Abstract
We introduce a three-player nonlocal game, with a finite number of classical questions and answers, such that the optimal success probability of1in the game can only be achieved in the limit of strategies using arbitrarily high-dimensional entangled states. Precisely, there exists a constant0<c≤1such that to succeed with probability1−εin the game it is necessary to use an entangled state of at leastΩ(ε−c)qubits, and it is sufficient to use a state of at mostO(ε−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 two-player games with similar (though quantitatively weaker) properties based on the representation theory of finitely presented groups andC∗-algebras respectively.
Highlights
The game is based on the coherent state exchange game of Leung et al (CJTCS 2013)
Our analysis shows that any near-optimal strategy for the game we construct must contain, within itself, the ability to “embezzle” an EPR pair from a product state – a task that, according to Fannes’ inequality, can only be achieved with arbitrarily high accuracy using a family of ancilla entangled states that have unbounded entanglement entropy
It remains an outstanding open question to determine the size of the smallest game such that the optimal success probability in the game can only be achieved in the limit of infinite-dimensional strategies
Summary
It remains an outstanding open question to determine the size of the smallest game such that the optimal success probability in the game can only be achieved in the limit of infinite-dimensional strategies. There are reasons to believe that the exponential trade-off between entanglement dimension and success probability demonstrated by our construction may be optimal. Even if one allows games whose size grows with ε−1 (equivalently, if one restricts to “not too small” values of ε), the best scaling known remains exponential The construction of the game, and its analysis, combines known rigidity results for the GHZ game and the Magic Square games.
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