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REVIEW 3 major objections 6 minor 44 references

No Bound Randomness in Quantum Nonlocality

T0 review · 3 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Any amount of quantum nonlocality suffices to certify randomness against a quantum adversary, once randomness is averaged over all input settings.

desk verdict Central no-bound-randomness claim is plausible and worth taking seriously, but Theorem 3 has a real proof gap in the passage from joint to local non-disturbance; the theorem is unproven as written and needs a repair before the paper's main conclusion can be trusted. read the letter →

arxiv 2509.08623 v1 pith:ZDSGYDEI submitted 2025-09-10 quant-ph

Ravishankar Ramanathan , Yuan Liu , Yutian Wu , Stefano Pironio This is my paper · ORCID
classification quant-ph MSC 81P1381P94 PACS 03.65.Ud03.67.Dd
keywords quantumnonlocalitydevice-independentrandomnessguessingprobabilityboundamplificationBellinequalitiesCHSHdetectionefficiency
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 establishes that quantum nonlocality is both necessary and sufficient for device-independent randomness, provided the protocol does not fix a single measurement setting. It proves that for any nonlocal quantum behavior, an adversary cannot perfectly guess the outputs on average over all inputs: the average guessing probability is strictly below 1. This rules out “bound randomness” in quantum nonlocality and makes the average guessing probability a faithful and monotonic measure of nonlocality, unlike the fixed-setting guessing probability. The same perspective lowers the detection-efficiency threshold for randomness certification from the I_4422 inequality and yields a new analytic formula for the average guessing probability in a CHSH test.

What carries the argument

The carrying mechanism is a classical joint probability distribution P_JPD assembled by ordering Alice's and Bob's inputs and applying their POVMs sequentially to a single copy of the marginal state rho_AB. Perfect predictability by Eve forces each joint measurement to leave rho_AB invariant, via the gentle measurement lemma, so the sequential marginals reproduce the original quantum probabilities and yield a local model, contradicting nonlocality. For the CHSH computation, the key object is the double-tilted CHSH expression I_CHSH^(alpha,phi) = alpha cos(phi/2) A0 + alpha sin(phi/2) A1 + I_CHSH; its quantum maximum bounds the tilted correlator combination, and translating the input bias p i

What would settle it

Construct any nonlocal bipartite behavior p and full-support input distribution mu_XY with an explicit tripartite quantum strategy achieving average guessing probability exactly 1; or, more surgically, exhibit a perfect-guessing tripartite strategy in which the joint measurement leaves rho_AB invariant but a single local measurement (Pi_x tensor identity or identity tensor Lambda_y) changes it. Either would invalidate Theorem 3.

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Extended reading notes

Core claim

The central claim is Theorem 3: no nonlocal quantum behavior exhibits bound randomness against a quantum adversary. If a tripartite quantum strategy allowed Eve to guess Alice and Bob's outputs perfectly for every input pair, the proof constructs a classical joint distribution—obtained by running all of their measurements sequentially on one copy of the shared marginal state—whose marginals reproduce the observed behavior, making the behavior local. Since a local behavior cannot win any nonlocal game above its classical value, perfect guessing is incompatible with nonlocality. Theorem 2 then states that for any nonlocal bipartite behavior p and any full-support input distribution mu_XY, the

Load-bearing premise

The proof rests on the step where perfect guessing is taken to imply that Alice's and Bob's individual measurements do not disturb their shared marginal state; the paper explicitly proves the joint non-disturbance and asserts the individual form needed for the sequential local model.

Editorial extensions

If this is right

  • Any nonlocal quantum behavior, however small the violation, certifies positive min-entropy in a device-independent randomness amplification protocol against a quantum adversary.
  • The average guessing probability, unlike the fixed-input guessing probability, is faithful and monotonic under single-copy WCCPI operations and therefore serves as a proper nonlocality measure.
  • For the I_4422 inequality, randomness can be certified for every detection efficiency above 61.8%, matching the nonlocality detection threshold instead of the higher fixed-setting threshold found previously.
  • The analytic CHSH bound extends the classical fixed-setting formula to biased input distributions and recovers the known formula as the p=1 limit, with numerical tightness confirmed at high NPA level.
  • The local-model argument does not rely on pseudo-telepathy games, so every nonlocal game, and by extension multipartite nonlocal behaviors, supports randomness amplification in this protocol structure.

