Quantum Information Processing

, Volume 15, Issue 7, pp 2993–3004 | Cite as

Temporal correlations and device-independent randomness



Leggett–Garg inequalities (LGI) are constraints on certain combinations of temporal correlations obtained by measuring one and the same system at two different instants of time. The usual derivations of LGI assume macroscopic realism per se and noninvasive measurability. We derive these inequalities under a different set of assumptions, namely the assumptions of predictability and no signaling in time (NSIT). As a novel implication of this derivation, we find application of LGI in randomness certification. It turns out that randomness can be certified from temporal correlations, even without knowing the details of the experimental devices, provided the observed correlations violate LGI but satisfy NSIT.


Temporal correlation Leggett–Garg inequality Device-independent randomness 



S.K.C. acknowledges fruitful discussions with Pankaj Agrawal. We also thank Guruprasad Kar for stimulating discussions. MB thankfully acknowledges comments from C. Brukner, C. Emary and A. J. Leggett. S.K.C. acknowledges support from the Council of Scientific and Industrial Research, Government of India (Scientists’ Pool Scheme). SM acknowledges support from the DST project SR/S2/PU-16/2007.


  1. 1.
    Leggett, A.J., Garg, A.: Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett. 54, 857 (1985)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Benatti, F., Ghirardi, G., Grassi, R.: Testing macroscopic quantum coherence. II Nuovo Cimento B 110, 593–610 (1995)ADSCrossRefGoogle Scholar
  3. 3.
    Leggett, A.J.: Realism and the physical world. Rep. Prog. Phys. 71, 022001 (2008)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Kofler, J., Brukner, C.: Conditions for quantum violation of macroscopic realism. Phys. Rev. Lett. 101, 090403 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    Kofler, J., Brukner, C.: Condition for macroscopic realism beyond the Leggett–Garg inequalities. Phys. Rev. A 87, 052115 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Ballentine, L.E.: Realism and quantum flux tunneling. Phys. Rev. Lett. 59, 1493 (1987)ADSCrossRefGoogle Scholar
  7. 7.
    Cliffton, R.: Symposium on the Foundations of Modern Physics. World Scientific, Singapore (1990)Google Scholar
  8. 8.
    Foster, S., Elby, A.: A SQUID no-go theorem without macrorealism: What SQUID’s really tell us about nature. Found. Phys. 21, 773 (1991)ADSCrossRefGoogle Scholar
  9. 9.
    Elby, A., Foster, S.: Why SQUID experiments can rule out non-invasive measurability. Phys. Lett. A 166, 17–23 (1992)ADSCrossRefGoogle Scholar
  10. 10.
    Jordan, A., Korotkov, A., Buttiker, M.: Leggett–Garg inequality with a kicked quantum pump. Phys. Rev. Lett. 97, 026805 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    Williams, N.S., Jordan, A.N.: Weak values and the Leggett–Garg inequality in solid-state qubits. Phys. Rev. Lett. 100, 026804 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    Palacios-Laloy, A., et al.: Experimental violation of a Bells inequality in time with weak measurement. Nat. Phys. 6, 442 (2010)CrossRefGoogle Scholar
  13. 13.
    Dressel, J., Broadbent, C., Howell, J., Jordan, A.: Experimental violation of two-party Leggett–Garg inequalities with semiweak measurements. Phys. Rev. Lett. 106, 040402 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Goggin, M.E., et al.: Violation of the Leggett–Garg inequality with weak measurements of photons. Proc. Natl. Acad. Sci. USA 108, 1256–1261 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    Souza, A.M., Oliveira, I.S., Sarthour, R.S.: A scattering quantum circuit for measuring Bell’s time inequality: a nuclear magnetic resonance demonstration using maximally mixed states. New J. Phys. 13, 053023 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Athalye, V., Roy, S.S., Mahesh, T.S.: Investigation of the Leggett–Garg inequality for precessing nuclear spins. Phys. Rev. Lett. 107, 130402 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Knee, G.C., et al.: Violation of a Leggett–Garg inequality with ideal non-invasive measurements. Nat. Commun. 3, 606 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    Wilde, M., Mizel, A.: Addressing the clumsiness loophole in a Leggett–Garg test of macrorealism. Found. Phys. 42, 256–265 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Suzuki, Y., Iinuma, M., Hofmann, H.F.: Violation of Leggett–Garg inequalities in quantum measurements with variable resolution and back-action. New J. Phys. 14, 103022 (2012)CrossRefGoogle Scholar
  20. 20.
    Devi, Usha, Karthik, A.R., Sudha, H.S., Rajagopal, A.K.: Macrorealism from entropic Leggett–Garg inequalities. Phys. Rev. A 87, 052103 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    Groen, J.P., et al.: Partial-measurement backaction and nonclassical weak values in a superconducting circuit. Phys. Rev. Lett. 111, 090506 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    Emary, C., Lambert, N., Nori, F.: Leggett–Garg inequalities. Rep. Prog. Phys. 77, 016001 (2014)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Leggett, A.J., Garg, A.: Comment on realism and quantum flux tunneling. Phys. Rev. Lett. 59, 1621 (1987)ADSCrossRefGoogle Scholar
