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Genuine Multipartite Nonlocality for Arbitrary Input: Maximal Randomness Generation and Robust Self-Testing

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A Bell inequality certifies genuine multipartite nonlocality for arbitrary odd inputs per party and extracts maximal device-independent randomness.

desk verdict New GMNL inequality for arbitrary odd settings per party, with claimed analytical SOS and maximal m-bit randomness, but the SOS generality is the part that still needs direct checking. read the letter →

arxiv 2606.10936 v1 pith:SXNPMLY7 submitted 2026-06-09 quant-ph

Rajdeep Paul , Ranendu Adhikary , Alok Kumar Pan This is my paper · ORCID
classification quant-ph
keywords genuinemultipartitenonlocalityBellinequalitydevice-independentrandomnessself-testingsum-of-squaresdecompositionquantumentanglement
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 introduces a Bell inequality designed to detect genuine multipartite nonlocality across any number of parties when each party has an arbitrary odd number of measurement settings. An analytical sum-of-squares decomposition establishes the optimal quantum bound without restricting Hilbert space dimension or assuming special forms for the measurement operators. Reaching this bound allows self-testing of the shared entangled state and the observables up to local isometries, confirmed through a swap-based scheme. The same violation yields the maximum m bits of global device-independent randomness, and the inequality's structure improves noise tolerance as the number of settings grows.

What carries the argument

The new multipartite Bell inequality together with its analytical sum-of-squares decomposition that certifies the optimal quantum bound without dimension restrictions.

What would settle it

A quantum strategy that achieves a violation strictly larger than the value certified by the sum-of-squares decomposition, or an experimental run reaching the claimed bound yet failing to pass the swap-based self-testing verification for the state and measurements.

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Extended reading notes

Core claim

We introduce a Bell inequality capable of identifying genuine multipartite nonlocality in an arbitrary m-partite scenario with an arbitrary odd number of measurements per party. Since the multi-setting nature precludes Jordan's lemma, we construct an analytical sum-of-squares decomposition to obtain the optimal quantum violation without assuming any bound on the Hilbert space dimension. This enables self-testing of the shared entangled state and the corresponding measurement observables up to local isometries via a swap-based certification scheme, extraction of maximal global device-independent randomness of m bits at the optimal quantum violation, and improved robustness to noise as the num

Load-bearing premise

An analytical sum-of-squares decomposition exists and certifies the optimal quantum bound for the new inequality without any dimension restriction or post-hoc assumptions on the measurement operators.

Editorial extensions

If this is right

  • Self-testing of the shared entangled state and measurement observables up to local isometries becomes possible at the optimal violation.
  • Maximal global device-independent randomness of m bits can be extracted when the quantum bound is achieved.
  • Noise robustness of the certification increases with the number of measurement settings.
  • The inequality applies to any number of parties and any odd number of inputs without requiring Jordan's lemma.

Reading between the lines

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

  • The dimension-independent certification method could support device-independent protocols involving larger numbers of parties than previously feasible.
  • The noise-robustness trend suggests the inequality may remain practical in experiments even when measurement settings are increased for other purposes.
  • The self-testing result provides a concrete route to certify multipartite entanglement sources without assuming finite dimension.
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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

2 major / 2 minor

Summary. The paper introduces a Bell inequality for detecting genuine multipartite nonlocality (GMNL) in m-partite scenarios with an arbitrary odd number of measurement settings per party. It constructs an analytical sum-of-squares (SOS) decomposition to certify the optimal quantum violation without any Hilbert-space dimension bound, enabling a swap-based self-testing scheme for the shared state and observables up to local isometries, extraction of m bits of global device-independent randomness at the optimal violation point, and improved noise robustness as the number of settings increases.

