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REVIEW 3 major objections 4 minor 47 references

Every Bell inequality in the (n,m,2) scenario has a canonical correlator form left invariant by the no-signalling L2 projection.

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2026-08-03 23:16 UTC pith:X6C6R3K4

load-bearing objection Main theorem is right and the projection formula is useful, but the abstract overstates the result and the I_3322 canonical form in Eq. (36) is wrong as written. the 3 major comments →

arxiv 2511.06624 v2 pith:X6C6R3K4 submitted 2025-11-10 quant-ph

No-signalling-projection-invariant Bell inequalities

Soumyadip Patra , Jitendra Prakash , Aaditya Paudel , Peter Bierhorst This is my paper
classification quant-ph
keywords Bell inequalitiesno-signallingL2 projectioncorrelatorsfinite statisticsweak signallingcanonical formdevice-independent
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.

The paper aims to establish that the standard preprocessing step of projecting experimental Bell data onto the no-signalling set does not have to distort the Bell violation, provided the inequality is written in a canonical correlator form. The central claim is that in any n-party, m-setting, binary-outcome Bell experiment, the least-squares projection onto the no-signalling affine hull preserves all full correlators and all uniformly-averaged marginal correlators. Consequently every Bell inequality can be rewritten so that its value is exactly the same before and after the projection. This makes experimental Bell values a cleaner, signalling-robust measure of nonlocal violation, and it yields a simple three-map closed formula for the projection that avoids expensive dense linear algebra. A weighted extension adapts the result to experiments with non-uniform measurement settings.

Core claim

The paper proves that in the (n,m,2) Bell scenario, the coefficient vectors of all full correlators and all uniformly-averaged marginal correlators lie in the kernel of the matrix encoding the no-signalling and normalisation constraints. Therefore the L2 projection onto the affine hull of the no-signalling polytope leaves every such correlator's value unchanged. Since these correlators span the space of linear functionals, any Bell expression admits a canonical correlator form that is termwise invariant under the projection, so the projected estimate attains exactly the same Bell value as the raw empirical distribution. The paper also gives a closed-form, three-step linear map for computing

What carries the argument

The central object is the uniformly-averaged marginal correlator (UMC): for a subset of k parties, the average over all settings choices of the remaining n−k parties of the k-party parity expectation. Its coefficient vector is shown to be orthogonal to every equality constraint defining the no-signalling affine hull, and that orthogonality is the engine of the projection invariance. The closed formula for the projection is the composition of three sparse linear maps: conditional probabilities → parity correlators → uniformly averaged correlators → probabilities.

Load-bearing premise

The method depends on the empirical data lying close enough to the no-signalling set that the affine projection produces a valid, non-negative probability distribution; otherwise enforcing non-negativity breaks the claimed projection invariance.

What would settle it

Take a real or simulated Bell dataset with large spurious signalling (e.g., 40% differences between marginals) and compare the canonical Bell value after the affine L2 projection against the value after a constrained projection onto the no-signalling polytope: if they differ, the practical invariance does not extend to the constrained case. More directly, for any random empirical distribution f, compute the canonical Bell expression before and after the affine projection using the paper's closed formula; a discrepancy beyond numerical precision would disprove the central invariance claim.

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

If this is right

  • Any Bell expression for the (n,m,2) scenario has a canonical form whose value is unchanged when the empirical data is L2-projected onto the no-signalling affine hull.
  • The projection can be computed via a sparse three-map pipeline instead of forming a dense Gram matrix and its inverse, scaling more easily to higher party and setting counts.
  • Experimental Bell violations reported in the canonical form become insensitive to weak finite-sample signalling, providing a standardised metric across experiments.
  • For experiments with non-uniform sampling of measurement settings, a weighted L2 projection leaves a correspondingly weighted correlator form invariant.
  • Full-correlator inequalities such as CHSH and Mermin are already in canonical form and thus automatically projection-invariant.

