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REVIEW 4 major objections 5 minor 45 references

Experimentally friendly approach towards nonlocal correlations in multisetting N -partite Bell scenarios

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Pith's one-line read A single family of lifted CHSH Bell inequalities estimates the nonlocal fraction and nonlocality strength of N-qubit states to within experimental error.

desk verdict A genuinely useful single-inequality estimator for nonlocal fraction and strength, but the advertised accuracy is contradicted by the paper's own GHZ4 numbers — worth serious refereeing, not a desk reject. read the letter →

arxiv 2009.11691 v2 pith:T3PZT3SY submitted 2020-09-24 quant-ph

Artur Barasiński , Antonín Černoch , Wies{l}aw Laskowski , Karel Lemr , Tamás Vértesi , Jan Soubusta This is my paper · ORCID
classification quant-ph
keywords nonlocalfractionnonlocalitystrengthliftedCHSHinequalitiesBellpolytopemultipartiteentanglementwitnessingrandommeasurementsN-qubitstatesscenarios
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

Measuring how nonlocal a multipartite quantum state is — the fraction $P_V$ of random measurement settings that violate local realism, and the average nonlocality strength $\bar S$ (the robustness of that violation to white noise) — normally requires knowing all tight Bell inequalities for the scenario, i.e. the full boundary of the local polytope, and such complete lists are unknown except for the simplest cases. The paper proposes that a single family of lifted CHSH inequalities, $I^{(N)}_{\mathrm{opt}}$, captures enough of this statistical content for $N$-qubit pure states that the estimated $P_V$ and $\bar S$ land within errors comparable to ordinary measurement noise. Across the $2\times2\times2$ scenario, all 46 facet classes were ranked, and $I^{(3)}_{\mathrm{opt}}$ was the closest to the full-polytope value, within about 5.3 percentage points for GHZ and W states; in four- and five-qubit archetypal states the agreement is similar and tightens as the number of settings grows. A three-qubit photonic experiment confirms the estimate for a noise-affected GHZ state. If the claim holds, experimental determination of $P_V$ and $\bar S$ no longer requires solving the Bell polytope, and the same data certify genuine multipartite entanglement without aligned reference frames.

What carries the argument

The machine that carries the argument is the family of lifted CHSH Bell inequalities $$$I^{{(N)}}$_{\mathrm{opt}}=\left\langle ($I^{{(2)}}$_{\mathrm{opt}}-2)\prod_{i=3}^{N}(1-$E^{{(i)}}$_0)\right\rangle\le0,$$ where $I^{(2)}_{\mathrm{opt}}=E^{(1)}_0E^{(2)}_0+E^{(1)}_1E^{(2)}_0+E^{(1)}_0E^{(2)}_1-E^{(1)}_1E^{(2)}_1$ is the CHSH expression and $E^{(i)}_j$ is the observable of party $i$ for setting $j$. The product over the remaining $N-2$ parties means every violation of the lifted inequality traces back to a two-qubit CHSH violation, yet the lifted family is itself tight, is violated by every pure entangled $N$-qubit state, and carries a much lower sampling cost: counting all equivalent forms gives time complexity $O(m^4N^2\,2^{m(N-2)})$ against $O(m^2N\,2^{mN})$ for linear programming over the polytope.

What would settle it

A concrete falsifier is to scan random four-qubit pure states in the $2\times2\times2\times2$ scenario, computing $P_V$ by linear programming over the full local polytope and comparing with the $I^{(4)}_{\mathrm{opt}}$-only estimate; if any state shows a discrepancy well beyond the roughly 10-percentage-point gap already seen for the Dicke state $|D_2^4\rangle$, the conjecture that the family is generally sufficient is refuted.

