Certified randomness between mistrustful players

TL;DR

This paper proves that in nonlocal games with complete support, achieving a superclassical score guarantees a quantifiable amount of randomness in players' outputs, ensuring unpredictability even against adversaries with full knowledge.

Contribution

It introduces an explicit function relating game scores to the probability of correctly guessing outputs, establishing a quantitative link between nonlocal game performance and certifiable randomness.

Findings

Achieving superclassical scores certifies randomness in outputs.

A nonzero function F_G bounds the guessing probability based on game score.

Global randomness certification necessarily involves local randomness.

Abstract

It is known that if two players achieve a superclassical score at a nonlocal game $G$, then their outputs are certifiably random - that is, regardless of the strategy used by the players, a third party will not be able to perfectly predict their outputs (even if he were given their inputs). We prove that for any complete-support game $G$, there is an explicit nonzero function $F_G$ such that if Alice and Bob achieve a superclassical score of $s$ at $G$, then Bob has a probability of at most $1 - F_G ( s )$ of correctly guessing Alice's output after the game is played. Our result implies that certifying global randomness through such games must necessarily introduce local randomness.