Where Pith is reading between the lines

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

  • Editorial inference: Because the projection is linear and closed-form, the same three-map pipeline could be embedded in real-time device-independent randomness generation to denoise data before probability estimation, though the statistical effect on the downstream randomness certification is not analysed here.
  • Editorial inference: The uniform averaging over remote settings is a design choice; other averages over the remote parties' settings would produce different projection-invariant functionals, potentially tailored to specific drift patterns in a particular experiment.
  • Editorial inference: The kernel-membership result suggests a broader principle: any linear functional whose coefficient vector is orthogonal to the equality constraints is projection-invariant. The correlators are a spanning set, but not the only one, so other invariant quantities may be constructed.
  • Editorial inference: For the constrained projection onto the no-signalling polytope (where non-negativity is enforced), the invariance does not hold in general; the method is therefore best suited to high-count experiments where signalling is small enough that the affine projection already yields valid probabilities.

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 / 4 minor

Summary. The paper considers the (n,m,2) Bell scenario and studies the L2 projection onto A = aff(P_NS), the affine hull of the no-signalling polytope. Its main result (Proposition 1 and Corollary 2) is that the uniformly-averaged marginal correlator (UMC) coefficient vectors lie in the kernel of the equality-constraint matrix A_eq, so all full correlators and UMC terms are invariant under the affine projection. From this it derives a closed-form projection formula via three linear maps, and proposes a 'canonical form' of any Bell expression whose value is unchanged by the affine projection. Examples include CHSH, tilted CHSH, I_3322, and a LOSR-GTNL witness. The paper also sketches an extension to weighted L2 projections. The main derivation is elementary and essentially correct, but the paper contains an inaccurate abstract-level claim about projecting onto the polytope rather than its affine hull, and a concrete error in the I_3322 canonical form.

Significance. If repaired, the paper would make a useful contribution: it gives a clean, elementary proof of the kernel membership of UMC terms, a closed-form and computationally sparse projector, and a clear recipe for making Bell expressions insensitive to the affine projection step. The derivation is parameter-free and no circularity is present. However, the advertised invariance is only for the unconstrained affine-hull projection; the abstract's 'onto the no-signalling polytope' wording is false when non-negativity is imposed. In addition, the I_3322 example in §4.2 is not actually the UMC canonical form. These issues do not invalidate the central theorem but do affect the paper's main applications, so the manuscript needs substantial correction before it can be accepted.

major comments (3)
  1. [Abstract and §2.2 (Eq. (7), discussion after it)] The abstract states that the L2 projection is 'onto the no-signalling polytope' and that the violation of the canonical Bell inequality is unchanged. This is not what is proved: the projection is onto the affine hull A = aff(P_NS), and P_NS = A ∩ R^d_+. Equation (7) minimises over A, not over P_NS. As the paper itself notes after Eq. (7), the affine projection can contain negative probabilities; when non-negativity is enforced by the convex QP over A ∩ R^d_+, the canonical Bell value is no longer invariant. The headline claim should therefore be restricted to the affine projection, and the practical relevance should be qualified by the closeness assumption that the empirical vector is near P_NS.
  2. [§4.2, Eq. (36)] Equation (36) is not the UMC canonical form of the I_3322 inequality. The 'like-for-like substitution' described in §4.2 requires replacing C^1_x and C^2_y in (35) by (1/3)Σ_y C^1_{xy} and (1/3)Σ_x C^2_{xy}. The coefficient vector of (36) does not satisfy the no-signalling condition p(a_2|x=0,y=0)=p(a_2|x=2,y=0): for a_2=0, the sum of the coefficients over a_1 is -1 for the settings block p_{00} and 0 for p_{20}. Hence (36) is not invariant under Π_A and cannot be the canonical form claimed. This is a concrete error in a worked application, and it indicates that the examples in Section 4 need to be re-derived with the explicit UMC denominators.
  3. [§5, Eqs. (39)-(44)] The weighted projection section claims that the D-invariant quantity ⟨c^I_{u_I}, p⟩_D is a non-uniform average of correlators that still reduces to the standard correlator on no-signalling behaviours. For a general positive diagonal D with entries reflecting settings probabilities, this is not correct: ⟨c^I_{u_I}, p⟩_D equals C^I_{u_I} times (1/m^{n-|I|}) Σ_{x_{\bar I}} D_{(u_I,x_{\bar I})} (assuming D is settings-dependent), which differs from C^I_{u_I} unless a special normalisation condition is imposed. The remark after Eq. (44) assumes such a normalisation without stating it, and the claimed generalisation of the Section 4 canonical-form construction to weighted projections is therefore unsupported as written.
minor comments (4)
  1. [Throughout] The phrase 'onto the no-signalling polytope' appears in the abstract and in Section 1; the precise object is the affine hull A. Please use 'affine hull' consistently and state the non-negativity caveat prominently.
  2. [§2.3 and Appendix A] In Appendix A, the proof of Proposition 1 writes 'for all i∈[m]' in the recap of the no-signalling rows; this should be i∈[n]. There are also typos such as 'scenarios scenarios' in Section 1 and 'a I' in the definition of x_I.
  3. [§3, proof accompanying Eq. (20)] The argument that (20) gives the orthogonal projection is terse. It would be clearer to state explicitly that the UMC/full-correlator moments separate points on A (which follows from the inversion identity (16)), so matching all these moments with a vector in A characterises the projection.
  4. [§4, first paragraph] The phrase 'termwise invariant under the projection' could be misread as applying to arbitrary Bell expressions. Please clarify that termwise invariance holds for the canonical UMC/full-correlator terms after the rewriting, not for the original coefficients in (24).

