold, the paper provides a useful quantitative framework for determining the minimal measurement globality needed to certify different entanglement classes, and it supplies explicit witness constructions that are directly usable in experiments. The main new conceptual finding is Proposition 1, which shows that the entanglement detection length can be as small as 2 while the GME detection length is maximal; this is a clean separation result. The SDP formulations and the explicit witness/ Bell-inequality examples in Appendix F are concrete and, apart from the issues noted below, reproducible. The paper also makes a welcome connection between detection length and noise robustness, although the bit-flip analysis needs correction.","major_comments":[{"comment":"The derivation of Proposition 3 assumes that every k-length witness with Tr(W)=d can be written as W = I - sum_i P_i with all non-identity P_i being k-qubit Pauli operators, and that bit-flip errors shrink the expectation of every such term by the same factor (1-2epsilon)^k. This is false for the paper's own witnesses. In Eq. (18), W1 contains S_1 = X1Z2 (weight 2) alongside S_2 = Z1X2Z3 (weight 3), and W2 is a product of factors with mixed-weight terms. An error on a party where the Pauli term acts as identity does not affect that term, and a physical bit-flip error leaves X-type measurement outcomes unchanged while flipping Z/Y outcomes. The correct per-term shrinkage depends on the weight and the Pauli content, so Eqs. (15)-(16) do not follow for the witnesses used in the comparison. Consequently, the inequality p*(W1)>p*(W2) in Eq. (20) is not established by the given argument.","section":"Section IV.B / Appendix A (Prop. 3, Eqs. 15-16)"},{"comment":"The normalization used in the p* formulas is inconsistent with the displayed witnesses. As written, W1 in Eq. (18) has Tr(W1)=1, not d, and its expectation on the cluster state is Tr(W1 rho)=-1/d, not -1. The formula p*(W1)=1-1/[2(1-2epsilon)^3] corresponds to Tr(W rho)=-1, so it does not apply to the displayed W1. Similarly, W2 as defined has trace d^{K-1} rather than d. Therefore the expressions for p*(W1) and p*(W2) in Eq. (20) and the claimed window 0.0857<epsilon<0.0905 for N=11 cannot be reproduced from the displayed constructions. The bit-flip comparison must be redone with correctly normalized witnesses and the correct state expectation values.","section":"Eq. (18) and Eq. (20)"},{"comment":"The constraint set is written as 'W = P + Q^{T_S}, exists S subset [n]' inside a minimization. This makes the feasible set a union over bipartitions S, which is not convex, so as written it is not a valid semidefinite program. Please clarify whether the intended procedure is to solve one SDP for each S (or use a max-type formulation), and state the resulting computational cost. The same ambiguity appears in the GME SDP in Appendix C if 'for all S' is meant as a single constraint.","section":"Section IV.A, Eq. (11)"},{"comment":"The proof asserts that 'all the biproduct states in R(rho) are {a|psi_n> | |a|=1}' without justification. This claim is load-bearing for the conclusion l_GME(rho)=n, since it is used to rule out a biseparable decomposition of rho. Please provide a complete argument, for example by writing a general vector alpha|psi_n>+beta|GHZ_n> in the Schmidt decomposition across each possible bipartition and showing that it is biproduct only when beta=0 (up to global phase).","section":"Appendix A, Proposition 1"}],"minor_comments":[{"comment":"The text describes bit-flip errors as physical qubit flips during a Z-basis measurement, but the proof of Proposition 3 uses a classical outcome-flip model in which every single-qubit outcome flips independently with probability epsilon. Please make the noise model consistent and specify whether the analysis applies to measurement-outcome flips or to a bit-flip channel on the qubits.","section":"Section IV.B"},{"comment":"The mathematical notation in Eq. (20) is garbled (e.g., 'C r 2' and 'C+3 r 1'); please rewrite the inequalities with proper superscripts, roots, and parentheses so that the claimed window can be verified.","section":"Eq. (20)"},{"comment":"The number of qubits is denoted by both n and N in Proposition 2 and its proof; please unify the notation.","section":"Appendix A and Proposition 2"},{"comment":"There are numerous typographical errors, including 'expactation' (Appendix A), 'Bell's equalities' (Section II), 'Wernern' (Appendix E), and inconsistent hyphenation of 'bipartitioned'. