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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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 and Joseph Bowles, "Self-testing of quantum systems: a review", Quantum 4, 337 (2020).

[2] J. William Helton, Hamoon Mousavi, Seyed Sajjad Nezhadi, Vern I. Paulsen, and Travis B. Russell, "Synchronous Values of Games", Annales Henri Poincaré (2024).

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

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

[5] Matthias Christandl, Nicholas Gauguin Houghton-Larsen, and Laura Mancinska, "An Operational Environment for Quantum Self-Testing", Quantum 6, 699 (2022).

[6] Shubhayan Sarkar, "An Operational Notion of Classicality Based on Physical Principles", Foundations of Physics 53 2, 47 (2023).

[7] Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou, "An Operator-Algebraic Formulation of Self-testing", Annales Henri Poincaré (2023).

[8] Zhengfeng Ji, Debbie Leung, and Thomas Vidick, "A three-player coherent state embezzlement game", Quantum 4, 349 (2020).

[9] Salman Beigi, "Separation of quantum, spatial quantum, and approximate quantum correlations", Quantum 5, 389 (2021).

[10] Shubhayan Sarkar, Debashis Saha, Jędrzej Kaniewski, and Remigiusz Augusiak, "Self-testing quantum systems of arbitrary local dimension with minimal number of measurements", npj Quantum Information 7 1, 151 (2021).

[11] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, "MIP*=RE", arXiv:2001.04383, (2020).

[12] Patryk Lipka-Bartosik, Henrik Wilming, and Nelly H. Y. Ng, "Catalysis in Quantum Information Theory", arXiv:2306.00798, (2023).

[13] Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner, and Henrik Wilming, "Embezzling entanglement from quantum fields", arXiv:2401.07292, (2024).

[14] Honghao Fu, Daochen Wang, and Qi Zhao, "Parallel self-testing of EPR pairs under computational assumptions", arXiv:2201.13430, (2022).

[15] Rui Chao and Ben W. Reichardt, "Quantum dimension test using the uncertainty principle", arXiv:2002.12432, (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-25 04:27:45) and SAO/NASA ADS (last updated successfully 2024-05-25 04:27:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

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