# A three-player coherent state embezzlement game

Zhengfeng Ji1, Debbie Leung2, and Thomas Vidick3

1Centre for Quantum Software and Information, University of Technology Sydney, Australia
2University of Waterloo and the Perimeter Institute, Canada. Email: \textttwcleung@uwaterloo.ca
3California Institute of Technology, USA. Email: \textttvidick@cms.caltech.edu

### 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.

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### Cited by

[1] Andrea Coladangelo and Jalex Stark, "Unconditional separation of finite and infinite-dimensional quantum correlations", arXiv:1804.05116.

[2] William Slofstra, "A group with at least subexponential hyperlinear profile", arXiv:1806.05267.

[3] Richard Cleve, Benoit Collins, Li Liu, and Vern Paulsen, "Constant gap between conventional strategies and those based on C*-dynamics for self-embezzlement", arXiv:1811.12575.

[4] Joseph Fitzsimons, Zhengfeng Ji, Thomas Vidick, and Henry Yuen, "Quantum proof systems for iterated exponential time, and beyond", arXiv:1805.12166.

[5] Andrea Coladangelo and Jalex Stark, "An inherently infinite-dimensional quantum correlation", Nature Communications 11, 3335 (2020).

[6] Louis Mathieu and Mehdi Mhalla, "Separating pseudo-telepathy games and two-local theories", arXiv:1806.08661.

[7] Andrea Coladangelo, "A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations", arXiv:1904.02350.

[8] Oded Regev and Thomas Vidick, "Bounds on Dimension Reduction in the Nuclear Norm", arXiv:1901.09480.

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