Parallel Repetition for Entangled Games via Conditional Relative Entropy

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 $$\exp [-\mathrm{\Omega }({\epsilon }^{3}n\mathrm{/}L)],L:=\max \{1,\log (\mathrm{\mid }\mathcal{A}\mathrm{\mid }\mathrm{\mid }\mathcal{B}\mathrm{\mid })\},$$ 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 $\exp [-\mathrm{\Omega }({\delta }^{5}n\mathrm{/}({\epsilon }^{2}L))]$ 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.