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

We describe a two-player non-local game, with a fixed small number of questions and answers, such that an $ϵ$-close to optimal strategy requires an entangled state of dimension $2^{\mathrm{\Omega }({ϵ}^{-1\mathrm{/}8})}$. 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.