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"Please explain the meaning of the '#' entries and clearly indicate which numerical values are newly computed in this paper and which are taken from reference [19].","section":"Appendix D, Tables II and III"},{"comment":"The classical bounds beta_C = 4 for the constructed Bell inequalities are quoted without derivation; please specify the method used to compute the classical maximum (e.g., enumeration of local deterministic strategies).","section":"Eqs. (23)-(24)"}],"recommendation":"major_revision","confidential_remarks":"The paper's main advertised result about bit-flip robustness (shorter witnesses being more noise-tolerant) is not supported by the current derivation, and the normalization issues make Eq. (20) unreproducible. However, the conceptual framework, Proposition 1, Proposition 2, and the SDP constructions appear sound and are likely salvageable with a corrected bit-flip analysis. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe detection length framework is worth knowing about, and this paper extends it in a useful direction. The genuinely new result is Proposition 1: a family of n-qubit states where entanglement can be certified with 2-body marginals while GME requires an n-body measurement, and the bipartite detection length depends on whether the bipartition separates the two particles in |ψ_n>. The argument is short but correct, and it sharpens the earlier framework from ref [2]. The SDP constructions of length-constrained witnesses and Bell inequalities are also practical and clearly explained; the Bell inequalities for 4-qubit cluster and ring states improve the classical/quantum ratio over the naive stabilizer construction, which is a concrete deliverable.\n\nThe soft spot is Proposition 3 and the noise-robustness comparison that leans on it. The proof assumes a k-length witness can be written as I − Σ_i P_i with every P_i a k-qubit Pauli, and then shrinks every term by (1−2ε)^k under bit-flip errors. That is not what the paper's own witnesses look like. W1 in Eq. (18) contains S1 = X1Z2 (weight 2) alongside S2 = Z1X2Z3 (weight 3), so the uniform shrinkage factor does not apply; a bit-flip on a qubit where the term acts as identity leaves that term alone, and a bit-flip does not flip an X-outcome if the error is the physical σ_x. The claimed window (0.0857, 0.0905) for N=11 also doesn't reproduce from the printed W1, W2: with the normalization as written, Tr(W1)=1, not d, and the p* formulas in Eq. (20) use Tr(Wρ)=−1 and −2K, which corresponds to a different normalization. So the quantitative shorter-witness-robustness claim is not established. This is a load-bearing flaw in one advertised contribution, but it does not sink the framework or Proposition 1. The fix is straightforward in principle: redo the noise analysis term by term, restricted to witnesses with uniform Pauli weight and a basis-consistent error model.\n\nWho is this for? People working on entanglement certification experiments who want a quantitative handle on how many parties need to measure together, and who can use the SDP to construct length-minimized witnesses. The paper deserves a serious referee — the framework and Prop 1 are solid — but Proposition 3 needs correction before publication.\n\nBest,","headline":"Solid extension of detection length framework with a correct gap result, but the bit-flip noise-robustness claim rests on an unjustified uniform-shrinkage assumption.","tokens_in":22830,"tokens_out":4497,"would_cite":true,"duration_ms":38124,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","90C22"],"pacs":["03.65.Ud","03.67.Mn"],"model":"deepseek-v4-flash","headline":"For one family of states, detection length is 2 for plain entanglement and n for genuine multipartite entanglement; SDP-built witnesses achieve prescribed lengths, and shorter witnesses can beat longer ones under bit-flip noise.","keywords":["detection length","entanglement witness","genuine multipartite entanglement","Bell inequality","nonlocality","semidefinite programming","noise robustness","bit-flip error"],"falsifier":"Recompute the bit-flip noise tolerance of the cluster-state witness $W_1$ of Eq. (18) term by term: each stabilizer term $S_i$ contains some number of $Z$ factors and some of $X$ factors, and a bit-flip on a measured qubit only changes the sign of a $Z$ outcome, so the decay factor should be $(1-2\\epsilon)^{\\#Z}$ rather than $(1-2\\epsilon)^k$; if the resulting $p^*(W_1)$ fails to exceed $p^*(W_2)$ in the window of Eq. (20), the claimed shorter-witness robustness is refuted. Independently, the Proposition 1 claim can be probed numerically by SDP: minimize $\\mathrm{Tr}(W\\rho)$ over witnesses using only marginals of size at most $n-1$ and check whether the optimum is nonnegative, which would confirm $l_{GME}=n$.","tokens_in":21761,"feed_emoji":"⚛️","tokens_out":14833,"duration_ms":120699,"temperature":0.7,"pith_summary":"Detection length is the size of the largest joint measurement an experiment must perform to certify a quantum property, and this paper asks how that minimum varies across entanglement classes and nonlocality. The paper's central result is a construction of genuinely entangled states for which the entanglement detection length is the smallest possible, $l_{Ent}=2$, while the genuine-multipartite-entanglement detection length is the largest possible, $l_{GME}=n$: two-body marginals fully certify that the state is entangled, yet certifying that it is genuinely multipartite entangled requires one $n$-party measurement. It also extends detection lengths to bipartition entanglement, entanglement depth, and intactness, and proves the ordering $2\\le l_{Ent}\\le l_{Bipa}(\\rho,G)\\le l_{GME}(\\rho)$ together with $2\\le l_{Ent}\\le l_{Nol}(\\rho)$. Methodologically it gives a semidefinite-programming construction of entanglement witnesses and Bell inequalities with prescribed measurement length, which converts directly into upper bounds on detection length for any given state, and it tables explicit witnesses for W, Dicke, cluster, ring, and Bell-product states. Finally, it analyzes noise robustness and claims that under bit-flip measurement errors a shorter witness can be more noise tolerant than a longer one in a specific parameter window.","feed_headline":"For some states, certifying genuine entanglement needs all n parties","feed_subtitle":"Two-body marginals certify plain entanglement, yet genuine multipartite entanglement needs all n parties.","key_machinery":"The carrier of the argument is the compatibility set $\\mathcal{C}(\\rho,\\mathcal S)$ — the set of all density matrices sharing $\\rho$'s marginals on a family $\\mathcal S$ of subsystems — with detection length defined as the smallest possible value of $\\max_{S\\in\\mathcal S}|S|$ over families $\\mathcal S$ whose compatibility set contains only states with the target property. Proposition 1 works by showing that the marginal on qubits $\\{1,2\\}$ is NPT entangled, so two-body detection suffices for plain entanglement, while a carefully chosen state sharing all $(n-1)$-body marginals is biseparable, forcing $l_{GME}=n$. For the constructive half, the machinery is the decomposable-witness semidefinite program of Eq. (11), which optimizes $\\mathrm{Tr}(W\\rho)$ over length-restricted witnesses $W=\\sum_{M_j\\in\\mathcal M} H_{M_j}\\otimes I_{\\overline{M_j}}$ with $W=P+Q^{T_S}$, turning a negative optimum into a witness of length $l(\\mathcal M)$ and hence an upper bound on the detection length; the same SDP, constrained to $X/Z$-type terms and mapped to observables, yields Bell inequalities of prescribed length. The noise claim rests on the identity $\\alpha(\\rho)=1-(1-2\\epsilon)^k(1-\\mathrm{Tr}(W\\rho))$ for a length-$k$ witness under per-qubit bit-flip error.","core_discovery":"On the paper's own terms, the discovery is Proposition 1: for the mixed state $\\rho = p|\\psi_n\\rangle\\langle\\psi_n| + (1-p)|GHZ_n\\rangle\\langle GHZ_n|$ with $1/2<p<1$, where $|\\psi_n\\rangle = (|100\\cdots0\\rangle+|010\\cdots0\\rangle)/\\sqrt{2}$, the state $\\rho$ is genuinely entangled with $l_{GME}(\\rho)=n$ while $l_{Ent}(\\rho)=2$, so the two detection lengths are as far apart as possible; moreover for any bipartition $G$, $l_{Bipa}(\\rho,G)=2$ exactly when $G$ separates particles 1 and 2, and equals $n$ otherwise. The paper presents this as a negative answer to the question of whether entanglement and GME detection lengths always coincide, and proves it by showing the compatibility set $\\mathcal{C}(\\rho,\\mathcal{S}_{n-1})$ contains a biseparable state while the two-body marginal $\\rho_{\\{1,2\\}}$ is an NPT (negative partial transpose) entangled state. It further gives Proposition 2, an explicit two-parameter family of genuinely entangled states with $l_{GME}=n$ that for allowed parameter values violates no Bell inequality and hence has $l_{Nol}=+\\infty$, and Proposition 3, a formula for the expectation value of a length-$k$ witness under global depolarizing noise plus bit-flip measurement errors, from which it concludes that shorter witnesses can tolerate more bit-flip noise than longer ones in a parameter window such as $0.0857<\\epsilon<0.0905$ for the 11-qubit cluster state.","pith_inferences":["The maximal-gap construction suggests an operational reading the paper leaves implicit: the gap $l_{GME}-l_{Ent}$ measures how much measurement globality a state hides, and could serve as a resource quantifier for certification tasks in which parties trust only few-body measurements.","Because the proof of Proposition 1 only uses that the state lives in the two-dimensional span of a two-excitation state and the GHZ state, the same maximal gap should obtain for any mixture supported on such a subspace; testing three-parameter mixtures would show whether the gap is generic or special to two-dimensional supports.","The noise analysis points to an actionable selection rule not stated by the authors: when measurement errors are dominated by bit flips, the optimal witness length balances the decay factor $(1-2\\epsilon)^k$ against the ideal noise tolerance, so an experimenter could choose the length that maximises $p^*$ at the measured $\\epsilon$.","The SDP-to-Bell pipeline is effectively an automated search for short Bell inequalities with large quantum-to-classical gaps; applying it beyond four-qubit graph states could reveal whether the $1/\\sqrt{2}$ ratio at length three is typical or an artefact of small systems."],"forward_implications":["For the states of Proposition 1, no protocol that only ever measures fewer than $n$ parties together can certify genuine multipartite entanglement, no matter how many two-body marginals it collects; the trade-off between detection capability and measurement globality is intrinsic to the state, not a deficiency of a particular witness.","The hierarchy $2\\le l_{Ent}\\le l_{Bipa}(\\rho,G)\\le l_{GME}(\\rho)$ and $2\\le l_{Ent}\\le l_{Nol}(\\rho)$ gives a universal lower bound for experiments: nonlocality detection is never cheaper in measurement globality than entanglement detection, and the same holds for bipartition entanglement relative to plain entanglement.","The SDP pipeline converts any target state into an explicit entanglement witness or Bell inequality of prescribed length, so detection-length statements become experimentally testable observables; the paper tables concrete witnesses for W, Dicke, cluster, ring, and Bell-product states.","The Bell inequalities derived for the 4-qubit cluster and ring states have quantum-to-classical ratio $1/\\sqrt{2}$, tolerating depolarizing noise up to $p<1-1/\\sqrt{2}$, which improves on the naive stabilizer-based inequalities with ratio $3/4$ while using only 3-body terms.","Under the bit-flip model of Proposition 3, the noise tolerance of a length-$k$ witness shrinks as $(1-2\\epsilon)^k$, so for the 11-qubit cluster state a 3-length witness outperforms the near-global witness for bit-flip rates $0.0857<\\epsilon<0.0905$ — a concrete regime where fewer-body measurements are the more robust experimental choice."],"supporting_citations":[{"why":"Establishes the detection-length framework and compatibility set that this paper extends, and proves the no-gap results for GHZ, symmetric, cluster, and ring states.","marker":"[2]"},{"why":"Supplies the genuinely entangled state with a fully local model used in Proposition 2, along with the parameter condition under which it violates no Bell inequality.","marker":"[31]"},{"why":"Shows multipartite Werner states are entangled exactly when they violate the PPT criterion, which fixes their entanglement detection length in this paper's analysis.","marker":"[30]"},{"why":"Provides the decomposable and fully decomposable witness formalism on which the SDP constructions of Eqs. (11) and (C1) are built.","marker":"[19]"},{"why":"Gives the stabilizer-based