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Some symmetric Bell inequalities can only reach their maximum quantum violation through asymmetric strategies operating in the minimal Hilbert-space dimension.

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2026-08-03 12:26 UTC pith:HFJCHRRS

load-bearing objection A credible and interesting phenomenon, but the central claim leans on unproven numerical upper bounds and a couple of overclaims; worth refereeing, but needs a rigorous revision before I'd treat the main theorem as established. the 3 major comments →

arxiv 2601.02893 v4 pith:HFJCHRRS submitted 2026-01-06 quant-ph

Trading symmetry for Hilbert-space dimension in Bell-inequality violation

Hsin-Yu Hsu , Gelo Noel M. Tabia , Kai-Siang Chen , Mu-En Liu , Tam\'as V\'ertesi , Nicolas Brunner , Yeong-Cherng Liang This is my paper
classification quant-ph
keywords Bell inequalityquantum nonlocalityparty-permutation symmetryHilbert-space dimensionasymmetric quantum strategyself-testingquantum correlationssemidefinite programming
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether a symmetry of a Bell test—invariance under exchanging the two parties—can be kept when one insists on using the smallest possible quantum system to get the maximal violation. The authors find that for several symmetric Bell inequalities, the answer is no: any strategy that respects the symmetry and uses qubits is strictly worse than an asymmetric two-qubit strategy. They show this for a family of three-setting inequalities and for nine four-setting facet-defining inequalities, and they prove a general consequence: if an asymmetric correlation maximizes a symmetric inequality, the set of quantum maximizers contains a flat one-parameter region, which makes self-testing from the maximal violation alone impossible. For the CGLMP family, by contrast, no such trade-off appears: symmetric strategies of minimal dimension already achieve the maximum.

Core claim

For several symmetric (party-permutation-invariant) Bell inequalities, including the family IS(α) in the (2,3,2) scenario and nine symmetric facet-defining inequalities in the (2,4,2) scenario, the maximal quantum violation can only be attained by an asymmetric quantum strategy of minimal dimension. Numerical evidence shows that symmetric qubit strategies strictly underperform the global maximum, while explicit asymmetric two-qubit strategies match the upper bound from the Navascués-Pironio-Acín hierarchy. Consequently, the set of quantum maximizers contains a flat one-parameter region (Prop. 7, Cor. 8), making self-testing from the maximal violation alone impossible. In contrast, the CGLMP-

What carries the argument

The central object is the family IS(α) of symmetric Bell inequalities in correlator form, together with nine symmetric facet-defining inequalities in the (2,4,2) scenario. The key mechanism is Proposition 7: because the Bell functional is linear and the quantum set is convex, an asymmetric maximizer of a symmetric inequality forces a one-parameter flat boundary in the quantum set. The numerical bound on symmetric qubit strategies comes from a dimension-bounded SDP hierarchy in which states are sampled from the symmetric/antisymmetric subspace and Bob's POVMs are forced equal to Alice's; lower bounds come from explicit two-qubit strategies with degenerate measurements.

Load-bearing premise

The load-bearing premise is that the modified QDimSum hierarchy (Appendix A1b) correctly upper-bounds the maximum of a symmetric Bell inequality over symmetric quantum strategies of bounded local dimension; if that bound is not tight or is flawed, the observed gaps between symmetric and asymmetric qubit values could be numerical artifacts.