Significance. If the analytical SOS construction holds without gaps or implicit restrictions, the result provides a meaningful extension of device-independent protocols to multi-setting GMNL scenarios, overcoming the inapplicability of Jordan's lemma and enabling maximal randomness generation beyond prior GMNL limitations.

major comments (2)
  1. [SOS decomposition section (following inequality definition)] The central claims of optimal quantum violation, self-testing, and maximal m-bit randomness all rest on the analytical SOS decomposition (presented after the inequality definition). The decomposition must be shown explicitly to hold for arbitrary m and any odd number of settings without dimension bounds or post-hoc assumptions on the measurement operators; any algebraic gap would invalidate the optimality and downstream results.
  2. [Self-testing section] The swap-based self-testing scheme (in the self-testing section) is stated to confirm the existence of the required local isometries at the optimal violation. This relies directly on the SOS-certified bound; the scheme should include an explicit check that the isometry construction remains valid for general odd settings without finite-dimensional reductions.
minor comments (2)
  1. [Abstract and introduction] Notation for the number of parties (m) and settings per party should be introduced consistently in the abstract and early sections to avoid confusion with the claimed m bits of randomness.
  2. [Robustness analysis section] The robustness plots would benefit from explicit labeling of the noise parameter and the number of settings used in each curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comments point by point below, clarifying the generality of the constructions presented.

read point-by-point responses

  1. Referee: [SOS decomposition section (following inequality definition)] The central claims of optimal quantum violation, self-testing, and maximal m-bit randomness all rest on the analytical SOS decomposition (presented after the inequality definition). The decomposition must be shown explicitly to hold for arbitrary m and any odd number of settings without dimension bounds or post-hoc assumptions on the measurement operators; any algebraic gap would invalidate the optimality and downstream results.

    Authors: The analytical SOS decomposition is constructed explicitly in the section following the Bell inequality definition and holds for arbitrary m and any odd number of settings. The SOS terms are defined algebraically in terms of the general Bell operator such that their sum equals the operator minus the claimed quantum bound, forming an identity that holds in the operator algebra without any reference to Hilbert-space dimension or post-hoc assumptions on the measurement operators (beyond the standard requirement that operators on different parties commute). This algebraic identity certifies the bound directly and is free of gaps, as the cancellation is complete for the general odd-setting case. revision: no

  2. Referee: [Self-testing section] The swap-based self-testing scheme (in the self-testing section) is stated to confirm the existence of the required local isometries at the optimal violation. This relies directly on the SOS-certified bound; the scheme should include an explicit check that the isometry construction remains valid for general odd settings without finite-dimensional reductions.

    Authors: The swap-based self-testing scheme is built directly upon the dimension-independent SOS bound. The local isometries are defined in terms of the measurement operators and the shared state using the algebraic relations obtained from the SOS decomposition. These relations are independent of the specific odd number of settings and do not invoke any finite-dimensional reductions or assumptions; the mapping to the ideal state and observables (up to local isometries) follows uniformly from the same operator identities for arbitrary odd settings. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; central claims rest on explicit analytical construction of new inequality and SOS

full rationale

The manuscript introduces a novel Bell inequality for GMNL with arbitrary odd inputs per party and explicitly constructs an analytical SOS decomposition to certify the quantum bound without Hilbert-space dimension restrictions or post-hoc operator assumptions. This construction underpins the self-testing via swap scheme and the m-bit randomness extraction. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior work appear in the abstract or described derivation chain. The result is presented as derived from the new inequality rather than equivalent to its inputs by construction, making the paper self-contained.

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

The central claims rest on the existence of an analytical SOS decomposition for the new inequality and on standard quantum-mechanical assumptions about Hilbert-space operators; no free parameters or new entities are mentioned in the abstract.

assumptions (1)
  • domain assumption Standard framework of Bell inequalities and quantum mechanics in finite-dimensional Hilbert spaces
    The work assumes the usual operator-algebraic setting for nonlocality without additional justification.