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Extended reading notes

Core claim

The paper's central claim is that both the nonlocal fraction $P_V(\rho)$ and the average nonlocality strength $\bar S(\rho)$ of $N$-qubit pure states can be estimated from the violations of a single tight family of Bell inequalities, $I^{(N)}_{\mathrm{opt}}$, obtained by lifting the CHSH expression so that two parties have two measurement settings while the other $N-2$ parties have one. Every violation of $I^{(N)}_{\mathrm{opt}}$ has a two-qubit CHSH origin, yet the family is tight, is violated by any pure entangled $N$-qubit state, and is shown numerically to dominate the statistics: in the $2\times2\times2$ scenario it is the best of the 46 facet classes for both $P_V$ and $\bar S$, and in the four- and five-qubit scenarios its estimates of the polytope values agree to a few percentage points, with the gap shrinking as the number of settings grows. The authors also show that for a randomly sampled pure state and randomly chosen bases, the $I^{(N)}_{\mathrm{opt}}$ family detects nonlocality almost 100% of the time once each party can choose three settings, underestimating the full-polytope typical fraction by only 3–4 percentage points. Their three-qubit photonic experiment on a GHZ state with visibility $v\approx0.96$ yields $P_V$ and $\bar S$ consistent with the theoretical predictions, and its $P_V=56\pm5\%$ value detects genuine three-qubit entanglement.

Load-bearing premise

The whole procedure rests on the unproved working conjecture that the lifted CHSH family keeps tracking the full nonlocal fraction and nonlocality strength for every $N$-qubit pure state and measurement scenario, with errors staying close to experimental noise rather than growing for untested states or settings.

Editorial extensions

If this is right

  • Nonlocal fraction and nonlocality strength for $N$-qubit pure states become experimentally accessible in arbitrary multisetting scenarios without enumerating the full local polytope.
  • Randomly chosen pure states and random measurement bases already violate the lifted CHSH family almost always when each party has at most three settings, so Bell violation is typical rather than fine-tuned.
  • A single inequality family can witness genuine multipartite entanglement without aligned reference frames; the three-qubit GHZ experiment detects genuine tripartite entanglement with $P_V=56\pm5\%$ against the $2(\pi-3)\approx28.3\%$ threshold.
  • The convergence of $P_V$ to unity with increasing settings is explained statistically: more settings mean more equivalent lifted CHSH inequalities, not fundamentally new Bell facets.
  • The $I^{(N)}_{\mathrm{opt}}$ family serves as a first approximation to the geometry of the local polytope, with the majority of multipartite nonlocality having an effective two-setting CHSH origin.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: because $I^{(N)}_{\mathrm{opt}}$ detects only two-qubit CHSH-type violations, its estimate of the nonlocality strength should systematically under-represent genuinely multipartite violation channels; the paper's own GHZ$_4$ and GHZ$_5$ strength distributions, which deviate most from the full polytope, fit this pattern.
  • Inference: combining $I^{(N)}_{\mathrm{opt}}$ with the next-best facet families $I_5$ and $I_6$ could close most of the residual gap (up to 10 percentage points for the four-qubit Dicke state) at still moderate experimental cost.
  • Inference: the near-100% violation probability for random states suggests using $I^{(N)}_{\mathrm{opt}}$ as a fast device-independent entanglement probe before any expensive tomography, flagging only states whose estimated $P_V$ approaches the multipartite threshold for a full-polytope check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes that the nonlocal fraction P_V and the average nonlocality strength S of N-qubit pure states can be estimated experimentally from violations of a single family of lifted CHSH Bell inequalities, I_opt^(N), instead of requiring the full (generally unknown) set of tight Bell inequalities. Numerical evidence is given for GHZ, W, cluster, Dicke, and random states in three-, four-, and five-qubit scenarios, together with an experimental demonstration for the three-qubit GHZ state. The paper further argues that this approach enables detection of genuine multipartite entanglement without aligned reference frames and that its imprecision is of similar magnitude to typical measurement errors.

Significance. If the central conjecture were established, the paper would provide a practical and scalable route to experimentally quantifying nonlocality and certifying multipartite entanglement in settings where complete Bell polytope knowledge is unavailable. The experimental implementation for GHZ3 is a genuine strength, and the numerical exploration over multiple archetypal states provides useful evidence. However, the central claim is explicitly left as a conjecture, and several of the paper's own numerical results show estimation errors substantially larger than the quoted experimental uncertainties, so the significance is conditional on a substantial revision of the accuracy claim and its scope.

