Advertisement

Hierarchical axioms for quantum mechanics

  • S. Aravinda
  • Anirban PathakEmail author
  • R. Srikanth
Regular article
  • 13 Downloads
Part of the following topical collections:
  1. Topical Issue: Quantum Correlations

Abstract

The origin of nonclassicality in quantum mechanics (QM) has been investigated recently by a number of authors with a view to identifying axioms that would single out quantum mechanics as a special theory within a broader framework such as convex operational theories. In these studies, the axioms tend to be logically unconnected in the sense that no specific ordering of the axioms is implied. Here, we identify a hierarchy of five nonclassical features that separate QM from a classical theory. By hierarchy is meant an axiomatic scheme where the succeeding axioms can be regarded as superstructure built on top of the structure provided by the preceding axioms. In a sense, the latter are necessary, but not sufficient, for the succeeding axioms. In our scheme, the axioms briefly are: (Q1) incompatibility and uncertainty; (Q2) contextuality; (Q3) entanglement; (Q4) nonlocality and (Q5) indistinguishability of identical particles. Such a hierarchy isn’t obvious when viewed from within the quantum mechanical framework, but, from the perspective of generalized probability theories (GPTs), relevant toy GPTs are introduced at each layer when useful to illustrate the action of the nonclassical features associated with the particular layer.

