Unbounded Violation of Tripartite Bell Inequalities

Abstract

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.

References

  1. 1

    Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: Restructuring quantum information’s family tree. http://arXiv.org/list/quant-ph/0606225, 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

    Article  ADS  Google 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

    Article  ADS  Google 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

    MATH  Article  Google 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

    Article  ADS  MathSciNet  Google 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

    Article  ADS  MathSciNet  Google 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

    Article  ADS  Google Scholar 

  9. 9

    Barrett J., Hardy L. and Kent A. (2005). No Signalling and Quantum Key Distribution. Phys. Rev. Lett. 95: 010503

    Article  ADS  Google 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

    MATH  Article  MathSciNet  Google Scholar 

  12. 12

    Bombal F., Pérez-García D. and Villanueva I. (2005). Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55: 441–450

    Article  Google 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

    Article  ADS  MathSciNet  Google 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

    Article  ADS  MathSciNet  Google Scholar 

  15. 15

    Buhrman H., Cleve R. and Dam W.v. (2001). Quantum Entanglement and Communication Complexity. SIAM J.Comput. 30: 1829–1841

    MATH  Article  MathSciNet  Google Scholar 

  16. 16

    Carando D. (1999). Extendible Polynomials on Banach Spaces. J. Math. Anal. Appl. 233: 359–372

    MATH  Article  MathSciNet  Google Scholar 

  17. 17

    Carne T.K. (1980). Banach lattices and extensions of Grothendieck’s inequality. J. London Math. Soc. 21(3): 496–516

    MATH  Article  MathSciNet  Google 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

    MATH  Article  MathSciNet  Google 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

    Article  ADS  Google Scholar 

  20. 20

    Davie A.M. (1973). Quotient algebras of uniform algebras. J. London Math. Soc. 7: 31–40

    MATH  Article  MathSciNet  Google 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

    MATH  Article  MathSciNet  Google Scholar 

  22. 22

    Defant A. and Floret K. (1993). Tensor Norms and Operator Ideals. North-Holland, Amsterdom

    MATH  Google 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

    Article  ADS  Google 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

    MATH  Article  ADS  MathSciNet  Google Scholar 

  25. 25

    Diestel J., Jarchow H. and Tonge A. (1995). Absolutely Summing Operators. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  26. 26

    Effros, E.G., Ruan, Z.-J.: Operator Spaces. London Math. Soc. Monographs New Series, Oxford: Clarendon Press, 2000

  27. 27

    Einstein A., Podolsky B. and Rosen N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. Phys. Rev 47: 777

    MATH  Article  ADS  Google Scholar 

  28. 28

    Ekert A. (1991). Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67: 661

    MATH  Article  ADS  MathSciNet  Google Scholar 

  29. 29

    Fine A. (1982). Hidden Variables, Joint Probability and the Bell Inequalities. Phys. Rev. Lett. 48: 291

    Article  ADS  MathSciNet  Google 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

    Article  MathSciNet  Google Scholar 

  31. 31

    Gordon Y. and Lewis D.R. (1974). Absolutely summing operators and local unconditional structures. Acta Math. 133: 27–48

    MATH  Article  MathSciNet  Google 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

    Article  ADS  MathSciNet  Google Scholar 

  33. 33

    Grothendieck A. (1953). Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. SO Paulo 8: 1–79

    MathSciNet  Google Scholar 

  34. 34

    Gurvits L. and Barnum H. (2003). Separable balls around the maximally mixed multipartite quantum states. Phys. Rev. A 68: 042312

    Article  ADS  Google Scholar 

  35. 35

    Hayden, P.: The maximal p-norm multiplicativity conjecture is false. http://arXiv.org/abs/0707.3291, 2007

  36. 36

    Hayden P., Leung D.W. and Winter A. (2006). Aspects of generic entanglement. Commun. Math. Phys. 265(1): 95–117

    MATH  Article  ADS  MathSciNet  Google 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

    MATH  Article  MathSciNet  Google 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, 1999

  39. 39

    Kirwan P. and Ryan R.A. (1998). Extendibility of Homogeneous Polynomials on Banach Spaces. Proc. Amer. Math. Soc. 126(4): 1023–1029

    MATH  Article  MathSciNet  Google 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

    MATH  Article  MathSciNet  Google 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

    Article  ADS  MathSciNet  Google Scholar 

  42. 42

    Le Merdy C. (1998). The Schatten space S 4 is a Q-algebra. Proc. Amer. Math. Soc. 126(3): 715–719

    MATH  Article  MathSciNet  Google Scholar 

  43. 43

    Ledoux M. and Talagrand M. (1991). Probability in Banach Spaces. Springer-Verlag, Berlin-Heidelberg-New York

    MATH  Google Scholar 

  44. 44

    Lust-Picard F. and Pisier G. (1991). Noncommutative Khintchine and Paley inequalities. Ark. Mat. 29(2): 241–260

    Article  MathSciNet  Google 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, 1981

  46. 46

    Masanes, L.: Extremal quantum correlations for N parties with two dichotomic observables per site. http://arXiv.org/list/quant-ph/0512100, 2005

  47. 47

    Masanes Ll., Acin A. and Gisin N. (2006). General properties of Nonsignaling Theories. Phys. Rev. A. 73: 012112