Reading between the lines

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

  • If the theorem is correct, other multi-input Bell inequalities with apparently high randomness thresholds may also certify randomness at their nonlocality threshold once protocols average over all input pairs; I_4422 is one worked example.
  • The same average-guessing-probability quantity could be applied to device-independent quantum key distribution and to expansion protocols beyond amplification, potentially closing gaps left by fixed-setting security analyses.
  • A concrete experimental test would be to measure the average min-entropy directly in an existing Bell test with a small but nonzero violation: positive min-entropy would confirm the mechanism, while a finding of zero would challenge it.
  • The sequential-measurement construction suggests a resource-theoretic reading: quantum nonlocality and certifiable randomness may coincide as resources once the protocol structure is optimized, rather than being distinct operational resources.
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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

3 major / 6 minor

Summary. The paper studies device-independent randomness certification against quantum adversaries and distinguishes two protocol structures: spot-checking, where a fixed input pair is used for generation, and amplification-style protocols, where all input pairs are used. The authors introduce the average guessing probability P_g^(q)(p, mu_XY) and claim (Theorem 2/3) that any nonlocal bipartite quantum behavior satisfies P_g^(q)(p, mu_XY)<1 for any full-support input distribution, i.e., quantum nonlocality is sufficient for randomness amplification and there is no 'bound randomness' against quantum adversaries. They then argue that the average guessing probability is a faithful and monotonic measure of nonlocality (Appendix D), use it to show that the I_4422 inequality certifies randomness at its nonlocality detection-efficiency threshold (Appendix C), provide an analytical formula for the average guessing probability in CHSH (Appendix E), and demonstrate improved amplification rates via a numerical entropy calculation (Appendix F). The central proof is an immunization argument that builds a classical joint distribution from a hypothetical perfect-guessing tripartite strategy.

Significance. If the central theorem holds, it is a conceptually important result: it complements the recent finding that maximum nonlocality can be useless for fixed-setting spot-checking protocols by showing that the same resource is sufficient in amplification protocols with randomized inputs. It also gives a concrete operational meaning to the average guessing probability as a nonlocality measure and yields a practical improvement for the I_4422 inequality. The paper contains explicit calculations, a candidate analytical formula, an SOS decomposition for a new Bell expression, and numerical SDP checks. However, the main theorem's proof has a load-bearing gap, and the advertised analytic CHSH result relies on an imported conjectured formula. These issues currently prevent acceptance.

