Communications in Mathematical Physics

, Volume 279, Issue 2, pp 455–486 | Cite as

Unbounded Violation of Tripartite Bell Inequalities

  • D. Pérez-GarcíaEmail author
  • M. M. Wolf
  • C. Palazuelos
  • I. Villanueva
  • M. Junge
Open Access


We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.


Operator Space Bell Inequality Quantum Information Theory Trilinear Form Tensor Norm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: Restructuring quantum information’s family tree., 2006
  2. 2.
    Acin A., Brunner N., Gisin N., Massar S., Pironio S. and Scarani V. (2007). Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98: 230501 CrossRefADSGoogle Scholar
  3. 3.
    Acin A., Gisin N. and Masanes L. (2006). From Bell’s Theorem to Secure Quantum Key Distribution. Phys. Rev. Lett. 97: 120405 CrossRefADSGoogle Scholar
  4. 4.
    Acin A., Gisin N., Masanes L. and Scarani V. (2004). Bell’s inequalities detect efficient entanglement. Int. J. Quant. Inf. 2: 23 zbMATHCrossRefGoogle Scholar
  5. 5.
    Acín A., Gisin N. and Toner B. (2006). Grothendieck’s constant and local models for noisy entangled quantum states. Phys. Rev. A 73: 062105 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Acin A., Scarani V. and Wolf M.M. (2002). Bell inequalities and distillability in N-quantum-bit systems. Phys. Rev. A 66: 042323 CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Alon N. and Naor A. (2004). Approximating the cut-norm via Grothendieck’s inequality. Proceedings of the 36th Annual ACM Symposium on Theory of Computing. ACM, New York, 72–80 Google Scholar
  8. 8.
    Aspect A., Grangier P. and Roger G. (1981). Experimental Tests of Realistic Local Theories via Bell’s Theorem. Phys. Rev. Lett. 47: 460 CrossRefADSGoogle Scholar
  9. 9.
    Barrett J., Hardy L. and Kent A. (2005). No Signalling and Quantum Key Distribution. Phys. Rev. Lett. 95: 010503 CrossRefADSGoogle Scholar
  10. 10.
    Bell J.S. (1964). On the Einstein-Poldolsky-Rosen paradox. Physics 1: 195 Google Scholar
  11. 11.
    Blei R.C. (1979). Multidimensional extensions of Grothendieck’s inequality and applications. Ark. Mat. 17: 51–68 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bombal F., Pérez-García D. and Villanueva I. (2005). Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55: 441–450 CrossRefGoogle Scholar
  13. 13.
    Brukner C., Zukowski M., Pan J.-W. and Zeilinger A. (2004). Violation of Bell’s inequality: criterion for quantum communication complexity advantage. Phys. Rev. Lett. 92: 127901 CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Buhrman H., Christandl M., Hayden P., Lo H.-K. and Wehner S. (2006). Security of quantum bit string commitment depends on the information measure. Phys. Rev. Lett. 97: 250501 CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Buhrman H., Cleve R. and Dam W.v. (2001). Quantum Entanglement and Communication Complexity. SIAM J.Comput. 30: 1829–1841 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Carando D. (1999). Extendible Polynomials on Banach Spaces. J. Math. Anal. Appl. 233: 359–372 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Carne T.K. (1980). Banach lattices and extensions of Grothendieck’s inequality. J. London Math. Soc. 21(3): 496–516 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Castillo J.M.F., García R. and Jaramillo J.A. (2001). Extension of Bilinear Forms on Banach Spaces. Proc. Amer. Math. Soc. 129(12): 3647–3656 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Clauser J.F., Horne M.A., Shimony A. and Holt R.A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett. 23: 880 CrossRefADSGoogle Scholar
  20. 20.
    Davie A.M. (1973). Quotient algebras of uniform algebras. J. London Math. Soc. 7: 31–40 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Defant A., Daz J.C., Garcia D. and Maestre M. (2001). Unconditional basis and Gordon-Lewis constants for spaces of polynomials. J. Funct. Anal. 181: 119–145 zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Defant A. and Floret K. (1993). Tensor Norms and Operator Ideals. North-Holland, Amsterdom zbMATHGoogle Scholar
  23. 23.
    Deuar P., Munro W.J. and Nemoto K. (2000). Upper Bound on the region of Separable States near the Maximally Mixed State. J. Opt. B: Quantum Semiclass. Opt 2: 225 CrossRefADSGoogle Scholar
  24. 24.
    Devetak I., Junge M., King C. and Ruskai M.B. (2006). Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266: 37–63 zbMATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Diestel J., Jarchow H. and Tonge A. (1995). Absolutely Summing Operators. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  26. 26.
    Effros, E.G., Ruan, Z.-J.: Operator Spaces. London Math. Soc. Monographs New Series, Oxford: Clarendon Press, 2000Google Scholar