  24. 24.
    Leggett, A.J.: Experimental approaches to the quantum measurement paradox. Found Phys. 18, 939–952 (1988)ADSCrossRefGoogle Scholar
  25. 25.
    Maroney, O. J. E.: Detectability, Invasiveness and the Quantum Three Box Paradox. arXiv:1207.3114
  26. 26.
    Leggett, A.J.: (Festschrift for David Bohm) (1987), eds. B. J. Hiley and D. Peat (Routledge and Kegan Paul)Google Scholar
  27. 27.
    Wilde, M., Mizel, A.: Addressing the clumsiness loophole in a Leggett–Garg Test of macrorealism. Found. Phys. 42, 256 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Foster, S., Elby, A.: A SQUID No-Go theorem without macrorealism: What SQUID’s really tell us about nature. Found. Phys. 21, 773 (1991)ADSCrossRefGoogle Scholar
  29. 29.
    Benatti, F., Ghirardi, G.C., Grassi, R.: On some recent proposals for testing macrorealism versus quantum mechanics. Found. Phys. Lett. 7, 105–126 (1994)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Cavalcanti, E.G., Wiseman, H.M.: Bell nonlocality, signal locality and unpredictability (or what Bohr could have told einstein at Solvay had he known about Bell experiments). Found. Phys. 42, 1329–1338 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Knuth, D.: The Art of Computer Programming. Semi-numerical Algorithms, vol. 2. Addison-Wesley, Boston (1981)Google Scholar
  32. 32.
    Jennewein, T., Achleitner, U., Weihs, G., Weinfurter, H., Zeilinger, A.: A fast and compact quantum random number generator. Rev. Sci. Instrum. 71, 1675 (2000)ADSCrossRefGoogle Scholar
  33. 33.
    Stefanov, A., Gisin, N., Guinnard, O., Guinnard, L., Zbinden, H.: Optical quantum random number generator. J. Mod. Opt. 47, 595–598 (2000)ADSGoogle Scholar
  34. 34.
    Atsushi, U., et al.: Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photon. 2, 728–732 (2008)CrossRefGoogle Scholar
  35. 35.
    Pironio, S., et al.: Random numbers certified by Bells theorem. Nature (London) 464, 1021–1024 (2010)ADSCrossRefGoogle Scholar
  36. 36.
    Acin, A., Massar, S., Pironio, S.: Randomness versus nonlocality and entanglement. Phys. Rev. Lett. 108, 100402 (2012)ADSCrossRefGoogle Scholar
  37. 37.
    Dhara, C., Torre, G., Acin, A.: Can observed randomness be certified to be fully intrinsic? Phys. Rev. Lett. 112, 100402 (2014)ADSCrossRefGoogle Scholar
  38. 38.
    Torre, G., Hoban, M.J., Dhara, C., Prettico, G., Acin, A.: Maximally nonlocal theories cannot be maximally random. arXiv:1403.3357 (2014)
  39. 39.
    Scarani, V.: The device-independent outlook on quantum physics (Lecture notes on the power of Bell’s theorem). arXiv:1303.3081
  40. 40.
    Chaturvedi, A., Banik, M.: Measurement-device-independent randomness from local entangled states. arXiv:1401.1338
  41. 41.
    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Fond. Phys. 24, 379–385 (1994)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Spekkens, R.W.: Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A 71, 052108 (2005)ADSCrossRefGoogle Scholar
  43. 43.
    Harrigan, N., Rudolph, T.: Ontological models and the interpretation of contextuality. arXiv:0709.4266
  44. 44.
    Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)ADSCrossRefGoogle Scholar
  45. 45.
    Wigner, E.P.: On hidden variables and quantum mechanical probabilities. Am. J. Phys. 38, 1005 (1970)ADSCrossRefGoogle Scholar
  46. 46.
    Yearsley, J. M.: The Leggett–Garg inequalities and non-invasive measurability. arXiv:1310.2149
  47. 47.
    Barbieri, M.: Multiple-measurement Leggett–Garg inequalities. Phys. Rev. A 80, 034102 (2009)ADSCrossRefGoogle Scholar
  48. 48.
    Koenig, R., Renner, R., Schaffner, C.: The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55, 4337 (2009)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964)Google Scholar
  50. 50.
    Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014)ADSCrossRefGoogle Scholar
  52. 52.
    Kochen, S., Specker, E.: The problem of hidden of variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)MathSciNetMATHGoogle Scholar
  53. 53.
    Cirelson, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93–100 (1980)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Budroni, C., Moroder, T., Kleinmann, M., Gühne, O.: Bounding temporal quantum correlations. Phys. Rev. Lett. 111, 020403 (2013)ADSCrossRefGoogle Scholar
  55. 55.
    Budroni, C., Emary, C.: Temporal quantum correlations and Leggett–Garg inequalities in multilevel systems. Phys. Rev. Lett. 113, 050401 (2014)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Shiladitya Mal
    • 1
  • Manik Banik
    • 2
    • 3
  • Sujit K. Choudhary
    • 4
  1. 1.S. N. Bose National Centre for Basic SciencesSalt Lake, KolkataIndia
  2. 2.Optics and Quantum Information GroupThe Institute of Mathematical SciencesChennaiIndia
  3. 3.Physics and Applied Mathematics UnitIndian Statistical InstituteKolkataIndia
  4. 4.Institute of PhysicsBhubaneswarIndia

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