major comments (4)
  1. [Abstract and §3.1.3, Table 3] The abstract's statement that the imprecision is 'of similar magnitude as the potential measurement errors' is contradicted by the paper's own numbers. For the GHZ4 state in the 2x2x2x2 scenario, S_Iopt = 0.1067 while the full-polytope value is S_L4 = 0.1578, a relative underestimate of about 32%, whereas the experimental uncertainties in Table 7 are about 0.001 in S. For the D2_4 Dicke state, Fig. 5 shows a P_V gap of up to 10 percentage points, compared with experimental P_V errors of ±2 to ±5 points in Table 7. These are not edge cases but canonical states, so the advertised accuracy does not hold generally; the claim should be restricted to the states and settings where the error is actually comparable to the measurement noise, or replaced by a quantitative state-dependent bound.
  2. [§3.1, after Eq. (5)] The load-bearing assumption that I_opt^(N) is sufficient to estimate P_V and S is explicitly presented as a 'working conjecture' with no proof of optimality. The numerical evidence covers selected archetypal states and random samples, but the state-dependence of the approximation error (e.g., the GHZ4 S gap and the D2_4 P_V gap) shows that the conjecture cannot be read as a universal statement over all N-qubit pure states. The paper should either prove a bound (even a loose one) for a well-defined state family, or clearly delimit the scope of the conjecture and avoid unconditional wording such as 'we show' in the abstract and conclusions.
  3. [Tables 1–6] The Monte Carlo estimates of P_V and S are reported without statistical error bars or sample sizes per state (Table 5 gives 1.2x10^5 random states but not the number of random settings per state). Without such error bars, the deviations between P_Iopt, P_L3, and P_L4 cannot be distinguished from sampling fluctuations, and the claim that I_opt provides a 'very good approximation' is not quantitatively supported. Please include standard errors (e.g., binomial errors for P_V) or specify the sampling procedure so the gaps can be assessed.
  4. [§4.2, Eq. (9) and Table 7] The experimental test covers only the three-qubit GHZ state and uses the visibility v obtained from tomography to fix the theoretical state in Eq. (9); the experiment therefore validates the estimation procedure for one state and one noise model, but does not test the general claim for arbitrary N-qubit states or for states like GHZ4 and D2_4 where the approximation error is largest. The text should explicitly state that the experimental agreement does not resolve the conjecture for those cases, and should discuss whether the white-noise model in Eq. (9) is adequate for the random-setting measurements performed.
minor comments (5)
  1. [Throughout] There are several typographical issues: 'Uniwersity' in the affiliation, 'politope' in §3.1.1, 'gubit' in §4.2, and '60,141' with a comma in Table 4. These should be corrected.
  2. [§3.2, Table 6] The quantity 'u.s.' in Table 6 is undefined; please spell out 'unable to see' or otherwise clarify what it denotes. Also, 'Rosier et al.' in §3.2 should be 'de Rosier et al.' to match the reference list.
  3. [Figs. 6 and 8] The dashed lines referenced to Ref. [20] are not explained in the captions; please state explicitly that they represent the full-polytope results for the corresponding states.
  4. [§4.2] The text says the experimental data were processed with n = 5x10^4 random settings, but the error bars in Table 7 are quoted as ±2 to ±5 p.p. for P_V and ~0.001 for S; please explain how these error bars were derived from the Monte Carlo simulation of Poissonian noise and how they account for finite sampling.
  5. [§3.1, complexity estimate] The formula for the number of equivalent inequalities R = m^2(m-1)^2 N(N-1) 2^{m(N-2)} appears to count ordered pairs of CHSH parties; if so, the factor 1/2 is missing and the complexity estimate should be adjusted accordingly. Please clarify the counting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the single-inequality estimates are benchmarked, not derived, from the linear-programming polytope values.

full rationale

The paper's central estimator P_V^{I_opt} and S^{I_opt} is computed by evaluating one family of lifted CHSH inequalities, while the 'exact' values P_V^{L_N} and S^{L_N} are taken from prior linear-programming analyses (Refs. [15,20]). These are independent methods, and the inequality family is not fitted to those benchmarks: Eq. (5) defines I_opt^(N) directly in terms of CHSH observables, and the comparisons in Tables 1-5 and Figures 1-8 are Monte Carlo estimates of how much of the local-polytope violation is captured by this single family. The authors explicitly flag the absence of a proof: 'As there is no proof of the optimality of I_opt^(N), our working conjecture is supported by the numerical results presented below' (Section 3.1). The selection of I_opt as the best of 46 facet classes for N=3 creates an in-sample selection effect for the three-qubit claim, but the N=4 and N=5 tests are out-of-sample, and the random-state typicality tests are independent. Self-citations to Refs. [15], [18], and [20] supply benchmarks and prior properties, but those citations are not load-bearing: the paper recomputes the relevant nonlocal fractions itself for the states it studies, and the cited linear-programming results use a different algorithm. The skeptical concern that the claimed accuracy is contradicted by the paper's own numbers (e.g., S^{I_opt}(GHZ4)=0.1067 versus S^{L4}(GHZ4)=0.1578, and up to 10 p.p. gap for D2_4) is a correctness or validity criticism, not a circularity: because I_opt is a valid Bell inequality, any violation of I_opt implies violation of local realism, so the estimate is a lower bound by construction, and an inaccurate approximation does not make the derivation circular. Overall, no equation reduces the estimated quantities to their inputs, no fitted parameter is renamed as a prediction, and no load-bearing uniqueness claim is imported from self-citations.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central estimate relies on the unproved sufficiency of I_opt^(N) and on linear-programming benchmarks from prior work (some self-cited) for the full-polytope values. The only fitted number in the paper is the visibility v used to model the experimental state, which does not enter the main theoretical claims.

