Non-closure of the Set of Quantum Correlations via Graphs

  • Ken Dykema
  • Vern I. PaulsenEmail author
  • Jitendra Prakash


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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  1. 1.
    Dykema K., Paulsen V.: Synchronous correlation matrices and Connes’ embedding conjecture. J. Math. Phys. 57, 015214 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dykema K., Paulsen V., Prakash J.: The delta game. Quantum Inf. Comput. 18, 599–616 (2018)MathSciNetGoogle Scholar
  3. 3.
    Fritz T.: Tsirelson’s problem and Kirchberg’s conjecture. Rev. Math. Phys. 24, 1250012 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kim S.-J., Paulsen V.I., Schafhauser C.: A synchronous game for binary constraint systems. J. Math. Phys. 59(3), 032201 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Mančinska L., Roberson D.E.: Quantum homomorphisms. J. Combin. Theory Ser. B 118, 228–267 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Navascués M., Guryanova Y., Hoban M.J., Acín A.: Almost quantum correlations. Nat. Commun. 6, 6288 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    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)ADSCrossRefGoogle Scholar
  10. 10.
    Ozawa N.: About the Connes embedding conjecture: algebraic approaches. Jpn. J. Math. 8, 147–183 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Paulsen V.I., Severini S., Stahlke D., Todorov I.G., Winter A.: Estimating quantum chromatic numbers. J. Funct. Anal. 270, 2188–2222 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Paulsen V.I., Todorov I.G.: Quantum chromatic numbers via operator systems. Q. J. Math. 66, 677–692 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Roberson D.E.: Variations on a theme: Graph homomorphisms, Ph.D. thesis, University of Waterloo (2013)Google Scholar
  14. 14.
    Slofstra, W.: The set of quantum correlations is not closed, preprint, available at arXiv:1703.08618.
  15. 15.
    Tsirelson B.S.: Some results and problems on quantum Bell-type inequalities. Hadron. J. Suppl. 8, 329–345 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Tsirelson, B.S.: Bell inequalities and operator algebras (2006), available at Problem statement for website of open problems at TU Braunschweig

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTexas A & M UniversityCollege StationUSA
  2. 2.Department of Pure Mathematics, Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada

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