A three-player coherent state embezzlement game

Abstract

We introduce a three-player nonlocal game, with a finite number of classical questions and answers, such that the optimal success probability of 1 in the game can only be achieved in the limit of strategies using arbitrarily highdimensional entangled states. Precisely, there exists a constant 0 < c ≤ 1 such that to succeed with probability 1 - " in the game it is necessary to use an entangled state of at least W("-c) qubits, and it is sufficient to use a state of at most O("-1) qubits. The game is based on the coherent state exchange game of Leung et al. (CJTCS 2013). In our game, the task of the quantum verifier is delegated to a third player by a classical referee. Our results complement those of Slofstra (arXiv:1703.08618) and Dykema et al. (arXiv:1709.05032), who obtained twoplayer games with similar (though quantitatively weaker) properties based on the representation theory of finitely presented groups and C*-algebras respectively.