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 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.

Self-testing of quantum systems: a review

“…All these proofs rely on the representation theory of C * -algebras. A relatively simpler proof, using embezzling entanglement [vDH03] and selftesting, is presented in [Col20].…”

Separation of quantum, spatial quantum, and approximate quantum correlations

“…This result is an improvement over [6] that shows this separation for larger parameters (n A , n B , m A , m B ) = (5, 6, 3, 3), but is not comparable to [9,14].…”

“…This result then improved in [9] and [14] for the parameters (n A , n B , m A , m B ) = (5, 5, 2, 2). Later, Coladangelo [6] gave a quite simple proof of this separation based on the ideas of self-testing and entanglement embezzlement [21]. Coladangelo's proof of C qs = C qc although simpler, is for the parameters (n A , n B , m A , m B ) = (5,6,3,3) and does not improve the parameters of any of the previous results.…”

An inherently infinite-dimensional quantum correlation

“…Such works have also shown that answering foundational questions about the theory of entanglement is not only of theoretical significance, but brings forward important insights that result in potential applications in quantum information protocols, for example in the certification of high-dimensional entanglement 16,25,26 or high-dimensional states and measurements 27 .…”

“…In a related line of work, Slofstra 12 , and the subsequent [13][14][15][16] , provide non-local games which require arbitrarily highdimensional strategies to attain arbitrarily close to optimal winning probabilities. However, for each of these games, any sequence of ideal strategies approaching the optimal winning probability does not have a well-defined limit, and the optimal correlation cannot be attained exactly (not even in infinite dimensions).…”