  27. 27.
    Einstein A., Podolsky B. and Rosen N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. Phys. Rev 47: 777 zbMATHCrossRefADSGoogle Scholar
  28. 28.
    Ekert A. (1991). Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67: 661 zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Fine A. (1982). Hidden Variables, Joint Probability and the Bell Inequalities. Phys. Rev. Lett. 48: 291 CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Floret K. and Hunfeld S. (2001). Ultratability of ideals of homogeneous polynomials and multilinear mappings. Proc. Amer. Math. Soc. 130: 1425–1435 CrossRefMathSciNetGoogle Scholar
  31. 31.
    Gordon Y. and Lewis D.R. (1974). Absolutely summing operators and local unconditional structures. Acta Math. 133: 27–48 zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Gross D., Audenaert K. and Eisert J. (2007). Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48: 052104 CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Grothendieck A. (1953). Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. SO Paulo 8: 1–79 MathSciNetGoogle Scholar
  34. 34.
    Gurvits L. and Barnum H. (2003). Separable balls around the maximally mixed multipartite quantum states. Phys. Rev. A 68: 042312 CrossRefADSGoogle Scholar
  35. 35.
    Hayden, P.: The maximal p-norm multiplicativity conjecture is false., 2007
  36. 36.
    Hayden P., Leung D.W. and Winter A. (2006). Aspects of generic entanglement. Commun. Math. Phys. 265(1): 95–117 zbMATHCrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Jarchow H., Palazuelos C., Pérez-García D. and Villanueva I. (2007). Hahn–Banach extension of multilinear forms and summability. J. Math. Anal. Appl. 336: 1161–1177 zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Junge, M.: Factorization theory for Spaces of Operators. Habilitationsschrift Kiel, 1996; see also: Preprint server of the university of southern Denmark 1999, IMADA preprint: PP-1999-02, 1999Google Scholar
  39. 39.
    Kirwan P. and Ryan R.A. (1998). Extendibility of Homogeneous Polynomials on Banach Spaces. Proc. Amer. Math. Soc. 126(4): 1023–1029 zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Kumar A. and Sinclair A. (1998). Equivalence of norms on operator space tensor products of C*-algebras. Trans. Amer. Math. Soc. 350(5): 2033–2048 zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Laskowski W., Paterek T., Zukowski M. and Brukner C. (2004). Tight Multipartite Bell’s Inequalities Involving Many Measurement Settings. Phys. Rev. Lett. 93: 200401 CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Le Merdy C. (1998). The Schatten space S 4 is a Q-algebra. Proc. Amer. Math. Soc. 126(3): 715–719 zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Ledoux M. and Talagrand M. (1991). Probability in Banach Spaces. Springer-Verlag, Berlin-Heidelberg-New York zbMATHGoogle Scholar
  44. 44.
    Lust-Picard F. and Pisier G. (1991). Noncommutative Khintchine and Paley inequalities. Ark. Mat. 29(2): 241–260 CrossRefMathSciNetGoogle Scholar
  45. 45.
    Marcus, M.B., Pisier, G.: Random Fourier Series with Applications to Harmonic Analysis. Annals of Math. Studies 101, Princeton NJ: Princeton Univ. Press, 1981Google Scholar
  46. 46.
    Masanes, L.: Extremal quantum correlations for N parties with two dichotomic observables per site., 2005
  47. 47.
    Masanes Ll., Acin A. and Gisin N. (2006). General properties of Nonsignaling Theories. Phys. Rev. A. 73: 012112 CrossRefADSGoogle Scholar
  48. 48.
    Masanes, L., Winter, A.: Unconditional security of key distribution from causality constraints., 2006
  49. 49.
    Mermin N.D. (1990). Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65: 1838 zbMATHCrossRefADSMathSciNetGoogle Scholar
  50. 50.
    Montanaro A. (2007). On the distinguishability of random quantum states. Commun. Math. Phys. 273: 619–636 CrossRefADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    Munoz G.A., Sarantopoulos Y. and Tonge A. (1999). Complexifications of real Banach spaces, polynomials and multilinear maps. Studia Math 134: 1–33 zbMATHMathSciNetGoogle Scholar
  52. 52.
    Nagata K., Laskowski W. and Paterek T. (2006). Bell inequality with an arbitrary number of settings and its applications. Phys. Rev. A 74: 062109 CrossRefADSGoogle Scholar
  53. 53.
    Pérez-García D. (2006). The trace class is a Q-algebra. Ann. Acad. Sci. Fenn. Math. 31(2): 287–295 zbMATHMathSciNetGoogle Scholar
  54. 54.
    Pérez-García D. (2004). Deciding separability with a fixed error. Phys. Lett. A. 330: 149 CrossRefADSzbMATHGoogle Scholar
  55. 55.