free parameters (1)
  • white-noise visibility v = 0.96 (from maximum-likelihood state tomography, with stated value 0.97±0.01)
    In the experimental section (§4.2), the reconstructed state is modeled as ρ_expt = v|ϑ⟩⟨ϑ| + (1−v)ρ_white. Theoretical curves in Table 7 are computed for v=1, 0.97, 0.96, 0.95, and the experimental points agree best with v=0.96, so the comparison is a consistency check rather than an independent prediction.
assumptions (3)
  • standard math The complete set of tight Bell inequalities is unknown for m1×m2×m3 scenarios with settings >2; linear programming over the behavior space yields the exact nonlocal fraction P_V^LN.
    The benchmark values L3, L322, etc. in Tables 1-2 and Figs. 1-4 are taken from Refs. [15,20], which use linear programming. This is a background method accepted in the field, not derived here.
  • ad hoc to paper The lifted CHSH family I_opt^(N) is sufficient to estimate P_V and S for all N-qubit pure states.
    Section 3.1 states: 'As there is no proof of the optimality of I_opt^(N), our working conjecture is supported by the numerical results presented below.' This is the load-bearing unproved assumption.
  • domain assumption Experimental imperfections in the prepared GHZ state can be modeled as white noise admixture.
    Section 4.2 assumes ρ_expt = v|ϑ⟩⟨ϑ| + (1−v)ρ_white. This is a modeling approximation, not a derived fact.

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Cite this review

Pith. "Pith review of Experimentally friendly approach towards nonlocal correlations in multisetting N -partite Bell scenarios." pith.science (2026). https://pith.science/paper/T3PZT3SY

@misc{pith2026200911691,
  author       = {Pith},
  title        = {Pith review of: Experimentally friendly approach towards nonlocal correlations in multisetting N -partite Bell scenarios},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3PZT3SY}},
  note         = {Machine review of arXiv:2009.11691}
}
abstract

In this work, we study a recently proposed operational measure of nonlocality by Fonseca and Parisio~[Phys. Rev. A 92, 030101(R) (2015)] which describes the probability of violation of local realism under randomly sampled observables, and the strength of such violation as described by resistance to white noise admixture. While our knowledge concerning these quantities is well established from a theoretical point of view, the experimental counterpart is a considerably harder task and very little has been done in this field. It is caused by the lack of complete knowledge about the facets of the local polytope required for the analysis. In this paper, we propose a simple procedure towards experimentally determining both quantities for $N$-qubit pure states, based on the incomplete set of tight Bell inequalities. We show that the imprecision arising from this approach is of similar magnitude as the potential measurement errors. We also show that even with both a randomly chosen $N$-qubit pure state and randomly chosen measurement bases, a violation of local realism can be detected experimentally almost $100\%$ of the time. Among other applications, our work provides a feasible alternative for the witnessing of genuine multipartite entanglement without aligned reference frames.

Figures

Figures reproduced from arXiv: 2009.11691 by the authors.

Figure 1
Figure 1. Nonlocal fraction calculated for (a) the three-qubit [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Nonlocal fraction calculated for randomly gener [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Nonlocality strength distributions for three-qubit [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Comparison between the nonlocal fraction calcu [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Comparison between the nonlocal fraction calcu [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Nonlocality strength distributions for four-qubit [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Comparison between the nonlocal fraction calcu [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 9
Figure 9. Figure 9: Scheme of the experimental setup for GHZ state [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Comparison between theoretical and experimental [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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Reviewed August 27, 2026 · model on record in the stance chip above.