Abstract
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 2Ω(ϵ−1/8). Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick \cite{ji2018three}. 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 \cite{slofstra2019set, dykema2017non, musat2018non} involved representation theoretic machinery for finitely-presented groups and C∗-algebras.
Highlights
A non-local game consists of a one-round interaction between a trusted referee, or verifier, and two or more spatially isolated players, or provers
We focus on the study of non-local games as witnesses of highdimensional entanglement
We focus on dimension witnesses that can certify entanglement of arbitrarily high dimension
Summary
A non-local game consists of a one-round interaction between a trusted referee, or verifier, (who asks questions) and two or more spatially isolated players, or provers (who provide answers). In a subsequent work [Slo18], Slofstra provides a finitely-presented group whose hyperlinear profile is at least subexponential As a corollary, this yields a two-player non-local game, with question and answer sets of finite size, with the property that a 1 − winning probability requires dimension at least 2Ω( −c) to attain for some constant 0 < c < 1. The caveat of such a non-local game is that its description is quite involved and the size of question and answer sets is large It is not clear whether a winning probability of 1 in the game can be attained in the limit of finite-dimensional strategies or not ( it can be attained in the commuting-operator model). We sketch the main ideas in the design of our two-player non-local game
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