What would settle it

Find an explicit symmetric two-qubit strategy (a party-symmetric state with identical measurements on both sides) that attains the value 1/3(13+4√13) ≈ 9.1407 for IS(2), or that exceeds the local bound 2α+5 for some α in (1.975, 3]. Equivalently, run a higher-level SDP hierarchy with the symmetry constraints and show that the upper bound drops below the asymmetric two-qubit value for IS(2).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For the symmetric inequalities IS(α) (with α in the relevant range) and the nine listed four-setting inequalities, any quantum strategy reaching the maximum must break the party-exchange symmetry; respecting the symmetry forces a higher-dimensional system or a suboptimal violation.
  • The set of quantum correlations that maximally violate these inequalities contains a one-parameter flat boundary, so the maximal violation alone cannot be used to self-test a specific reference strategy (Corollary 8).
  • For J^42_{4422}, the hierarchy of values 0.5682 (qubit SQS) < 0.6012 (qubit symmetric correlations) < 0.6722 (general qubit maximum) implies that a symmetric correlation can still certify that the underlying qubit strategy is asymmetric, giving a semi-device-independent witness of asymmetry.
  • The CGLMP-type symmetric inequalities I22_dd do not exhibit a trade-off for d = 2,...,19: a symmetric strategy of dimension d achieves the maximal violation, so symmetry and minimal dimensionality can coexist there.
  • There exist asymmetric Bell inequalities (the two-parameter family Ir0,r1) whose maximal violation is attained by a symmetric correlation, showing that the direction of the trade-off is not universal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The flat-region phenomenon identified here suggests that device-independent protocols relying on the maximal violation of a symmetric Bell inequality may need to add extra constraints (e.g., on the number of outcomes or on other moments) to pin down the underlying state and measurements; otherwise the one-parameter family of maximizing correlations leaves genuine freedom.
  • The mirror-symmetric construction (Eqs. 29–30) provides a template for building further examples of asymmetric strategies that produce symmetric correlations; a systematic search over such strategies could reveal more inequalities with a symmetry–dimension trade-off, possibly in simpler Bell scenarios.
  • The numerical gap reported for the three dm=3 inequalities in Table I (where lower and upper bounds differ) suggests that a higher-level SDP or an analytic sum-of-squares certificate could either close the gap or reveal that those examples require an even larger symmetric dimension than currently found.
  • If the trade-off holds generally, then in experiments aiming at maximal Bell violation with minimal-dimensional systems, one should expect the optimal implementations to be lopsided; this has practical consequences for how measurement settings are chosen in device-independent experiments.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper investigates, for symmetric (party-permutation-invariant) Bell inequalities, whether the maximal quantum violation can always be attained by a symmetric quantum strategy (SQS) acting on the minimal local Hilbert-space dimension. It proves structural results (mirror-symmetric strategies give symmetric correlations; asymmetric maximizers force a one-dimensional flat boundary of the quantum set and preclude self-testing) and presents a mix of analytic and numerical evidence. For the CGLMP family the authors find SQSs matching NPA upper bounds for dimensions 2–19, indicating no trade-off. For the IS(α) family, nine (2,4,2) facet inequalities, and a nine-setting inequality I9, they report a gap between the maximal quantum value and the value attainable by SQSs in the minimal dimension, concluding that some symmetric Bell inequalities are maximally violated only by asymmetric minimal-dimension strategies. The I9 result is supported by an analytic proof (Prop. 9); the other trade-off claims rely on numerical SDP bounds from a modified QDimSum hierarchy described in Appendix A1b.

Significance. If fully established, the paper's central claim would be a clean example of asymmetry as a resource in Bell scenarios: some symmetric Bell inequalities force asymmetric minimal-dimension strategies, which in turn implies flat regions of the quantum correlation set and limits self-testing from maximal violation. The paper contains several valuable analytic contributions: Prop. 5 and Prop. 6 characterize mirror-symmetric qubit strategies; Prop. 7 and Cor. 8 are simple but important implications for geometry and self-testing; Prop. 9 gives a rigorous proof of a symmetry–dimension trade-off for I9. The explicit quantum strategies for IS(2) and J42, and the matching of CGLMP SQS values with NPA bounds, are also useful data points. However, the numerical upper bounds for SQSs in Fig. 2 and Table I are load-bearing and currently lack rigorous certification.