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Pith. "Pith review of Genuine Multipartite Nonlocality for Arbitrary Input: Maximal Randomness Generation and Robust Self-Testing." pith.science (2026). https://pith.science/paper/SXNPMLY7

@misc{pith2026260610936,
  author       = {Pith},
  title        = {Pith review of: Genuine Multipartite Nonlocality for Arbitrary Input: Maximal Randomness Generation and Robust Self-Testing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXNPMLY7}},
  note         = {Machine review of arXiv:2606.10936}
}
read the original abstract

Bell nonlocality provides the foundation for device-independent (DI) certification of quantum devices. We introduce a Bell inequality capable of identifying genuine multipartite nonlocality (GMNL) in an arbitrary m-partite scenario with an arbitrary odd number of measurements per party. Since the multi-setting nature of this inequality precludes the use of Jordan's Lemma, we construct an analytical sum-of-squares (SOS) decomposition to obtain the optimal quantum violation without assuming any bound on the Hilbert space dimension. This, in turn, enables self-testing of the shared entangled state and the corresponding measurement observables, up to local isometries, whose existence we confirm using a swap-based certification scheme. In addition, we show that our framework enables the extraction of maximal global DI randomness (m bits) at the optimal quantum violation, thereby exceeding previous limitations in the GMNL regime. Finally, we demonstrate that the architecture of our inequality yields improved robustness to noise as the number of measurement settings grows, ensuring experimental feasibility.

Figures

Figures reproduced from arXiv: 2606.10936 by the authors.

Figure 1
Figure 1. depicts the relationship between the lower bound on the fidelity and the relative observed violation. Employing the swap circuit, both the state and the measurement observ￾ables can be robustly extracted whenever the fidelity exceeds 50%. For n = 11, this condition is met at r = 0.968 (0 ≤ β ≤ 0.0358) for the state and at r = 0.992 (0 ≤ β ≤ 0.0358) for the observables. At the point where the state is extracted, the … view at source ↗
Figure 2
Figure 2. FIG. 2: Self-testing circuit for n [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

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Reference graph

Works this paper leans on

65 extracted references · 2 canonical work pages

  1. [1]

    Further we define δm 1,n = nX i1=1 {A1 i1 ,A 1 i1+1}, δ m 2,n = n−1X i=3 {A1,A i}+ n−2X i1=2 nX i=i1+2 {A1 i1,A 1 i }

    Considering∆ m n = Pn i1=1 A1 i1 , we obtain (∆m n )2 =n11 d + nX i1=1 {A1 i1 ,A 1 i1+1}+ n−1X i=3 {A1,A i}+ n−2X i1=2 nX i=i1+2 {A1 i1 ,A 1 i }. Further we define δm 1,n = nX i1=1 {A1 i1 ,A 1 i1+1}, δ m 2,n = n−1X i=3 {A1,A i}+ n−2X i1=2 nX i=i1+2 {A1 i1,A 1 i }. 8 Then it follows that −δm 1,n =n11 d +δ m 2,n −(∆ m n )2. Hence, we have, (Γm n )Q ≤n m−2 q...

  2. [2]

    The derivation of local value ofΓ m n To derive the local value ofΓm n , we first focus on the simplest bipartite Bell scenario, with two spatially separated parties, each choosing from a finite set of dichotomic measurements. In this framework, the Bell expression under study can be expressed as Γ2 n =(A 1 1 −A 1 2)⊗A 2 1 +(A 1 2 −A 1 3)⊗A 2 2 +· · ·+(A ...

  3. [3]

    Recall that a bi-local model corresponds to correlations that are local with respect to a fixed bi-partition of themparties

    The derivation of Bi-local value ofΓ m n We are now in a position to evaluate the maximum bi-local value of the multipartite Bell expressionΓm n . Recall that a bi-local model corresponds to correlations that are local with respect to a fixed bi-partition of themparties. We show, by mathematical induction on the number of partiesm, that the maximal bi-loc...

  4. [4]

    Detailed calculation form=3andn=3 ConsiderAlice 1,Alice 2, andAlice 3 have three possible measurement choicesx 1 ∈ {A 1 1,A 1 2,A 1 3},x 2 ∈ {A 2 1,A 2 2,A 2 3}andx 3 ∈ {A3 1,A 3 2,A 3 3}respectively. Now, the Bell functionalΓ 3 3 has the following form, Γ3 3 =((A 1 1 −A 1 2)⊗A 2 1 +(A 1 2 −A 1 3)⊗A 2 2 +(A 1 3 −A 1 1)⊗A 2 3)⊗A 3 1 +((A1 1 −A 1 2)⊗A 2 2 +...

