Parallel Repetition for Entangled Games via Conditional Relative Entropy

Abstract

We give a proof of the cubic parallel-repetition bound for finite two-player entangled games using quantum conditional relative entropy. For a game with entangled value $1-\epsilon <1$ and answer alphabets $\mathcal{A},\mathcal{B}$, its $n$-fold repeated value is at most \[ \exp[-Ω(\varepsilon^3n/L)],\qquad L:=\max\{1,\log(|\mathcal A||\mathcal B|)\}, \] for every $n\ge 1$. This recovers Song's cubic rate. We also prove a threshold bound: the probability of winning a fraction at least $1-\epsilon +\delta$ is at most $\exp [-\Omega ({\delta }^{5}n/({\epsilon }^{2}L))]$ for $0<\delta \le \epsilon$. Our information charge is the decrease in relative entropy when a question reference is discarded. Our comparison states preserve the original conditional question law. A chain rule bounds the coordinate charges using the cost of conditioning on checked wins and retaining checked answers. Our main lemma converts small charges into nearby pure-state descriptions and local corrections, each depending only on one player's question. Together with Song's sampler, these enable approximate local preparation of the conditioned workspace.

arXiv comment35 pages, version 1