Circularity Check

0 steps flagged

No significant circularity: the invariance theorem is proved from the explicit constraint rows, not assumed.

full rationale

The derivation chain is self-contained. The UMC functional is defined independently in Eq. (10) as an explicit averaging of parity-weighted probabilities; Proposition 1 proves its coefficient vector lies in ker(A_eq) by direct dot products against the no-signalling rows (8) and normalisation rows (9), and Corollary 2 derives Π_A-invariance from the self-adjoint idempotence of the orthogonal projector. The closed projection formula (20) is justified by the elementary Fourier-inversion identity (16)-(19), which expresses any probability behaviour in terms of its correlators, so no fitted parameter is later relabelled as a prediction. The canonical Bell form is not defined to be invariant; any Bell expression is first rewritten in the UMC/full-correlator basis, and then invariance follows termwise from Proposition 1. References to prior work ([22], [23], [25]) supply motivation, alternative derivations, or the negative-probability example; they are not load-bearing, and the present paper supplies its own proof. Self-citations in the reference list ([6], [13], [19], [33]) occur in application/background contexts and do not support the central theorem. The acknowledged limitation in Section 2.2—that the affine projection Π_A can contain negative entries and that enforcing non-negativity changes the optimization—is a genuine scope restriction on the abstract's 'polytope' wording and on the practical denoising claim, but it is not a circular reduction: the stated theorems are about Π_A and remain valid for that map. Any algebraic issue in the I_3322 example would be an implementation error, not circularity. Therefore no circular step is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters or invented entities. The paper's results rest on standard linear algebra of the no-signalling polytope, the Fourier/Hadamard representation of binary-outcome behaviours, and the modelling assumption that Euclidean denoising is meaningful.

axioms (4)
  • domain assumption The set of UMC coefficient vectors for nonempty I spans the kernel of A_eq (equivalently, the correlator representation (16) is bijective on the no-signalling affine space).
    Used implicitly to justify that matching UMC values determines the projection; standard in the Bell literature (Fourier analysis on the outcome hypercube), not explicitly proven as a spanning statement.
  • domain assumption The empirical frequency f is normalised (Σ_a f(a|x) = 1 for each settings vector x).
    Required for the I=∅ UMC to equal 1 so that the projected behaviour is normalised; true for count-data frequencies.
  • domain assumption The L2 distance on probability vectors is the appropriate metric for removing weak signalling (vs. ML/KL).
    A modeling choice that motivates the whole projector; the paper argues for it in the case of blockwise drifts.
  • domain assumption The no-signalling equalities (3) together with normalisation (2) define the affine hull A; the listed conditions are complete.
    Standard characterisation of the (n,m,2) no-signalling set, cited to [33,34].

pith-pipeline@v1.3.0-alltime-deepseek · 23435 in / 34789 out tokens · 291049 ms · 2026-08-03T23:16:07.381458+00:00 · methodology

