A three-player coherent state embezzlement game

TL;DR

This paper introduces a three-player nonlocal game demonstrating that achieving perfect success probability requires arbitrarily high-dimensional entangled states, highlighting the complexity of quantum strategies in multi-player settings.

Contribution

It presents a novel three-player game based on coherent state exchange, showing the necessity of high-dimensional entanglement for near-perfect success, extending prior two-player results.

Findings

Optimal success probability of 1 requires arbitrarily high-dimensional entangled states.

Success with probability 1 - ε requires at least Ω(ε^{-c}) qubits of entanglement.

States of at most O(ε^{-1}) qubits suffice for near-perfect success.

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 high-dimensional entangled states. Precisely, there exists a constant $0 <c\leq 1$ such that to succeed with probability $1-\varepsilon$ in the game it is necessary to use an entangled state of at least $\Omega(\varepsilon^{-c})$ qubits, and it is sufficient to use a state of at most $O(\varepsilon^{-1})$ 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 and $C^*$-algebras respectively.