major comments (3)
  1. [Appendix B, Eqs. (B4)-(B13)] The proof of Theorem 3 hinges on the claim that the local measurements do not disturb rho_AB. Proposition 1 establishes only the joint non-disturbance (Phi_x⊗Psi_y)(rho_AB)=rho_AB for each pair. The sequential construction of P_JPD in Eq. (B10) requires the individual conditions Phi_{x_k}(rho_AB)=rho_AB and Psi_{y_l}(rho_AB)=rho_AB; without them the equality in Eq. (B13) holds only for the first pair and not for later pairs. Joint invariance does not imply local invariance for general channels; for example, for rho_AB=(|00><00|+|11><11|)/2, the unitary channels Phi_x=Psi_y=X(·)X satisfy joint invariance while local invariance fails. The sentence 'Evidently...' after Eq. (B11) is therefore not a proof. Additionally, Eq. (B6) uses rho_AB=tr_E[1⊗1⊗Gamma_{(x,y)}(rho_ABC)], which is not true for arbitrary POVM measurement channels unless Gamma is projective or a dilation argument is supplied.
  2. [Appendix E, Eq. (E5) and Theorem 4] The advertised analytical expression for the average guessing probability in CHSH, Eq. (8) in the main text and Eq. (E2) in the appendix, depends on the maximum quantum value of the double-tilted CHSH expression, Eq. (E5), imported from Ref. [34]. That reference is a preprint whose title indicates a conjectured analytical solution. The manuscript treats Eq. (E5) as an established bound, and the only verification offered is numerical NPA coincidence to 10^{-9}. Since the claim 'analytically compute the average guessing probability' is one of the paper's main results, the formula must either be proved or explicitly flagged as conditional on the conjecture.
  3. [Appendix D, Lemma 3] The monotonicity proof is not rigorous. In the input-substitution case, Eqs. (D10)-(D12) relabel x_i and x_j without a corresponding transformation of the input weights, and Eq. (D12) asserts an equality to a maximum that does not follow from the preceding optimization. The coarse-graining argument is also only sketched. Since Lemma 4's claim that the average guessing probability defines a nonlocality measure depends on Lemma 3, these arguments need to be made rigorous or replaced by a reference to a known resource-theoretic result.
minor comments (6)
  1. [Abstract and Section 'Detection efficiency thresholds'] The phrase 'the detection efficiency threshold for randomness generation is never lower than that for nonlocality detection' appears to be the opposite of what Theorem 3 implies. If every nonlocal behavior certifies randomness on average, the randomness threshold is no higher (in fact equal) to the nonlocality threshold. Please clarify.
  2. [Appendix C, Eq. (C1)] There is a typo in Eq. (C1): 'p(A_4-1)' should presumably be 'p(A_4=1)', and similarly for B_4. The formula should be checked carefully.
  3. [Figures 1 and 2] The captions use x=y∈{0,1,2,3} while the text uses {1,2,3,4}. Please standardize the indexing.
  4. [Theorem 2] The statement 'sufficient resource for DI randomness amplification' should be supported by an explicit protocol statement or a precise reference. P_g^(q)(p,mu)<1 is a necessary precondition, but by itself it does not automatically yield amplification of an SV source; the manuscript should make clear how the existing protocols of Refs. [5-11] convert this into full amplification.
  5. [Appendix A.2] There is a typo: 'adversary's adversary's side information' should be 'adversary's side information'.
  6. [Appendix E, Eq. (E13)] There are minor notation issues in the derivation, e.g., 'p^{A|X,Z,E}(a|A0,z=0,e=a)' is used before being defined, and some parentheses in the displayed f_phi formula are unbalanced in the main text.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 3 is a parameter-free derivation against external benchmarks; the only self-citation is non-load-bearing, and the flagged Appendix B gap is a correctness issue, not circularity.

full rationale

The central derivation chain (Theorem 2/3, Appendix B) assumes perfect guessing by Eve (B2), proves joint non-disturbance via the gentle measurement lemma (Proposition 1, B4), constructs a classical joint distribution P_JPD (B10), and claims that its marginals reproduce the quantum behavior (B13), forcing the behavior to be local. Nothing in this chain is fitted to data, and the conclusion is not contained in the definition of the average guessing probability; it is a substantive implication. The only self-citation, [12], provides the contrast example of a maximally nonlocal behavior with fixed-setting guessing probability 1; this is motivation/contrast and is not a premise of Theorem 3, so it is not load-bearing. The analytic CHSH result in Appendix E imports the double-tilted CHSH quantum maximum from external reference [34] and verifies tightness numerically via NPA/SDP duality; no fitted parameter is renamed as a prediction. The correctness caveat about Eq. (B12) is a genuine proof gap, not circularity: Proposition 1 proves joint non-disturbance (Π_x⊗Λ_y)(ρ_AB)=ρ_AB, while Eq. (B12) asserts that the individual local measurement channels leave ρ_AB invariant. That implication is not demonstrated and would be needed for the sequential construction of P_JPD. This is an omitted proof in the derivation, but it is not an equivalence-by-construction and does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data: the input probability p in Theorem 4 is a protocol choice, and alpha_min,phi in Eq. (E8) is the root of an explicit equation. The axioms listed are the load-bearing background facts: standard quantum-information lemmas used in the immunization proof, the unproven external conjecture [34] behind the analytic CHSH formula, the NPA hierarchy as a numerical proxy, and the implicit composition step from single-round bounds to protocol security. No new physical entities (particles, forces, dimensions) are introduced; the new mathematical objects (average guessing probability measure, the Appendix G Bell expression) are not entities of the schema's kind.

