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Bell inequalities with three outputs per party generically certify substantial noise-robust quantum randomness.

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2026-06-26 13:58 UTC pith:BXE2OQI2

load-bearing objection Numerical scan over structured three-outcome Bell families turns up many inequalities that certify solid min-entropy with decent noise tolerance and fewer settings than usual. the 2 major comments →

arxiv 2606.21371 v2 pith:BXE2OQI2 submitted 2026-06-19 quant-ph

Noise robustness of three outcome Bell certified quantum randomness

Raffaele D'Avino , Ignacio Perito , Piotr Mironowicz , Antonio Ac\'in , Remigiusz Augusiak This is my paper
classification quant-ph
keywords Bell inequalitiesdevice-independent randomnessthree outcomesnoise robustnessmin-entropyquantum certificationmulti-outcome measurements
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 examines device-independent certification of global randomness from Bell inequality violations in bipartite scenarios where each party has three possible measurement outcomes. It analyzes known expressions for noise robustness and introduces a systematic method to generate new ones within structured families, then conducts large-scale numerical optimization over them. The results show that a substantial fraction certify significant min-entropy, some reach near-maximal randomness with fewer input settings, and the vast majority maintain positive certified randomness under realistic noise. These observations indicate that strong randomness certification in multi-outcome settings is a generic feature of such families rather than a property of only specially engineered inequalities.

Core claim

In three-outcome bipartite Bell scenarios, a large number of inequalities generated from structured families certify substantial global min-entropy, with many exhibiting robustness to realistic noise levels and some achieving near-maximal randomness while using a reduced number of measurement settings.

What carries the argument

Systematic generation of structured families of Bell expressions evaluated through numerical sampling and optimization for min-entropy certification and noise tolerance.

Load-bearing premise

Numerical sampling and optimization over the chosen structured families is representative enough to establish that the observed randomness certification and noise robustness are generic features.

What would settle it

A broader enumeration or different sampling method that finds only a small minority of three-outcome Bell expressions certify any positive min-entropy or that most lose all certification under small noise.

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

If this is right

  • Many three-outcome Bell expressions certify significant min-entropy for global randomness.
  • Some expressions achieve near-maximal randomness with fewer measurement settings than typical cases.
  • The majority of nontrivial three-outcome certificates remain positive under realistic noise.
  • Strong device-independent randomness expansion arises generically inside suitably constructed families rather than only in engineered cases.

Where Pith is reading between the lines

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

  • Practical randomness sources could rely on simpler or fewer-setting inequalities without sacrificing certified output in noisy devices.
  • The generation method might scale to four or more outcomes to locate further improvements in the randomness-to-settings tradeoff.
  • Similar genericity could appear in other device-independent tasks such as entanglement certification or randomness expansion rates.

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 investigates device-independent certification of global randomness from Bell inequality violations in bipartite scenarios with three outcomes per party. It first evaluates the noise robustness of several known Bell expressions, then introduces a systematic method for generating new expressions within structured families and reports results from a large-scale numerical study. The central claims are that a substantial fraction of these inequalities certify significant min-entropy (with some achieving near-maximal values using fewer settings) and that the vast majority of nontrivial certificates remain robust to realistic noise, implying that strong randomness certification arises generically within such families rather than only for specially engineered inequalities.

Significance. If the numerical results hold under scrutiny, the work would indicate that multi-outcome Bell scenarios can deliver practical device-independent randomness expansion without requiring carefully tuned inequalities, potentially improving the trade-off between certified randomness and experimental complexity. The systematic generation approach for families of expressions is a methodological strength that could generalize to other scenarios. However, the absence of methodological details currently limits the ability to assess reproducibility and the strength of the genericity conclusion.

major comments (2)
  1. [Abstract and numerical study section] Abstract and the section describing the numerical study: the claims that 'a substantial number of inequalities certify significant amounts of min-entropy' and that 'the vast majority of nontrivial certificates exhibit robustness' rest on an unspecified large-scale numerical optimization whose methods (algorithm, convergence criteria, definition of the structured families, noise model, sample size, and definition of 'nontrivial') are not provided, preventing verification that the sampled expressions support the genericity assertion rather than reflecting selection bias in the generation procedure.
  2. [Systematic generation method section] The section on the systematic generation method: no explicit definition or parameterization of the 'structured families' is given, nor is there an analytical argument or exhaustive enumeration showing that the finite sampling is representative of the full family; this leaves the inference from observed instances to 'generically' as an unverified assumption that is load-bearing for the main conclusion.
minor comments (2)
  1. [Introduction] The introduction would benefit from a short quantitative comparison of the min-entropy values obtained here versus the best binary-outcome results under comparable noise levels.
  2. [Throughout] Notation for the three-outcome measurements and the Bell expressions should be standardized across sections to avoid ambiguity when comparing to prior multi-outcome literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and for identifying the need for greater methodological transparency. We agree that the original manuscript did not provide sufficient detail on the numerical procedures and family definitions, which limits reproducibility. We will revise the manuscript to include these elements explicitly, thereby strengthening the support for our claims about genericity.

read point-by-point responses

  1. Referee: [Abstract and numerical study section] Abstract and the section describing the numerical study: the claims that 'a substantial number of inequalities certify significant amounts of min-entropy' and that 'the vast majority of nontrivial certificates exhibit robustness' rest on an unspecified large-scale numerical optimization whose methods (algorithm, convergence criteria, definition of the structured families, noise model, sample size, and definition of 'nontrivial') are not provided, preventing verification that the sampled expressions support the genericity assertion rather than reflecting selection bias in the generation procedure.