Bell inequalities with bound ratio 3/4 for cluster and ring states that the paper's SDP-derived inequalities improve on.","marker":"[26]"},{"why":"Defines the noise-tolerance measure used to compare witnesses and supplies the known l_Nol = n result for GHZ states in Table I.","marker":"[20]"},{"why":"Shows the two-body-marginal compatibility set of any graph state contains a fully separable state, forcing entanglement detection length at least 3 for graph states.","marker":"[28]"},{"why":"Proves connected graph states are genuinely entangled, which underpins the l_GME = 3 claims for cluster and ring states.","marker":"[29]"}],"fun_headline_variants":["For some states, GME detection requires all n parties","Genuine entanglement may need all n parties to verify","Some states: GME detection length n, entanglement 2","Shorter detection witnesses tolerate more bit-flip noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The shorter-witness-beats-longer-witness noise claim rests on the assumption that every term of a length-$k$ witness is a full $k$-qubit Pauli product, so a bit-flip shrinks all terms by the identical factor $(1-2\\epsilon)^k$; the paper's own cluster-state witness mixes two- and three-qubit terms and treats $X$ and $Z$ outcomes alike even though a bit-flip does not change an $X$ outcome, so the uniform shrink factor does not apply to it.","fun_headline_variants_meta":{"raw":{"variants":["For some states, GME detection requires all n parties","Genuine entanglement may need all n parties to verify","Some states: GME detection length n, entanglement 2","Shorter detection witnesses tolerate more bit-flip noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00165,"raw_usage":{"total_tokens":6596,"prompt_tokens":1029,"completion_tokens":5567,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":645,"completion_tokens_details":{"reasoning_tokens":5500}},"tokens_in":645,"tokens_out":5567,"duration_ms":32724,"temperature":1.0,"reasoning_tokens":5500,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:00:59.897243+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the bit-flip noise tolerance of the cluster-state witness $W_1$ of Eq. (18) term by term: each stabilizer term $S_i$ contains some number of $Z$ factors and some of $X$ factors, and a bit-flip on a measured qubit only changes the sign of a $Z$ outcome, so the decay factor should be $(1-2\\epsilon)^{\\#Z}$ rather than $(1-2\\epsilon)^k$; if the resulting $p^*(W_1)$ fails to exceed $p^*(W_2)$ in the window of Eq. (20), the claimed shorter-witness robustness is refuted. Independently, the Proposition 1 claim can be probed numerically by SDP: minimize $\\mathrm{Tr}(W\\rho)$ over witnesses using only marginals of size at most $n-1$ and check whether the optimum is nonnegative, which would confirm $l_{GME}=n$.","supporting_citations":[{"cited_title":"detection length","cited_arxiv_id":null,"evidence_quote":"Establishes the detection-length framework and compatibility set that this paper extends, and proves the no-gap results for GHZ, symmetric, cluster, and ring states."},{"cited_title":"G¨ uhne, G","cited_arxiv_id":null,"evidence_quote":"Supplies the genuinely entangled state with a fully local model used in Proposition 2, along with the parameter condition under which it violates no Bell inequality."},{"cited_title":"Ichikawa, T","cited_arxiv_id":null,"evidence_quote":"Shows multipartite Werner states are entangled exactly when they violate the PPT criterion, which fixes their entanglement detection length in this paper's analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the decomposable and fully decomposable witness formalism on which the SDP constructions of Eqs. (11) and (C1) are built."},{"cited_title":"Zhao and Y","cited_arxiv_id":null,"evidence_quote":"Gives the stabilizer-based Bell inequalities with bound ratio 3/4 for cluster and ring states that the paper's SDP-derived inequalities improve on."},{"cited_title":"Santos, D","cited_arxiv_id":null,"evidence_quote":"Defines the noise-tolerance measure used to compare witnesses and supplies the known l_Nol = n result for GHZ states in Table I."},{"cited_title":"Zhang, Y","cited_arxiv_id":null,"evidence_quote":"Proves connected graph states are genuinely entangled, which underpins the l_GME = 3 claims for cluster and ring states."}],"review_version":1}