assumptions (6)
  • standard math Gentle measurement lemma (Winter, [30]): for a state sigma and 0 <= A <= 1, ||sigma - sqrt(A) sigma sqrt(A)||_1 <= 3 sqrt(1 - tr(A sigma)).
    Used in Appendix B, Eq. (B8), to prove Proposition 1 (non-disturbance of rho_AB under joint Alice-Bob measurements when Eve predicts perfectly).
  • standard math Trace-distance monotonicity under partial trace.
    Appendix B, Eq. (B6), used to pass from tripartite to bipartite non-disturbance in Proposition 1.
  • standard math Fine's theorem: existence of a joint probability distribution over all measurement outcomes for all settings is equivalent to Bell locality of the bipartite behavior.
    Implicit in Appendix B: the constructed P_JPD with the correct marginals implies the behavior is local. Not stated by name in the paper.
  • domain assumption The maximum quantum value of the double-tilted CHSH expression I_Q^(alpha,phi) (Eq. E5) taken from [34], whose own title flags it as a 'conjectured analytical solution'.
    Appendix E, Eqs. (E3)-(E9). The analytic average guessing probability formula (main text Eq. 8, Theorem 4) is derived from this value; the paper adds numerical NPA-3 verification of the final bound but does not prove the underlying extremal-point solution.
  • domain assumption NPA hierarchy gives outer approximations to the quantum set, used at levels 2 and 3 for numerical bounds on guessing probabilities and the I_4422 min-entropy.
    Appendix C Figs. 1-2, Appendix E Fig. 3, Appendix G Fig. 5. Numerical evidence; convergence is asymptotic, so the plots are not proofs of exact values.
  • domain assumption Protocol-level composition: single-round guessing probability P_g < 1 (or the per-round von Neumann entropy bounds of Appendix F) can be lifted to a secure finite-size randomness amplification protocol with positive rate using the existing frameworks of [5-11] and entropy accumulation.
    Main text Theorem 2 and Appendix F. The paper proves the single-round statement and computes entropic quantities, but does not present the full finite-size security proof of the composed protocol.

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Pith. "Pith review of No Bound Randomness in Quantum Nonlocality." pith.science (2026). https://pith.science/paper/ZDSGYDEI

@misc{pith2026250908623,
  author       = {Pith},
  title        = {Pith review of: No Bound Randomness in Quantum Nonlocality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDSGYDEI}},
  note         = {Machine review of arXiv:2509.08623}
}
read the original abstract

Recently it has been found that there exist maximally nonlocal quantum correlations that fail to certify randomness for any fixed input pair, rendering them useless for device-independent spot-checking randomness expansion schemes. Here we show that conversely, in DI randomness amplification protocols where all input pairs are used for randomness generation, any amount of quantum nonlocality is sufficient to certify randomness. This shows that no bound randomness exists in quantum nonlocality - any quantum nonlocal behavior is useful in a DI randomness generation task with appropriate modification of protocol structure. Secondly, we show that in contrast to the hitherto considered fixed-input guessing probability, the average guessing probability over all inputs is a faithful and monotonic measure of nonlocality. We use the average guessing probability to show that in contrast to findings in PRL 134, 090201, the detection efficiency threshold for randomness generation is never lower than that for nonlocality detection. Finally, we analytically compute the average guessing probability by a quantum adversary of a single player's measurement outputs in a standard CHSH Bell test, and use it to demonstrate an improvement in the generation rate in state-of-art amplification protocols.

Figures

Figures reproduced from arXiv: 2509.08623 by the authors.

Figure 1
Figure 1. FIG. 1: The horizontal axis represents the violation of the inequality [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Plot of the average min-entropy [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) Solid lines: The lower bound of the average guessing probability [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison between [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Solid lines: The guessing probability [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

Discussion (0). Continue with ORCID to comment.

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