    Authors: We agree that the methodological details were insufficiently specified. In the revised manuscript we will add a dedicated methods subsection that explicitly states: the optimization algorithm and solver employed, convergence criteria and tolerances, the precise mathematical parameterization used to define the structured families, the noise model (including how visibility or white noise is applied), the total number of sampled expressions, and the operational definition of 'nontrivial' certificates. These additions will enable independent verification and address concerns about possible selection bias. revision: yes

  2. Referee: [Systematic generation method section] The section on the systematic generation method: no explicit definition or parameterization of the 'structured families' is given, nor is there an analytical argument or exhaustive enumeration showing that the finite sampling is representative of the full family; this leaves the inference from observed instances to 'generically' as an unverified assumption that is load-bearing for the main conclusion.

    Authors: We accept that the original text lacked an explicit parameterization and a supporting argument for representativeness. The revised version will (i) provide the full mathematical definition and generation rules for each structured family, (ii) include a small-case exhaustive enumeration demonstrating that the sampling procedure captures the essential features of the family, and (iii) state the sampling strategy and any statistical checks performed. These changes will convert the genericity claim from an assumption into a substantiated inference. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; claims rest on external numerical evaluation.

full rationale

The paper's central results derive from analyzing known Bell expressions, introducing a systematic generation method for new expressions within structured families, and reporting outcomes of large-scale numerical optimization and sampling. No load-bearing step reduces a certified randomness quantity to a fitted parameter, self-defined input, or self-citation chain by construction. The inference to genericity within families is presented as an empirical observation from the sampled instances rather than a mathematical derivation that collapses to its own premises. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes only standard background assumptions of device-independent randomness certification; no new free parameters, ad-hoc axioms, or invented entities are introduced or fitted.

axioms (1)
  • domain assumption Bell inequality violations can be used to lower-bound min-entropy of measurement outcomes in a device-independent manner
    Standard assumption in the field of device-independent quantum information; invoked implicitly when the abstract discusses certification of global randomness.

pith-pipeline@v0.9.1-grok · 5711 in / 1195 out tokens · 25209 ms · 2026-06-26T13:58:33.354755+00:00 · methodology

0 comments
read the original abstract

We investigate device-independent certification of global randomness based on Bell inequality violations in bipartite scenarios with three outcomes per party. Our goal is to determine whether multi-outcome measurements allow one to surpass the amount of randomness achievable with binary outputs in realistic scenarios. We begin by analyzing several known Bell expressions and evaluating their robustness against noise for randomness certification. We then introduce a systematic method for generating new Bell expressions within structured families and perform a large-scale numerical study. We find that a substantial number of inequalities certify significant amounts of min-entropy. In particular, we identify simple inequalities that achieve near-maximal global randomness while involving a reduced number of measurement settings, thus improving the balance between certified randomness and number of inputs. Moreover, the vast majority of nontrivial certificates exhibit robustness against realistic noise, maintaining positive certified randomness away from the ideal regime. These results demonstrate that strong device-independent randomness expansion in multi-outcome scenarios is not restricted to carefully engineered inequalities, but arises generically within suitably constructed families of Bell expressions.

Figures

Figures reproduced from arXiv: 2606.21371 by Antonio Ac\'in, Ignacio Perito, Piotr Mironowicz, Raffaele D'Avino, Remigiusz Augusiak.

Figure 1
Figure 1. Figure 1: FIG. 1. Certified global randomness from the spot setting as [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Average certified randomness over all inputs under [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Certified global randomness from the spot setting as [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Average certified randomness over all inputs under [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Certified global randomness for six selected Bell ex [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Certified global randomness from the spot setting for [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Averaged certified global randomness over all inputs [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Averaged certified global randomness over all inputs [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bell inequalities tailored to optimal global randomness certification

    quant-ph 2026-06 unverdicted novelty 7.0

    Two families of Bell inequalities are constructed whose maximal quantum violations certify 2 log d random bits for arbitrary d using d x d maximally entangled states.

  2. Bell inequalities tailored to optimal global randomness certification

    quant-ph 2026-06 conditional novelty 6.0

    New d-outcome Bell inequalities certify 2 log d bits of global randomness at maximal violation; analytic self-testing of the optimal strategy is proven for d=3.

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