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

Andrea Coladangelo

Computing and Mathematical Sciences, Caltech

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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 [17]. 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 [26,16,19] involved representation theoretic machinery for finitely-presented groups and $C^*$-algebras.

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Cited by

[1] Ivan Šupić, "Nonlocality strikes again", Quantum Views 4, 38 (2020).

[2] Andrea Coladangelo and Jalex Stark, "An inherently infinite-dimensional quantum correlation", Nature Communications 11 1, 3335 (2020).

[3] Ivan Šupić and Joseph Bowles, "Self-testing of quantum systems: a review", arXiv:1904.10042.

[4] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, "MIP*=RE", arXiv:2001.04383.

[5] Zhengfeng Ji, Debbie Leung, and Thomas Vidick, "A three-player coherent state embezzlement game", arXiv:1802.04926.

[6] Shubhayan Sarkar, Debashis Saha, Jędrzej Kaniewski, and Remigiusz Augusiak, "Self-testing quantum systems of arbitrary local dimension with minimal number of measurements", arXiv:1909.12722.

[7] Rui Chao and Ben W. Reichardt, "Quantum dimension test using the uncertainty principle", arXiv:2002.12432.

[8] Salman Beigi, "Separation of quantum, spatial quantum, and approximate quantum correlations", arXiv:2004.11103.

The above citations are from Crossref's cited-by service (last updated successfully 2020-07-13 20:56:43) and SAO/NASA ADS (last updated successfully 2020-07-13 20:56:44). The list may be incomplete as not all publishers provide suitable and complete citation data.

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