Graphical abstract

Keywords

Topical issue 

References

  1. 1.
    S. Banerjee, A. Pathak, R. Srikanth, Physically inspired axioms for quantum mechanics (in press)Google Scholar
  2. 2.
    L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995)Google Scholar
  3. 3.
    R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000)Google Scholar
  4. 4.
    U. Leonhardt, Measuring the Quantum State of Light, Cambridge Studies in Modern Optics (Cambridge University Press, 1997)Google Scholar
  5. 5.
    M. Genovese, Phys. Rep. 413, 319 (2005)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    F. Verstraete, J. Dehaene, B. de Moor, J. Mod. Opt. 49, 1277 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    J. Bell, Physics 1, 195 (1964)CrossRefGoogle Scholar
  10. 10.
    J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880 (1969)ADSCrossRefGoogle Scholar
  11. 11.
    C. Brukner, S. Taylor, S. Cheung, V. Vedral, arXiv:quant-ph/0402127v1
  12. 12.
    J. Barrett, Phys. Rev. A 75, 032304 (2007)ADSCrossRefGoogle Scholar
  13. 13.
    J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, D. Roberts, Phys. Rev. A 71, 022101 (2005)ADSCrossRefGoogle Scholar
  14. 14.
    G. Brassard, H. Buhrman, N. Linden, A.A. Méthot, A. Tapp, F. Unger, Phys. Rev. Lett. 96, 250401 (2006)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Broadbent, A.A. Methot, Theor. Comput. Sci. 3, 358 (2006)Google Scholar
  16. 16.
    H. Barnum, J. Barrett, M. Leifer, A. Wilce, Phys. Rev. Lett. 99, 240501 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    H. Barnum, Stud. Hist. Philos. Sci. B 34, 343 (2003)MathSciNetGoogle Scholar
  18. 18.
    G. Chiribella, G.M. D’Ariano, P. Perinotti, Phys. Rev. A 81, 062348 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    P. Janotta, H. Hinrichsen, J. Phys. A: Math. Theor. 47, 323001 (2014)CrossRefGoogle Scholar
  20. 20.
    J. Oppenheim, S. Wehner, Science 330, 1072 (2010)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Banik, M.R. Gazi, S. Ghosh, G. Kar, Phys. Rev. A 87, 052125 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    R.W. Spekkens, Phys. Rev. A 71, 052108 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    R.W. Spekkens, Phys. Rev. A 75, 032110 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    C. Pfister, S. Wehner, Nat. Commun. 4, 1851 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012 (2001)
  26. 26.
    L. Hardy, Stud. Hist. Philos. Sci. B 34, 381 (2003)Google Scholar
  27. 27.
    P. Mana, arXiv:quant-ph/0305117 (2003)
  28. 28.
    M. Navascués, S. Pironio, A. Acn, New J. Phys. 10, 073013 (2008)ADSCrossRefGoogle Scholar
  29. 29.
    R. Clifton, J. Bub, H. Halvorson, Found. Phys. 33, 1561 (2003)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Grinbaum, Br. J. Philos. Sci. 58, 387 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    S. Popescu, D. Rohrlich, Found. Phys. 24, 379 (1994)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    B. Toner, Proc. R. Soc. A 465, 59 (2009)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    H.K. Lo, H.F. Chau, Science 283, 2050 (1999)ADSCrossRefGoogle Scholar
  34. 34.
    W.K. Wootters, W.H. Zurek, Nature 299, 802 (1982)ADSCrossRefGoogle Scholar
  35. 35.
    N. Gisin, Phys. Lett. A 242, 1 (1998)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    P. Busch, T. Heinosaari, J. Schultz, N. Stevens, EPL 103, 10002 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    S. Aravinda, R. Srikanth, A. Pathak, J. Phys. A: Math. Theor. 50, 465303 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    C.H. Bennett, G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore (1984), p. 175Google Scholar
  39. 39.
    S. Aravinda, A. Banerjee, A. Pathak, R. Srikanth, Int. J. Quantum Inf. 12, 1560020 (2014)MathSciNetCrossRefGoogle Scholar
  40. 40.
    I. Csiszar, J. Korner, IEEE Trans. Inf. Theory 24, 339 (1978)CrossRefGoogle Scholar
  41. 41.
    U. Vazirani, T. Vidick, Phys. Rev. Lett. 113, 140501 (2014)ADSCrossRefGoogle Scholar
  42. 42.
    J. Barrett, L. Hardy, A. Kent, Phys. Rev. Lett. 95, 010503 (2005)ADSCrossRefGoogle Scholar
  43. 43.
    S. Kochen, E.P. Specker, J. Math. Mech. 17, 59 (1967)MathSciNetGoogle Scholar
  44. 44.
    A.A. Klyachko, M. Ali Can, S. Binicioğlu, A.S. Shumovsky, Phys. Rev. Lett. 101, 020403 (2008)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    L. Hardy, arXiv:quant-ph/9906123 (1999)
  46. 46.
    R.F. Werner, Phys. Rev. A 40, 4277 (1989)ADSCrossRefGoogle Scholar
  47. 47.
    N. Brunner, N. Gisin, V. Scarani, New J. Phys. 7, 88 (2005)ADSCrossRefGoogle Scholar
  48. 48.
    G. Tóth, A. Acn, Phys. Rev. A 74, 030306 (2006)ADSCrossRefGoogle Scholar
  49. 49.
    M.L. Almeida, S. Pironio, J. Barrett, G. Tóth, A. Acn, Phys. Rev. Lett. 99, 040403 (2007)ADSCrossRefGoogle Scholar
  50. 50.
    R. Augusiak, M. Demianowicz, J. Tura, A. Acn, Phys. Rev. Lett. 115, 030404 (2015)ADSCrossRefGoogle Scholar
  51. 51.
    M.T. Quintino, T. Vértesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acn, N. Brunner, Phys. Rev. A 92, 032107 (2015)ADSCrossRefGoogle Scholar
  52. 52.
    R Srikanth, arXiv:1811.12409 (2018)
  53. 53.
    Ll Masanes, A. Acin, N. Gisin, Phys. Rev. A 73, 012112 (2006)ADSCrossRefGoogle Scholar
  54. 54.
    S. Aaronson, A. Arkhipov, The computational complexity of linear optics, in Proceedings of the forty-third annual ACM symposium on Theory of computing (ACM, 2011), pp. 333–342Google Scholar
  55. 55.
    M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, P. Walther, Nat. Photonics 7, 540 (2013)ADSCrossRefGoogle Scholar
  56. 56.
    H. Wang, W. Li, X. Jiang, Y.-M. He, Y.-H. Li, X. Ding, M.-C. Chen, J. Qin, C.-Z. Peng, C. Schneider, M. Kamp, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, J.P. Dowling, S. Hofling, C.-Y. Lu, J.-W. Pan, Phys. Rev. Lett. 120, 230502 (2018)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsIndian Institute of TechnologyMadras, ChennaiIndia
  2. 2.Jaypee Institute of Information Technology, A10, Sector 62NoidaIndia
  3. 3.Poornaprajna Institute of Scientific ResearchBangaloreIndia

Personalised recommendations