major comments (3)
  1. [Appendix A1b / Fig. 2 / Table I] The SQS upper bounds in Fig. 2 and Table I are obtained by a modified QDimSum hierarchy in which the algebraic constraints are learned from random sampling of symmetric/antisymmetric states and of identical POVMs for both parties. These constraints are not proven to be valid for all SQSs, and no dual feasible solutions are provided. For IS(2), the SQS bound is reported to equal the local bound 9, while the asymmetric two-qubit value is 9.1407; a single spurious sampled constraint would erase the claimed gap. The same issue affects all six dm=2 rows of Table I. The authors should either (i) prove the sampled constraints hold for every SQS of the relevant dimension, (ii) provide rigorous dual certificates for the reported SDP bounds, or (iii) explicitly relabel these as heuristic upper bounds and consequently limit the strength of the trade-off claim.
  2. [Section V.A / Eq. (26)] The central trade-off claim for IS(α) — in particular the statement that symmetric qubit strategies cannot even violate the inequality for α ∈ (1.975, 3] — rests entirely on the unproven numerical SQS bound from Appendix A1b. The asymmetric lower bound 9.1407 for IS(2) and its matching with the NPA hierarchy are convincing for the quantum maximum, but the SQS side is not. Without a certified or analytic upper bound on SQSs, the existence of a symmetry–dimension trade-off for this family is not rigorously established. Given that the abstract states 'we show that symmetric quantum strategies ... can only lead to a suboptimal Bell violation', this gap is load-bearing and should be addressed, either by added proof/certification or by softening the claim to 'provide numerical evidence'.
  3. [Table I / caption] For the three dm=3 inequalities (I8, I19, I13), the caption itself admits that the best lower bound falls short of the SDP upper bound. These rows therefore do not demonstrate a trade-off, only an upper bound on SQS values. For the six dm=2 rows, the lower bounds are said to match the upper bounds, but the explicit QS parameters are not published, making the matching non-reproducible. The paper would be strengthened by providing the QS parameters (e.g., in a supplementary file) or a reproducible script, especially because the entire trade-off claim for these inequalities rests on the numerical agreement.
minor comments (5)
  1. [Title/Abstract] The title contains a typo: 'Bell ineq uality' should be 'Bell inequality'.
  2. [Eq. (B9)] The symbol '⟳' is used without definition; the text says it refers to 'six remaining terms' but the reader must guess the precise symmetry convention. Please define it explicitly.
  3. [Table II] The entry 'I2233 [54]: Eq. (22)' appears inconsistent with the notation I22dd used elsewhere; please check the label.
  4. [Section VII.A] The proof of Prop. 7 assumes the symmetric group action commutes with the linear functional in the sense used in Eq. (54). This is fine, but it may be worth explicitly noting that Vσ maps Q to itself, which is needed for the conclusion that the convex combination remains in Q. This is implicit but should be stated.
  5. [Appendix B.4] The definition of the J42 inequality in Eq. (B9) uses a placeholder '⟳' in the sum, which obscures the actual expression. A fully explicit form is essential for reproducing the numerical values in Table I.

Circularity Check

0 steps flagged

No substantive circularity; the derivation relies on independent upper/lower bounds, and the paper's self-citations are building blocks rather than load-bearing.

full rationale

The central claims are not circular by construction. For IS(2), the asymmetric two-qubit strategy of Eq. (B8) is evaluated directly, giving (13+4√13)/3 ≈ 9.1407, and matched against an independent NPA upper bound; the SQS upper bound is obtained from a different numerical method (Appendix A1b). Lower bounds are explicit strategies, while upper bounds are SDP relaxations: no fitted parameter is renamed as a prediction, and no quantity used in a bound is defined in terms of the target bound. In Table I, the quantum and SQS bound columns are computed by distinct methods (NPA level 3 plus matching QSs vs. dimension-bounded QDimSum), and for the dm=3 rows the caption explicitly reports LB<UB, so the numerical evidence is not presented as tighter than it is. The self-citations ([40] symmetrization, [69] CGLMP/I22dd equivalence, [77] I3322c, [84] at-most-one CHSH violation) are used as lemmas or known algebraic identities, not as the target trade-off theorem; the paper proves Proposition 1 in Eqs. (15)-(17) rather than merely citing it, and no 'uniqueness' theorem from the same authors is imported to force the conclusion. The modified QDimSum sampling in Appendix A1b is a potential correctness/validity limitation (constraints are learned from random samples rather than certified by dual feasible solutions), but that is not circularity: a wrong bound would be an unsound numerical certificate, not a reduction of the output to the input. Hence no specific circular step can be exhibited.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The paper relies on standard quantum-information tools (Naimark dilation, purification, NPA/QDimSum hierarchies) and prior published results. The only fitted quantities are numerical parameters of explicit candidate strategies; they are not used as benchmarks for the derivations. No new physical entities are introduced.

