The RGB No-Signalling Game

Authors Xavier Coiteux-Roy, Claude Crépeau

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Author Details

Xavier Coiteux-Roy
  • Facoltà di scienze informatiche, Università della Svizzera italiana, Lugano, Switzerland
Claude Crépeau
  • School of Computer Science, McGill University, Montréal, Québec, Canada


We thank Gilles Brassard, Arne Hansen, Adel Magra, Alberto Montina, Louis Salvail, Stefan Wolf and Nan Yang for discussions about early versions of this work.

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Xavier Coiteux-Roy and Claude Crépeau. The RGB No-Signalling Game. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 != x != y != 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 non-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • No-Signalling
  • Quantum Entanglement
  • Non-Locality
  • Bell inequality
  • Semidefinite Programming
  • Non-locality Hierarchy


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