    Article  ADS  Google Scholar 

  48. 48

    Masanes, L., Winter, A.: Unconditional security of key distribution from causality constraints. http://arXiv.org/list/quant-ph/0606049, 2006

  49. 49

    Mermin N.D. (1990). Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65: 1838

    MATH  Article  ADS  MathSciNet  Google Scholar 

  50. 50

    Montanaro A. (2007). On the distinguishability of random quantum states. Commun. Math. Phys. 273: 619–636

    Article  ADS  MathSciNet  MATH  Google 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

    MATH  MathSciNet  Google 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

    Article  ADS  Google Scholar 

  53. 53

    Pérez-García D. (2006). The trace class is a Q-algebra. Ann. Acad. Sci. Fenn. Math. 31(2): 287–295

    MATH  MathSciNet  Google Scholar 

  54. 54

    Pérez-García D. (2004). Deciding separability with a fixed error. Phys. Lett. A. 330: 149

    Article  ADS  MATH  Google Scholar 

  55. 55

    Pérez-García D. (2004). A counterexample using 4-linear forms. Bull. Austral. Math. Soc 70(3): 469–473

    MATH  MathSciNet  Article  Google Scholar 

  56. 56

    Pérez-García D. and Villanueva I. (2003). Multiple summing operators on Banach spaces. J. Math. Anal. Appl. 285: 86–96

    MATH  Article  MathSciNet  Google 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)

  58. 58

    Pisier, G.: An Introduction to Operator Spaces, London Math. Soc. Lecture Notes Series 294, Cambridge: Cambridge University Press, Cambridge, 2003

  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

    Article  Google Scholar 

  61. 61

    Ruan Z.-J. (1988). Subspaces of C*-algebras. J. Funct. Anal. 76(1): 217–230

    MATH  Article  MathSciNet  Google Scholar 

  62. 62

    Rudolph O. (2000). A separability criterion for density operators. J. Phys. A: Math. Gen. 33: 3951

    MATH  Article  ADS  MathSciNet  Google Scholar 

  63. 63

    Rungta, P., Munro, W.J., Nemoto, K., Deuar, P., Milburn, G.J., Caves, C.M.: Qudit Entanglement. http://arXiv.org/list/quant-ph/0001075, 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

    Article  ADS  Google Scholar 

  66. 66

    Shor P.W. (2004). Equivalence of Additivity Questions in Quantum Information Theory. Commun. Math. Phys. 246: 453–472

    MATH  Article  ADS  MathSciNet  Google 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, 1989

  68. 68

    Toner B.F. and Bacon D. (2003). The Communication Cost of Simulating Bell Correlations. Phys. Rev. Lett. 91: 187904

    Article  ADS  MathSciNet  Google 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

    MATH  Article  MathSciNet  Google Scholar 

  70. 70

    Tsirelson B.S. (1993). Some results and problems on quantum Bell-type inequalities. Hadronic J. Supp. 8(4): 329–345

    MATH  MathSciNet  Google Scholar 

  71. 71

    Varopoulos N.T. (1975). A theorem on operator algebras. Math. Scand. 37(1): 173–182

    MATH  MathSciNet  Google 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

    Article  ADS  Google 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. 1992

  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

    Article  ADS  Google Scholar 

  75. 75

    Wassermann S. (1976). On tensor products of certain group C*-algebras. J. Funct. Anal. 23: 239–254

    MATH  Article  MathSciNet  Google Scholar 

  76. 76

    Werner R.F. and Wolf M.M. (2001). Bell inequalities and Entanglement. Quant. Inf. Comp. 1(3): 1–25

    MathSciNet  MATH  Google Scholar 

  77. 77

    Werner R.F. (1989). Quantum states with Einstein-Rosen-Podolsky correlations admitting a hidden- variable model. Phys. Rev. A 40: 4277

    Article  ADS  Google Scholar 

  78. 78

    Werner, R.F.: Quantum Information Theory - an Invitation. http://arXiv.org/list/quant-ph/0101061, 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

    Article  ADS  Google Scholar 

  80. 80

    Winter, A.: The maximum output p-norm of quantum channels is not multiplicative for any pi2. http://arXiv.org/abs/arXiv:0707.0402, 2007

  81. 81

    Zukowski M. (1993). Bell theorem involving all settings of measuring apparatus. Phys. Lett. A 177: 290

    Article  ADS  MathSciNet  Google Scholar 

  82. 82

    Zukowski, M.: All tight multipartite Bell correlation inequalities for three dichotomic observables per observer. http://arXiv.org/list/quant-ph/0611086, 2006

  83. 83

    Zukowski M. and Brukner C. (2002). Bell’s Theorem for General N-Qubit States. Phys. Rev. Lett. 88: 210401

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to D. Pérez-García.

Additional information

Communicated by M.B. Ruskai

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Pérez-García, D., Wolf, M.M., Palazuelos, C. et al. Unbounded Violation of Tripartite Bell Inequalities. Commun. Math. Phys. 279, 455–486 (2008). https://doi.org/10.1007/s00220-008-0418-4

Download citation

Keywords

  • Operator Space
  • Bell Inequality
  • Quantum Information Theory
  • Trilinear Form
  • Tensor Norm