Abstract
We prove that the set of quantum correlations for a bipartite system of 5 inputs and 2 outputs is not closed. Our proof relies on computing the correlation functions of a graph, which is a concept that we introduce.
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Dykema K., Paulsen V.: Synchronous correlation matrices and Connes’ embedding conjecture. J. Math. Phys. 57, 015214 (2016)
Dykema K., Paulsen V., Prakash J.: The delta game. Quantum Inf. Comput. 18, 599–616 (2018)
Fritz T.: Tsirelson’s problem and Kirchberg’s conjecture. Rev. Math. Phys. 24, 1250012 (2012)
Junge M., Navascues M., Palazuelos C., Perez-Garcia D., Scholz V.B., Werner R.F.: Connes embedding problem and Tsirelson’s problem. J. Math. Phys. 52, 012102 (2011)
Kim S.-J., Paulsen V.I., Schafhauser C.: A synchronous game for binary constraint systems. J. Math. Phys. 59(3), 032201 (2018)
Kruglyak, S.A., Rabanovich, V.I., Samoĭlenko, Yu.S.: On sums of projections, Funktsional. Anal. i Prilozhen. 36(3), 20–35, 96 (2002) (Russian, with Russian summary); English transl., Funct. Anal. Appl. 36(3), 182–195 (2002)
Mančinska L., Roberson D.E.: Quantum homomorphisms. J. Combin. Theory Ser. B 118, 228–267 (2016)
Navascués M., Guryanova Y., Hoban M.J., Acín A.: Almost quantum correlations. Nat. Commun. 6, 6288 (2015)
Navascués M., Pironio S., Acín A.: A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys. 10(7), 073013 (2008)
Ozawa N.: About the Connes embedding conjecture: algebraic approaches. Jpn. J. Math. 8, 147–183 (2013)
Paulsen V.I., Severini S., Stahlke D., Todorov I.G., Winter A.: Estimating quantum chromatic numbers. J. Funct. Anal. 270, 2188–2222 (2016)
Paulsen V.I., Todorov I.G.: Quantum chromatic numbers via operator systems. Q. J. Math. 66, 677–692 (2015)
Roberson D.E.: Variations on a theme: Graph homomorphisms, Ph.D. thesis, University of Waterloo (2013)
Slofstra, W.: The set of quantum correlations is not closed, preprint, available at arXiv:1703.08618.
Tsirelson B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Suppl. 8, 329–345 (1993)
Tsirelson, B.S.: Bell inequalities and operator algebras (2006), available at http://web.archive.org/web/20090414083019/http://www.imaph.tu-bs.de/qi/problems/33.html. Problem statement for website of open problems at TU Braunschweig
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Communicated by M. M. Wolf
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This work was supported by a grant from the Simons Foundation/SFARI (524187, K.D.). Supported in part by NSERC.
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Dykema, K., Paulsen, V.I. & Prakash, J. Non-closure of the Set of Quantum Correlations via Graphs. Commun. Math. Phys. 365, 1125–1142 (2019). https://doi.org/10.1007/s00220-019-03301-1
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DOI: https://doi.org/10.1007/s00220-019-03301-1