free parameters (3)
  • IS(α) strategy parameters p(α), s(α), t(α) = α=1.5: p=16/21, s=8/17, t=-√5/3; α=2: p=2(1+√13)/(3√13), s=(10√13-18)/61, t=-√((11-√13)/18); other α in [1.5,3] from num
    Parameters of the asymmetric two-qubit QS (Eq. B8) that achieves the quantum bound of IS(α). Values for α=1.5,2 are exact algebraic; the continuous curve in Fig. 2 relies on numerically optimized parameters not reported.
  • J42_4422 QS state angle α and Bloch vectors θk, φk = α=42.5092°, θ0=61.9767°, φ0=166.1570°, θ1=0°, θ2=54.3423°, φ2=41.5892°, θ3=52.2700°, φ3=-71.170°
    Parameters of the asymmetric two-qubit strategy of Eq. (28) giving a symmetric correlation with J42 value 0.6012; found by numerical optimization.
  • Unreported QS parameters for the six dm=2 trade-off inequalities in Table I
    The 'matching QS of dimension dm' for the quantum bounds in Table I is not given; only the J42 and I9 strategies are explicit.
axioms (7)
  • standard math Naimark dilation and state purification allow any quantum strategy to be converted to a purified PVM strategy
    Standard results cited to [61], used to establish Prop. 1 (Sec. IIIB).
  • domain assumption The NPA hierarchy (and Moroder/twirling variants) provides a convergent outer approximation of the quantum set
    Used throughout for upper bounds on quantum and SQS maxima; standard in the field.
  • domain assumption The symmetrization construction of Moroder et al. [40] (Eq. 16) preserves the produced correlation while making the strategy symmetric
    Core to Prop. 1; cited with proof sketch.
  • domain assumption For two-outcome Bell inequalities, it suffices to consider projective measurements to maximize the quantum violation
    Used to justify restricting symmetric qubit strategies to PVMs (footnote 4); cites [59,78].
  • domain assumption The self-testing result of Bowles et al. [91]: maximal violation of CHSH3 self-tests a two-qubit maximally entangled state and Pauli observables up to complex conjugation
    Used in the I9 proof (Appendix B5) to characterize any strategy reaching 12√2+3.
  • domain assumption Numerical SDP solvers return certified (or sufficiently precise) upper bounds
    Precision constraints: results quoted to 1e-6 or better; standard practice.
  • domain assumption Equivalence Id = 2d/(d-1) I22dd + 2 (from Liang's thesis [69])
    Used to relate CGLMP violation to the symmetric I22dd form; cited prior result.

pith-pipeline@v1.3.0-alltime-deepseek · 30772 in / 22903 out tokens · 191630 ms · 2026-08-03T12:26:55.886665+00:00 · methodology

0 comments
read the original abstract

In quantum information, asymmetry, i.e., the lack of symmetry, is a resource allowing one to accomplish certain tasks that are otherwise impossible. Similarly, in a Bell test using any given Bell inequality, the maximum violation achievable using quantum strategies respecting or disregarding a certain symmetry can be different. In this work, we focus on the symmetry involved in the exchange of parties and explore when we have to trade this symmetry for a lower-dimensional quantum strategy in achieving the maximal violation of given Bell inequalities. For the family of symmetric Collins-Gisin-Linden-Massar-Popescu inequalities, we provide evidence showing that there is no such trade-off. However, for several other Bell inequalities with a small number of dichotomic measurement settings, we show that symmetric quantum strategies in the minimal Hilbert space dimension can only lead to a suboptimal Bell violation. In other words, there exist symmetric Bell inequalities that can only be maximally violated by asymmetric quantum strategies of minimal dimension. In contrast, one can also find examples of asymmetric Bell inequalities that are maximally violated by symmetric correlations. The implications of these findings on the geometry of the set of quantum correlations and the possibility of performing self-testing therefrom are briefly discussed.

Figures

Figures reproduced from arXiv: 2601.02893 by Gelo Noel M. Tabia, Hsin-Yu Hsu, Kai-Siang Chen, Mu-En Liu, Nicolas Brunner, Tam\'as V\'ertesi, Yeong-Cherng Liang.

Figure 1
Figure 1. Figure 1: Schematic showing different pathways to obtain a p [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Maximal Bell value of IS(α) for α ∈ [1.5, 3] under various constraints. From bottom to top, we have, respectively, the local bound of 2α + 5 (red, dashed), the symmetric qubit upper bound (blue, dotted) computed using the method described in Appendix A, and the general quantum bound (yellow, solid), which is attainable using a two-qubit QS involving a degenerate observable for one of the parties. For concr… view at source ↗
Figure 3
Figure 3. Figure 3: Measurement directions ~ak (solid line) and~bk (dashed line) for k = 0, 1, 2, 3 corresponding to the QS described in Eq. (28). Notice that each ~bk may be obtained from the corresponding ~ak by performing a mirror reflection about the x − z plane (the pale blue plane), making it evident that the QS is asymmetric. Surprisingly, the resulting correlation is symmetric and gives a Bell value ≈ 0.6012 for the J… view at source ↗

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Reference graph

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