Parallel Repetition for Entangled Games with Gap Exponent Three
Abstract
We prove that every finite two-player game $G$ with entangled value $\omega^*(G)=1-\epsilon$ satisfies \[ \omega^*(G^{\otimes n}) \le\exp(-\Omega(\frac{\epsilon^3}{\epsilon+ \ell }n)) \] for every $n\ge1$, where $\ell:=\log(|A| |B|)$, and $A$ and $B$ are the answer alphabets. Compared with Chapter~6 of the OpenAI report [Ope26], this improves the gap exponent from thirteen to three and matches the cubic gap dependence in Holenstein's general classical bound [Hol09]: \[ \omega(G^{\otimes n})\le\exp(-\Omega( \frac{ (1-\omega(G))^3}{1+\ell} n)). \] The proof replaces the randomly shifted logarithmic grid used in quantum correlated sampling by smooth soft labels. This makes the relevant label infidelity quadratic in the distance between state descriptions and avoids a Jensen loss when averaging over questions. Together with the postselection argument, these improvements yield the cubic gap dependence stated above.
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