0 comments
read the original abstract

In this paper, we highlight how any Bell inequality for a configuration involving $n$ parties each performing one of $m$ binary-outcome measurements has a canonical form that is no-signalling-projection invariant. Specifically, the $L^2$-projection of weakly signalling data onto the no-signalling polytope leaves the violation of this canonical Bell inequality unchanged. Our methods allow us to derive a general closed formula for the projection and present a substantially more computationally simple procedure for its evaluation. We also show this can be generalised to non-standard projections of potential interest for certain applications. No-signalling projections serve as a preliminary step before undertaking any device-independent application involving Bell experiment data, such as hypothesis testing against local realism, random number generation and entanglement detection.

discussion (0)

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Works this paper leans on

47 extracted references · 1 linked inside Pith

  1. [1]

    Certified ran- domness in quantum physics

    A Acín and L Masanes. “Certified ran- domness in quantum physics”. In:Nature 540(7632) (Dec. 2016), pp. 213–219

    2016

  2. [2]

    Random numbers certi- fied by Bell’s theorem

    S Pironio et al. “Random numbers certi- fied by Bell’s theorem”. In:Nature464(7291) (Apr. 2010), pp. 1021–1024

    2010

  3. [3]

    Secure device-independent quantum key distribution with causally in- dependent measurement devices

    L Masanes et al. “Secure device-independent quantum key distribution with causally in- dependent measurement devices”. In:Nature Communications2(1) (Mar. 2011), p. 238

    2011

  4. [4]

    Advances in device- independent quantum key distribution

    V Zapatero et al. “Advances in device- independent quantum key distribution”. In: npj Quantum Information9(1) (Feb. 2023), p. 10. 13

    2023

  5. [5]

    Device-Independent En- tanglement Quantification and Related Ap- plications

    T Moroder et al. “Device-Independent En- tanglement Quantification and Related Ap- plications”. In:Phys. Rev. Lett.111 (3 July 2013), p. 030501

    2013

  6. [6]

    Strength of sta- tistical evidence for genuine tripartite nonlo- cality

    S Patra and P Bierhorst. “Strength of sta- tistical evidence for genuine tripartite nonlo- cality”. In:Phys. Rev. A110 (6 Dec. 2024), p. 062411

    2024

  7. [7]

    In:Bell Nonlocality

    V Scarani. In:Bell Nonlocality. Ox- ford University Press, Aug. 2019.isbn: 9780198788416

    2019

  8. [8]

    How to avoid (apparent) signaling in Bell tests

    M Smania et al. “How to avoid (apparent) signaling in Bell tests”. In:Quantum9 (June 2025), p. 1760

    2025

  9. [9]

    Analysis of assumptions of re- cent tests of local realism

    A Bednorz. “Analysis of assumptions of re- cent tests of local realism”. In:Phys. Rev. A 95 (4 Apr. 2017), p. 042118

    2017

  10. [10]

    Violation of No-Signaling on a Public Quantum Computer

    T Rybotycki et al. “Violation of No-Signaling on a Public Quantum Computer”. In:Ad- vanced Quantum Technologies8(6) (2025), p. 2400661

    2025

  11. [11]

    Bounding the Plau- sibility of Physical Theories in a Device- Independent Setting via Hypothesis Test- ing

    YC Liang and Y Zhang. “Bounding the Plau- sibility of Physical Theories in a Device- Independent Setting via Hypothesis Test- ing”. In:Entropy21(2) (2019)

    2019

  12. [12]

    Analysis of Assumptions in BIG Bell Test Experiments

    S Sołtan et al. “Analysis of Assumptions in BIG Bell Test Experiments”. In:Annalen der Physik532(10) (2020), p. 2000333

    2020

  13. [13]

    Experimentally generated randomness certified by the impossibility of superluminal signals

    P Bierhorst et al. “Experimentally generated randomness certified by the impossibility of superluminal signals”. In:Nature556(7700) (Apr. 2018), pp. 223–226

    2018

  14. [14]

    Experimental Low-Latency Device-Independent Quantum Random- ness

    Y Zhang et al. “Experimental Low-Latency Device-Independent Quantum Random- ness”. In:Phys. Rev. Lett.124 (1 Jan. 2020), p. 010505

    2020

  15. [15]

