# The RGB No-Signalling Game

## TL;DR

This paper introduces the RGB No-Signalling Game as a simple pedagogical example to explore nonlocal correlations, defining new classes of probability distributions, and demonstrating quantum strategies that outperform classical limits.

## Contribution

It presents the RGB Game as a pedagogical tool, introduces new definitions for reductions among multi-party distributions, and demonstrates a quantum strategy surpassing classical success probabilities.

## Key findings

- The RGB Game's winning strategy is equivalent to the PR-Box, exemplifying No-Signalling.
- A quantum strategy achieves an 11/12 success rate, exceeding the classical 8/9 maximum.
- New definitions for reductions among multi-party distributions and locality classes are proposed.

## Abstract

Introducing the simplest of all No-Signalling Games: the RGB Game where two verifiers interrogate two provers, Alice and Bob, far enough from each other that communication between them is too slow to be possible. Each prover may be independently queried one of three possible colours: Red, Green or Blue. Let $a$ be the colour announced to Alice and $b$ be announced to Bob. To win the game they must reply colours $x$ (resp. $y$) such that $a \neq x \neq y \neq b$. This work focuses on this new game mainly as a pedagogical tool for its simplicity but also because it triggered us to introduce a new set of definitions for reductions among multi-party probability distributions and related {locality classes}. We show that a particular winning strategy for the RGB Game is equivalent to the PR-Box of Popescu-Rohrlich and thus No-Signalling. Moreover, we use this example to define No-Signalling in a new useful way, as the intersection of two natural classes of multi-party probability distributions called one-way signalling. We exhibit a quantum strategy able to beat the classical local maximum winning probability of 8/9 shifting it up to 11/12. Optimality of this quantum strategy is demonstrated using the standard tool of semidefinite programming.