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Optimizing and Certifying Multipartite Permutationally Invariant Bell Inequalities

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Pith's one-line read Optimized multipartite Bell inequalities using only collective one- and two-body measurements yield exact asymptotic quantum-to-classical ratios that grow with the number of settings and approach coth(1).

desk verdict Solid, usable advance on PI Bell inequalities: explicit optimized rank-one families, certified large-N ratios that are exact rationals for m=2–6 and approach coth(1), with independent classical DP and SOS certificates that match. read the letter →

arxiv 2607.08462 v1 pith:IHYOEHLB submitted 2026-07-09 quant-ph

Jin-Fu Chen , Mengyao Hu , Jordi Tura This is my paper · ORCID
classification quant-ph
keywords BellnonlocalitypermutationallyinvariantinequalitiesmultipartitecorrelationsspinsqueezingHolstein–Primakoffapproximationsemidefiniteprogrammingcertificationquantum-to-classicalratio
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows how to build and certify multipartite Bell inequalities that stay useful when the number of parties becomes large. By restricting to permutationally invariant inequalities that use only one- and two-body correlators, and by forcing the two-body coefficients into a rank-one form, the authors obtain a scalable classical bound via dynamic programming and a scalable quantum value via spin-squeezed states plus the Holstein–Primakoff approximation. They then optimize the quantum-to-classical ratio in the infinite-N limit, obtaining simple rational numbers for any finite number of measurement settings and the continuum value coth(1) as that number goes to infinity. Semidefinite-programming certificates confirm that the leading-order quantum values are exact. The resulting inequalities can therefore detect many-body Bell nonlocality with collective measurements that are already natural in atomic ensembles, and more measurement settings produce stronger, more noise-tolerant violations.

What carries the argument

Rank-one two-body coefficients α_{k,l}=γ_k γ_l together with a paired antisymmetric structure on the coefficient vectors. This flattens the O(N^{2}) mean-field term, reduces classical optimization to dynamic programming over occupation numbers, maps the quantum problem onto an LMG-like spin Hamiltonian whose large-N limit is a single bosonic mode, and yields a one-dimensional continuum integral whose maximum is coth(1).

What would settle it

Compute the exact classical bound and the certified quantum value for one of the optimized inequalities at a large but finite N (say N=100–1000) and check whether the ratio remains strictly below the claimed infinite-N rational; or exhibit a full-rank two-body matrix that yields a larger asymptotic ratio under the same measurement constraints.

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Extended reading notes

Core claim

For a broad family of rank-one, paired permutationally invariant Bell inequalities with one- and two-body correlators, the infinite-N quantum-to-classical ratio optimizes to exact rationals for finite m (5/4, 9/7, 353/272, 275/211, 66637/51012 for m=2…6) and converges to coth(1) as m→∞; the same leading-order quantum values are certified by sum-of-squares and moment relaxations.

Load-bearing premise

The two-body coefficient matrix must be rank one; the authors argue that a full-rank positive-definite matrix cannot produce an asymptotic quantum-classical separation because both sides share the same O(N^{2}) term.

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Referee Report

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Summary. The manuscript constructs and optimizes multipartite permutationally invariant (PI) Bell inequalities in the (N,m,2) scenario that use only one- and two-body correlators. Restricting the two-body coefficient matrix to rank one (α_{k,l}=γ_k γ_l) and imposing a paired antisymmetric structure on coefficients and measurement angles, the authors obtain classical bounds by occupation-number dynamic programming (finite N) and continuum minimization (N→∞), and quantum values by symmetric-sector diagonalization, spin-squeezed variational states, and Holstein–Primakoff asymptotics. They maximize the quantum-to-classical ratio Δ_∞,m, obtaining exact rationals for m=2…6 (5/4, 9/7, 353/272, 275/211, 66637/51012) that approach coth(1) as m→∞, and certify the leading-order quantum values by sum-of-squares / moment SDP relaxations that match the variational results. Finite-N numerics confirm convergence of the ratios.

