A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

Abstract

We describe a two-player non-local game, with a fixed small number of questions and answers, such that an $\epsilon$-close to optimal strategy requires an entangled state of dimension $2^{\Omega ({\epsilon }^{-1/8})}$. Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick [arXiv:1802.04926]. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs involved representation theoretic machinery for finitely-presented groups and $C^{\ast }$-algebras.

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Cited by in corpus (11)

• Self-testing of quantum systems: a review
• Relativistic Quantum Fields Are Universal Entanglement Embezzlers
• Self-testing quantum systems of arbitrary local dimension with minimal number of measurements
• Separation of quantum, spatial quantum, and approximate quantum correlations
• An Operational Environment for Quantum Self-Testing
• An operator-algebraic formulation of self-testing
• A bound on the quantum value of all compiled nonlocal games
• Multipartite Embezzlement of Entanglement
• A three-player coherent state embezzlement game
• An operational notion of classicality based on physical principles
• The membership problem for constant-sized quantum correlations is undecidable