  5. [5]

    In this setting, all three partiesAlice k,∀k∈[3] each have five possible measurement settings, denoted byi k ∈ {A k 1,

    Detailed calculation form=3andn=5 Next, we examine the inequality in the case of five inputs. In this setting, all three partiesAlice k,∀k∈[3] each have five possible measurement settings, denoted byi k ∈ {A k 1, . . . ,Ak 5}, respectively. In this scenario, our inequality takes the form, Γ3 5 =((A1 1 −A 1 2)⊗A 2 1 +(A 1 2 −A 1 3)⊗A 2 2 +. . .+(A 1 5 −A 1...

  6. [6]

    Therefore, using Eqs. (C6) and (C23), we can express for an arbitrary number ofnsettings ⟨{A1 i1,A 1 i1+x}⟩=⟨{ eA1 i1 , eA1 i1+x}⟩=2(−1) x cos πx n ,∀x∈[n−i 1] (C24) To determine the relations between the observables for the remaining parties, in the tripartite scenario with arbitrary settingsn, Eq. (A7) can be recast as eA1 i1 ⊗A 2 (i1+i3−2)⊕n+1 ⊗A 3 i3 ...

  7. [7]

    Detailed calculation form=4andn=3 ConsiderAlice k having three possible measurement choicesik ∈ {A k 1,A k 2,A k 3},∀k∈[4], respectively. Now,Γ 4 3 has the following form, Γ4 3 = Γ3,1 3 ⊗A 4 1 + Γ3,2 3 ⊗A 4 2 + Γ3,3 3 ⊗A 4 3 (C33) where Γ3,1 3 =((A 1 1 −A 1 2)⊗A 2 1 +(A 1 2 −A 1 3)⊗A 2 2 +(A 1 3 −A 1 1)⊗A 2 3)⊗A 3 1 +((A 1 1 −A 1 2)⊗A 2 2 +(A 1 2 −A 1 3)⊗...

  8. [8]

    Detailed calculation form=4andn=5 Consider four partiesAlice k,∀k∈[4] to have three possible measurement choicesi k ∈ {A k 1,A k 2,A k 3}respectively. Now,Γ 4 5 has the following form, Γ4 5 = 5X i4=1 5X i3=1  5X i1=1 (A1 i1 −A 1 i1⊕5+1)⊗A 2 (i1+i3−2)⊕5+1  ⊗A 3 (i3+i4−2)⊕5+1 ⊗A 4 i4 (C44) Now, we will derive the optimal quantum value ofΓ4 5...

Show all 65 references
  1. [9]

    Therefore, using Eqs. (C6) and (C54), we can express for an arbitrary number ofnsettings ⟨{A1 i1,A 1 i1+x}⟩=⟨{ eA1 i1 , eA1 i1+x}⟩=2(−1) x cos πx n ,∀x∈[n−i 1] (C55) To analyze how the observables of the remaining parties are related, we can rewrite the self-testing condition ...

  2. [10]

    This is achieved by employing stabilizer characteristics and Property 2

    Single-Party Marginals We demonstrate that⟨A k ik ⟩=0. This is achieved by employing stabilizer characteristics and Property 2. ⟨A2 x⟩=⟨S i1i2i3 A2 xS i1i2i3 ⟩=⟨A 2 i2 A2 xA2 i2 ⟩=2c x−i2 ⟨A2 i2 ⟩ − ⟨A2 x⟩=⇒ ⟨A 2 x⟩=c x−i2 ⟨A2 i2 ⟩(D5) 18 Puttingi 2 =xin Eq. (D4), we obtain a ...

