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Threshold parallel repetition for finite-dimensional entangled games
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Abstract
For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a $v+\delta$ fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every $0\lt \delta\lt 1-v$ and k ≥ 1. Its universal rate is proportional to $\delta^5/(1+\log(|\mathcal A||\mathcal B|))$, where $\mathcal A,\mathcal B$ are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Rahul Jain, Parallel Repetition for Entangled Games via Conditional Relative Entropy (arXiv:2610.21410, 15 October 2026). Cites this paper's Theorem 1.1 and Corollary 5.2 and, in Section 1.1, compares its two threshold rates with Jain's bound exp(−Ω(δ⁵n/(ε²L))), where ε = 1 − ω*(G) and L = max{1, log(|A||B|)}, obtained using conditional relative entropy and Song's sampler.
Formalization
- OpenAI math release, Threshold repetition for entangled games: Lean scope notes (lean/docs/277.md, result 277). The release's Lean development covers a universal threshold bound with the weaker rate κδ¹³k/(1+log(|A||B|)) for every finite-dimensional joint strategy and their supremum; the paper's fifth-power universal rate and its distribution-dependent cubic bound are not in this scope.
Related prior literature mentioned by others
- Zhao Song, Parallel Repetition for Entangled Games with Gap Exponent Three (ECCC TR26-199, 2026). Jain's follow-up uses Song's smooth sampler, and the Open Mathematics Courses exposition identifies Song's conditioning estimate as the external input to this paper's universal δ⁵ threshold bound.
Possibly relevant literature (found by the agent, not checked)
- Russell Impagliazzo and Valentine Kabanets, Constructive proofs of concentration bounds (ECCC TR10-072; RANDOM-APPROX 2010). Shows that optimal direct-product and optimal threshold direct-product theorems are equivalent, which compares with the paper's transfer that samples tested coordinates to get threshold bounds from conditioning bounds.
- Falk Unger, A Probabilistic Inequality with Applications to Threshold Direct-product Theorems (ECCC TR09-078, September 2009). Gets threshold direct-product theorems, including for quantum XOR games, from co-moment bounds on binary variables, related to the paper's reduction of threshold decay to the joint law of the binary win indicators.
Digests and expositions
- Latent Seminar, Entangled games: conditioning and threshold repetition (2026-10-07). Explains the conditioning-to-threshold transfer, resolvent purifications and the support-dependent rate, distinguishes the manuscript's bounds from the Lean scope, and states that it does not independently verify the complete proofs. [family-level]
- Open Mathematics Courses / Claude Opus 5.5, Results on threshold repetition, purified effects and conditioned strategies (undated). Links a five-lesson exposition of the manuscript and leaves Song's conditioning estimate as hypothesis (S) for the universal fifth-power bound. [AI-generated per source]
Elsewhere
- Emergent Mind: family 277
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026,
author = {{OpenAI}},
title = {{Threshold parallel repetition for finite-dimensional entangled games}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf}{OAI:Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/277.md. The Lean statements have not been checked by people against the paper's claims.
Family 277: Threshold repetition for entangled games
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction $v+\delta$ of k independent repetitions decays exponentially in k, for $0\lt \delta\lt 1-v$. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.
This is the only paper in this family.