    Asymptotically optimal data analysis for rejecting local realism

    Y Zhang et al. “Asymptotically optimal data analysis for rejecting local realism”. In:Phys. Rev. A84 (6 Dec. 2011), p. 062118

    2011

  16. [16]

    Efficient quantification of ex- perimental evidence against local realism

    Y Zhang et al. “Efficient quantification of ex- perimental evidence against local realism”. In:Phys. Rev. A88 (5 Nov. 2013), p. 052119

    2013

  17. [17]

    Certifying quantum random- ness by probability estimation

    Y Zhang et al. “Certifying quantum random- ness by probability estimation”. In:Phys. Rev. A98 (4 Oct. 2018), p. 040304

    2018

  18. [18]

    Generation of quantum ran- domness by probability estimation with clas- sical side information

    E Knill et al. “Generation of quantum ran- domness by probability estimation with clas- sical side information”. In:Phys. Rev. Re- search2 (3 Sept. 2020), p. 033465

    2020

  19. [19]

    Asymptotically Optimal Adversarial Strategies for the Prob- ability Estimation Framework

    S Patra and P Bierhorst. “Asymptotically Optimal Adversarial Strategies for the Prob- ability Estimation Framework”. In:Entropy 25(9) (July 2023)

    2023

  20. [20]

    Efficient randomness cer- tification by quantum probability estima- tion

    Y Zhang et al. “Efficient randomness cer- tification by quantum probability estima- tion”. In:Phys. Rev. Res.2 (1 Jan. 2020), p. 013016

    2020

  21. [21]

    A simple low-latency real-time certifiable quantum random num- ber generator

    Y Zhang et al. “A simple low-latency real-time certifiable quantum random num- ber generator”. In:Nature Communications 12(1) (Feb. 2021), p. 1056

    2021

  22. [22]

    More randomness from the samedata

    JD Bancal et al. “More randomness from the samedata”.In:New Journal of Physics16(3) (Mar. 2014), p. 033011

    2014

  23. [23]

    Device-independent point es- timation from finite data and its application to device-independent property estimation

    PS Lin et al. “Device-independent point es- timation from finite data and its application to device-independent property estimation”. In:Phys. Rev. A97 (3 Mar. 2018), p. 032309

    2018

  24. [24]

    On the inequivalence of the CH and CHSH inequalities due to finite statistics

    MO Renou et al. “On the inequivalence of the CH and CHSH inequalities due to finite statistics”. In:Journal of Physics A: Mathe- matical and Theoretical50(25) (May 2017), p. 255301

    2017

  25. [25]

    D Rosset et al.Algebraic and geometric prop- erties of local transformations. 2020. arXiv: 2004.09405 [quant-ph]

    Pith/arXiv arXiv 2020

  26. [26]

    Efficientquantumkeydistribu- tion secure against no-signalling eavesdrop- pers

    AAcínetal.“Efficientquantumkeydistribu- tion secure against no-signalling eavesdrop- pers”. In:New Journal of Physics8(8) (Aug. 2006), p. 126

    2006

  27. [27]

    Practical device- independent quantum cryptography via en- tropy accumulation

    R Arnon-Friedman et al. “Practical device- independent quantum cryptography via en- tropy accumulation”. In:Nature Communi- cations9(1) (Jan. 2018), p. 459

    2018

  28. [28]

    Device-independent quantum key distribution with random key basis

    R Schwonnek et al. “Device-independent quantum key distribution with random key basis”. In:Nature Communications12(1) (May 2021), p. 2880. 14

    2021

  29. [29]

    Universal Security for Randomness Expansion from the Spot- Checking Protocol

    CA Miller and Y Shi. “Universal Security for Randomness Expansion from the Spot- Checking Protocol”. In:SIAM Journal on Computing46(4) (2017), pp. 1304–1335

    2017

  30. [30]

    Improved device- independent randomness expansion rates using two sided randomness

    R Bhavsar et al. “Improved device- independent randomness expansion rates using two sided randomness”. In:New Journal of Physics25(9) (July 2023), p. 093035

    2023

  31. [31]

    Device-independent ran- domness expansion with entangled pho- tons

    LK Shalm et al. “Device-independent ran- domness expansion with entangled pho- tons”. In:Nature Physics17(4) (Apr. 2021), pp. 452–456

