Naturally restricted subsets of nonsignaling correlations: typicality and convergence
Bibliographic record
Abstract
It is well-known that in a Bell experiment, the observed correlation between measurement outcomes – as predicted by quantum theory – can be stronger than that allowed by local causality, yet not fully constrained by the principle of relativistic causality. In practice, the characterization of the set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> of quantum correlations is carried out, often, through a converging hierarchy of outer approximations. On the other hand, some subsets of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> arising from additional constraints [e.g., originating from quantum states having positive-partial-transposition (PPT) or being finite-dimensional maximally entangled (MES)] turn out to be also amenable to similar numerical characterizations. How, then, at a quantitative level, are all these naturally restricted subsets of nonsignaling correlations different? Here, we consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations. Within the number of cases investigated, we have observed that (1) for a given number of inputs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math> (outputs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:math>), the relative volume of both the Bell-local set and the quantum set increases (decreases) rapidly with increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math>) (2) although the so-called macroscopically local set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> may approximate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> well in the two-input scenarios, it can be a very poor approximation of the quantum set when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x003E;</mml:mo></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:math> (3) the almost-quantum set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>Q</mml:mi><mml:mo stretchy="false">&#x007E;</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math> is an exceptionally-good approximation to the quantum set (4) the difference between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> and the set of correlations originating from MES is most significant when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>o</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math>, whereas (5) the difference between the Bell-local set and the PPT set generally becomes more significant with increasing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:math>. This last comparison, in particular, allows us to identify Bell scenarios where there is little hope of realizing the Bell violation by PPT states and those that deserve further exploration.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".