Threshold repetition for entangled games

For every finite two-player one-round game with entangled value v < 1, the probability of winning at least a fraction v + delta of k parallel repetitions decays exponentially in k, uniformly over all finite-dimensional entangled strategies and correlated question distributions. The proof: sampling tested coordinates turns a conditioning estimate into a threshold bound; resolvent purifications put conditioned states in common spaces with exact answer probabilities; an entropy telescope controls how they depend on the other player's question; one reference vector per support component yields legal one-game strategies and a delta^3 rate. A universal delta^5 rate rests on Song's conditioning estimate, a named hypothesis. The course is an exposition of OpenAI's proof (2026).

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Entangled games and threshold repetition
2. From conditioning to a threshold bound
3. A common purification of effects
4. Conditioning on a tested set
5. A cubic bound for a fixed question distribution

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Concentration and the discrete cube (The Gaussian moat problem)
• Entropy of finite random variables (The Gaussian moat problem)
• Entropy, relative entropy and Pinsker's inequality; Hoeffding's inequality (course The Gaussian moat problem); the spectral theorem for finite matrices and tensor products of finite-dimensional Hilbert spaces.