    2021

  32. [32]

    Device-independent random- ness expansion against quantum side in- formation

    WZ Liu et al. “Device-independent random- ness expansion against quantum side in- formation”. In:Nature Physics17(4) (Apr. 2021), pp. 448–451

    2021

  33. [33]

    Consistency and causality of interconnected nonsignaling resources

    P Bierhorst. “Consistency and causality of interconnected nonsignaling resources”. In: Journal of Physics A: Mathematical and Theoretical57(42) (Sept. 2024), p. 425301

    2024

  34. [34]

    Aspects of Quantum Nonlocal- ity

    S Pironio. “Aspects of Quantum Nonlocal- ity”. PhD thesis. Université Libre de Brux- elles, 2004

    2004

  35. [35]

    I-Divergence Geometry of Proba- bility Distributions and Minimization Prob- lems

    I Csiszar. “I-Divergence Geometry of Proba- bility Distributions and Minimization Prob- lems”. In:The Annals of Probability3(1) (1975), pp. 146–158

    1975

  36. [36]

    Information projec- tions revisited

    I Csiszar and F Matus. “Information projec- tions revisited”. In:IEEE Transactions on Information Theory49(6) (2003), pp. 1474– 1490

    2003

  37. [37]

    An Elementary Introduction to Information Geometry

    F Nielsen. “An Elementary Introduction to Information Geometry”. In:Entropy22(10) (2020)

    2020

  38. [38]

    A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations

    M Navascués et al. “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations”. In:New Jour- nal of Physics10(7) (July 2008), p. 073013

    2008

  39. [39]

    2053-2563

    PSkrzypczykandDCavalcanti.Semidefinite Programming in Quantum Information Sci- ence. 2053-2563. IOP Publishing, 2023.isbn: 978-0-7503-3343-6

    2053

  40. [40]

    Geometry of the set of quan- tum correlations

    KT Goh et al. “Geometry of the set of quan- tum correlations”. In:Phys. Rev. A97 (2 Feb. 2018), p. 022104

    2018

  41. [41]

    Extremal correlations of the tripartite no-signaling polytope

    S Pironio et al. “Extremal correlations of the tripartite no-signaling polytope”. In:Journal of Physics A: Mathematical and Theoretical 44(6) (Jan. 2011), p. 065303

    2011

  42. [42]

    Simple unified form for the major no-hidden-variables theorems

    ND Mermin. “Simple unified form for the major no-hidden-variables theorems”. In: Phys. Rev. Lett.65 (27 Dec. 1990), pp. 3373– 3376

    1990

  43. [43]

    Randomness versus Nonlocal- ity and Entanglement

    A Acín et al. “Randomness versus Nonlocal- ity and Entanglement”. In:Phys. Rev. Lett. 108 (10 Mar. 2012), p. 100402

    2012

  44. [44]

    Device-independent randomness generation from several Bell es- timators

    O Nieto-Silleras et al. “Device-independent randomness generation from several Bell es- timators”. In:New Journal of Physics20(2) (Feb. 2018), p. 023049

    2018

  45. [45]

    Symmetries of the Bell correlation inequalities

    C Śliwa. “Symmetries of the Bell correlation inequalities”. In:Physics Letters A317(3) (2003), pp. 165–168

    2003

  46. [46]

    A relevant two qubit Bell inequality inequivalent to the CHSH in- equality

    D Collins and N Gisin. “A relevant two qubit Bell inequality inequivalent to the CHSH in- equality”. In:Journal of Physics A: Math- ematical and General37(5) (Jan. 2004), p. 1775

    2004

  47. [47]

    Test of Genuine Multipar- tite Nonlocality

    YL Mao et al. “Test of Genuine Multipar- tite Nonlocality”. In:Phys. Rev. Lett.129 (15 Oct. 2022), p. 150401. A Proof of Proposition 1 Proposition.For all ˜uI∈S k, the UMC coefficient vectorc˜uI satisfiesc ˜uI∈ker(A eq). Before proceeding with the proof, we recap the coefficient vectors encoding the no-signalling and nor- 15 malisation conditions, and the...

    2022