  3. [11]

    ⟨A2 i2 ⟩=c i2−x⟨A2 x⟩(D6) Asc x−j =c j−x, we can write from Eq

    Similarly, using the previous approach, we obtain the following. ⟨A2 i2 ⟩=c i2−x⟨A2 x⟩(D6) Asc x−j =c j−x, we can write from Eq. (D5) and (D6)⟨A 2 x⟩=⟨A 2 i2 ⟩=0,∀x,i 2 ∈[n]. Similarly, we can write it for ⟨eA1 i1 ⟩=⟨A 3 i3 ⟩=0,∀i 1,i 3 ∈[n]. The next section will explore the ...

  4. [12]

    We now show that the double marginal must also be zero

    Double-Party Marginals In the preceding section, we established that all single marginals are equal to zero. We now show that the double marginal must also be zero. In this regard, our aim is to provef(b,c)=⟨11 d ⊗A 2 b ⊗A 3 c⟩=0,∀b,c∈[n]. We conjugate the operator O=11 d ⊗A 2...

  5. [13]

    (A7), we get

    Randomness evaluation for any oddninput andmparty From the SOS condition given in Eq. (A7), we get . eA1 i1 ⊗A 2 (i1+i3−2)⊕n+1 ⊗m−1 k=3 Ak fk(ik+1,ik) ⊗A m im |ψ⟩A1..Am =|ψ⟩ A1..Am ,∀i k∈[3,m−1] ∈[n] (D17) A1 i1 −A 1 i1⊕n+1 wi1 ⊗A 2 (i1+i3−2)⊕n+1 ⊗m−1 k=3 Ak fk(ik+1,ik) ⊗A m i...

  6. [14]

    Hence, putting Eq

    Form=3andn=3 For circuit implementation, note that eA1 1 =X A1 , eA1 2 =sin π 3 ZA1 −cos π 3 XA1 , eA1 3 =−sin π 3 ZA1 −cos π 3 XA1 (E1) A2 1 =X A2 ,A 2 2 =−sin π 6 XA2 +cos π 6 ZA2 ,A 2 3 =−sin π 6 XA2 −cos π 6 ZA2 (E2) A3 1 =X A3 ,A 3 2 =sin π 3 ZA3 −cos π 3 XA3 ,A 3 3 =−sin...

  7. [15]

    Robust self-testing of state For perfect and imperfect implementation, the output of the isometry is Φ(|ψ⟩A1A2A3 ⊗ |000⟩A1′ A2′ A3′ ) = 1 8 X a1,a2,a3∈{0,1} (ZA1)a1 (ZA2)a2 (ZA3)a3 (1+(−1) a1 XA1)(1+(−1) a2 XA2)(1+(−1) a3 XA3)|ψ⟩ A1A2A3 |a1a2a3⟩A1′ A2′ A3′ (F12) eΦ(|ψ⟩A1A2A3 ⊗...

  8. [16]

    Robust self-testing of observables The procedure will be the same as in the previous section; only here will we calculate the robustness with observables, i.e., XAk ,Z Ak (∀k∈[3]). Hence, the robustness ofX Ak is represented as ||eΦ(eXAk |ψ⟩A1A2A3 ⊗ |000⟩A1′ A2′ A3′ )−Φ(X Ak |...

  9. [17]

    X a1,a2,a3∈{0,1} (eZA3)a3 −(Z A3)a3 +(−1) a3 ((eZA3)a3 eXA3 −(Z A3)a3 XA3) 1+(−1) a2 XA2 +(−1)a1 XA1 +(−1) a1+a2 XA1 XA2 |ψ⟩A1A2A3 |a1a2a3⟩A1′ A2′ A3′ # ≤ 1 8

    Special case:Only the3rd party implements imperfect observables Now, if we consider only the third number of parties implement the imperfect observables (A3 i3 ), then||( eXA3 −X A3)|ψ⟩ A1A2A3 || ≤ δ,||(eZA3 −Z A3)|ψ⟩ A1A2A3 || ≤δ, where we consider that the error is the same,...

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    In this scenario, we introduce equal noise parameters for each of the observables ofA 3, assuming the state is perfect

    Robustness of genuine randomness in tripartite scenario In the experimental scenario, achieving the optimal quantum violation is nearly impossible due to noisy systems. In this scenario, we introduce equal noise parameters for each of the observables ofA 3, assuming the state ...

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