    Pérez-García D. (2004). A counterexample using 4-linear forms. Bull. Austral. Math. Soc 70(3): 469–473 zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Pérez-García D. and Villanueva I. (2003). Multiple summing operators on Banach spaces. J. Math. Anal. Appl. 285: 86–96 zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Pietsch, A.: Proceedings of the Second International Conference on Operator Algebras, Ideals and their Applications in Theoretical Physics (Leipzig), Stuttgart: Teubner-Texte, pp. 185–199 (1983)Google Scholar
  58. 58.
    Pisier, G.: An Introduction to Operator Spaces, London Math. Soc. Lecture Notes Series 294, Cambridge: Cambridge University Press, Cambridge, 2003Google Scholar
  59. 59.
    Pitowsky, I.: Correlation polytopes: their geometry and complexity. Math. Programming 50(3), (Ser. A) 395–414 (1991)Google Scholar
  60. 60.
    Rowe M. (2001). Experimental violation of a Bell’s inequality with efficient detection. Nature 409: 791 CrossRefGoogle Scholar
  61. 61.
    Ruan Z.-J. (1988). Subspaces of C*-algebras. J. Funct. Anal. 76(1): 217–230 zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Rudolph O. (2000). A separability criterion for density operators. J. Phys. A: Math. Gen. 33: 3951 zbMATHCrossRefADSMathSciNetGoogle Scholar
  63. 63.
    Rungta, P., Munro, W.J., Nemoto, K., Deuar, P., Milburn, G.J., Caves, C.M.: Qudit Entanglement., 2000
  64. 64.
    Ryan R.A. (2002). An introduction to Tensor Products of Banach spaces. Springer-Verlag, Berlin-Heidelberg-New York Google Scholar
  65. 65.
    Scarani V., Gisin N., Brunner N., Masanes L., Pino S. and Acin A. (2006). Secrecy extraction from no-signalling correlations. Phys. Rev. A 74: 042339 CrossRefADSGoogle Scholar
  66. 66.
    Shor P.W. (2004). Equivalence of Additivity Questions in Quantum Information Theory. Commun. Math. Phys. 246: 453–472 zbMATHCrossRefADSMathSciNetGoogle Scholar
  67. 67.
    Tomczak-Jaegermann, N.: Banach-Mazur Distances and Finite Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics 38, London: Longman Scientific and Technical, 1989Google Scholar
  68. 68.
    Toner B.F. and Bacon D. (2003). The Communication Cost of Simulating Bell Correlations. Phys. Rev. Lett. 91: 187904 CrossRefADSMathSciNetGoogle Scholar
  69. 69.
    Tonge A.M. (1978). The Von Neumann inequality for polynomials in several Hilbert-Schmidt operators. J. London Math. Soc. 18: 519–526 zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Tsirelson B.S. (1993). Some results and problems on quantum Bell-type inequalities. Hadronic J. Supp. 8(4): 329–345 zbMATHMathSciNetGoogle Scholar
  71. 71.
    Varopoulos N.T. (1975). A theorem on operator algebras. Math. Scand. 37(1): 173–182 zbMATHMathSciNetGoogle Scholar
  72. 72.
    Verstraete F. and Wolf M.M. (2002). Entanglement versus Bell Violations and Their Behavior under Local Filtering Operations. Phys. Rev. Lett. 89: 170401 CrossRefADSGoogle Scholar
  73. 73.
    Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1. Providence, RI: Amer. Math. Soc. 1992Google Scholar
  74. 74.
    Walther P., Aspelmeyer M., Resch K.J. and Zeilinger A. (2005). Experimental violation of a cluster state Bell inequality. Phys. Rev. Lett. 95: 020403 CrossRefADSGoogle Scholar
  75. 75.
    Wassermann S. (1976). On tensor products of certain group C*-algebras. J. Funct. Anal. 23: 239–254 zbMATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Werner R.F. and Wolf M.M. (2001). Bell inequalities and Entanglement. Quant. Inf. Comp. 1(3): 1–25 MathSciNetzbMATHGoogle Scholar
  77. 77.
    Werner R.F. (1989). Quantum states with Einstein-Rosen-Podolsky correlations admitting a hidden- variable model. Phys. Rev. A 40: 4277 CrossRefADSGoogle Scholar
  78. 78.
    Werner, R.F.: Quantum Information Theory - an Invitation., 2001
  79. 79.
    Werner R.F. and Wolf M.M. (2001). All multipartite Bell correlation inequalities for two dichotomic observables per site. Phys. Rev. A 64: 032112 CrossRefADSGoogle Scholar
  80. 80.
    Winter, A.: The maximum output p-norm of quantum channels is not multiplicative for any pi2., 2007
  81. 81.
    Zukowski M. (1993). Bell theorem involving all settings of measuring apparatus. Phys. Lett. A 177: 290 CrossRefADSMathSciNetGoogle Scholar
  82. 82.
    Zukowski, M.: All tight multipartite Bell correlation inequalities for three dichotomic observables per observer., 2006
  83. 83.
    Zukowski M. and Brukner C. (2002). Bell’s Theorem for General N-Qubit States. Phys. Rev. Lett. 88: 210401 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • D. Pérez-García
    • 1
    Email author
  • M. M. Wolf
    • 2
  • C. Palazuelos
    • 1
  • I. Villanueva
    • 1
  • M. Junge
    • 3
  1. 1.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain
  2. 2.Max Planck Institut für QuantenoptikGarchingGermany
  3. 3.Department of MathematicsUniversity of Illinois at Urbana-ChampaignIllinoisUSA

Personalised recommendations