Significance. The work supplies a scalable, symmetry-adapted pipeline for multipartite Bell inequalities that remain useful at large N, where the local polytope is otherwise intractable. Exact classical DP, closed-form continuum limits, explicit optimized rationals, and independent SDP certificates that agree to leading order in N are concrete strengths. The continuum limit coth(1) and the demonstration that more measurement settings improve the asymptotic ratio are clean, falsifiable predictions. Within the rank-one paired family the results are rigorous and immediately usable for collective-measurement experiments on spin-squeezed ensembles.

minor comments (4)
  1. Appendix A motivates the rank-one restriction by arguing that a full-rank positive-definite two-body matrix shares the O(N²) term and cannot produce asymptotic separation. A short explicit remark that higher-rank matrices with nontrivial kernels remain open would clarify the scope without weakening the claim.
  2. Table II and Eqs. (47)–(61) list optimized coefficients; stating the numerical precision or the exact rational form of intermediate Γ_j^* (e.g. 15/34 for m=4) in the main text would aid reproducibility.
  3. Figure 2 caption and panels (b)–(f) use both Δ_∞,m and Δ_opt_∞,m; a uniform notation would avoid minor confusion.
  4. The continuum derivation in Appendix E.4 is clear, but a one-sentence pointer in the main text that the continuum functional depends only on the final cumulative Γ_f would help readers who skip the appendix.

Circularity Check

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No significant circularity: classical and quantum sides are independent optimizations; ratios are maximized over free coefficients and certified by a separate SDP, not defined from the variational inputs.

full rationale

The load-bearing chain is self-contained. Classical bounds come from occupation-number dynamic programming (App. B) and a continuum minimization under γ·s=0 (App. C); quantum values come from an independent Holstein–Primakoff / spin-squeezed variational calculation (Sec. III, App. D). The infinite-N ratios are then obtained by maximizing an explicit functional of free coefficient sequences Γ and c (App. E, Eqs. E9, E21, E41), yielding exact rationals for finite m and coth(1) as m→∞—these are optimization outputs, not fits to data renamed as predictions. Rank-one two-body structure (Eq. 6) is a deliberate modeling restriction motivated by an O(N²)-sharing argument (App. A), not a uniqueness theorem imported from the authors. SDP/sum-of-squares certificates (App. F–G) are separate relaxations whose feasible values match the variational energies rather than being defined from them. Self-citations (prior PI Bell inequalities, tropical DP methods) supply background and tools; none force the numerical values of the new optimized ratios. No step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard quantum mechanics and Bell locality plus three modeling choices that define the optimized family: rank-one two-body coefficients, a paired antisymmetric structure for α and γ, and restriction of the quantum search to the fully symmetric spin sector (later certified). No free parameters are fitted to experimental data; coefficients are obtained by analytic or numerical maximization of an explicit ratio functional. No new physical entities are postulated.

assumptions (5)
  • ad hoc to paper Two-body coefficient matrix is rank-one: α_{k,l}=γ_k γ_l (Eq. 6).
    Appendix A argues this is necessary for an asymptotic quantum-classical gap; it is a modeling restriction that defines the family being optimized, not a theorem of Bell theory.
  • ad hoc to paper Paired (antisymmetric) structure of α, γ and measurement angles (Eqs. 44–45, C20).
    Imposed to make the large-N classical and quantum optimizations tractable and to enforce the flat-direction constraint γ·s=0; other structures are not explored.
  • domain assumption Quantum value is evaluated (variationally) in the fully symmetric spin-N/2 subspace.
    Standard for PI collective measurements; yields an upper bound on the true quantum value that is later certified from below by SOS (Appendix F).
  • domain assumption Holstein–Primakoff bosonic approximation captures the leading large-N ground-state energy of the LMG-like Bell operator.
    Standard semiclassical spin approximation; validated by finite-N exact diagonalization and moment certificates in Fig. 2.
  • standard math Local deterministic strategies and the classical polytope of the (N,m,2) Bell scenario.
    Standard Bell theory; used to define β_C via occupation numbers and dynamic programming.

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Pith. "Pith review of Optimizing and Certifying Multipartite Permutationally Invariant Bell Inequalities." pith.science (2026). https://pith.science/paper/IHYOEHLB

@misc{pith2026260708462,
  author       = {Pith},
  title        = {Pith review of: Optimizing and Certifying Multipartite Permutationally Invariant Bell Inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IHYOEHLB}},
  note         = {Machine review of arXiv:2607.08462}
}
abstract

Multipartite Bell nonlocality provides a device-independent probe of many-body quantum correlations, but its characterization is limited by the rapid growth of the underlying classical and quantum optimization problems. We develop a scalable method for constructing and certifying permutationally invariant Bell inequalities using only one- and two-body correlators. The construction gives families of inequalities with robust quantum violations for general $m$ measurements as the number of parties $N$ becomes large. To improve robustness against noise, we optimize the ratio of the quantum value to the classical bound for these families in the large-$N$ limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-$N$ ratios take simple rational values for finite $m$ and converge to $\coth(1)$ as $m\to\infty$. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.

Figures

Figures reproduced from arXiv: 2607.08462 by the authors.

Figure 1
Figure 1. FIG. 1. Diagram for the measurement angles and the spin [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Finite- [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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