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Weak Values: The Progression from Quantum Foundations to Tool

  • Andrew N. Jordan
  • Jeff TollaksenEmail author

Abstract

Since its introduction in 1988, the weak value has made a remarkable progression within the scientific world. This article will delineate each step of this progression, displayed in four courses. The first course is the notion of a weak value in itself, the second is the first experiments together with theoretical challenges, the third is acceptance as a phenomena and further experiments, and the fourth is the use of weak values as a tool for both the further understanding of quantum puzzles and for precision measurements. We will discuss recent developments in the field and argue that the notion of contextual values as generalized eigenvalues of an observable, contextualized to the measurement being done, is the next conceptual step beyond the weak value. [Editor’s note: for a video of the talk given by Prof. Jordan at the YA80 conference at Chapman, see quantum.chapman.edu/talk-8 and for Prof. Howell, see quantum.chapman.edu/talk-9.]

Keywords

Quantum Mechanic Color Average Weak Measurement Conditioned Average Counterfactual Statement 
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.

Notes

Acknowledgements

This work was supported by the NSF grant No. DMR-0844899, Army Research Office “Advanced Quantum Sensing” grant W911NF-12-1-0237 and grant W911NF-09-1-0417. We thank Michael Berry for pointing out Madelung’s work on the hydrodynamical formulation of quantum mechanics to us [47].

References

  1. 1.
    Y. Aharonov, P.G. Bergmann, J.L. Lebowitz, Phys. Rev. B 134, 1410 (1964) MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Y. Aharonov, S. Popescu, J. Tollaksen, in Quantum Theory: A Two-Time Success Story. Yakir Aharonov Festschrift, ed. by D. Struppa, J. Tollaksen (Springer, Milan, 2013). doi: 10.1007/978-88-470-5217-8_3 Google Scholar
  3. 3.
    Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, J. Phys. A, Math. Theor. 44, 365304 (2011) MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Nussinov, J. Tollaksen, Color transparency in QCD and post-selection in quantum mechanics. Phys. Rev. D 78, 036007 (2008) ADSCrossRefGoogle Scholar
  5. 5.
    Y. Aharonov, S. Popescu, J. Tollaksen, Each moment of time is a new universe, this volume Google Scholar
  6. 6.
    Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988) ADSCrossRefGoogle Scholar
  7. 7.
    Y. Aharonov, L. Vaidman, in Time in Quantum Mechanics, ed. by J.G. Muga, R. Sala Mayato, I.L. Egusquiza (Springer, Berlin, 2002). arXiv:quant-ph/0105101v2 Google Scholar
  8. 8.
    Y. Aharonov, S. Popescu, J. Tollaksen, A time-symmetric formulation of quantum mechanics. Phys. Today 63(11), 27–32 (2010) CrossRefGoogle Scholar
  9. 9.
    Y. Aharonov, J. Tollaksen, New insights on time-symmetry in quantum mechanics, in Visions of Discovery, ed. by R.Y. Chiao, M.L. Cohen, A.J. Leggett, W.D. Phillips, C.L. Harper Jr. (Cambridge University Press, Cambridge, 2009) Google Scholar
  10. 10.
    A.J. Leggett, Phys. Rev. Lett. 62, 2325 (1989) ADSCrossRefGoogle Scholar
  11. 11.
    A. Peres, Phys. Rev. Lett. 62, 2326 (1989) ADSCrossRefGoogle Scholar
  12. 12.
    Y. Aharonov, A. Botero, Phys. Rev. A 72, 052111 (2005) ADSCrossRefGoogle Scholar
  13. 13.
    R.E. Kastner, Found. Phys. 29, 851 (1999) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Y. Aharonov, E. Cohen, D. Grossman, A.C. Elitzur. arXiv:1206.6224
  15. 15.
    Y. Aharonov, S. Popescu, J. Tollaksen, Phys. Today 63, 27–32 (2011) CrossRefGoogle Scholar
  16. 16.
    N.S. Williams, A.N. Jordan, Phys. Rev. Lett. 100, 026804 (2008) ADSCrossRefGoogle Scholar
  17. 17.
    L. Hardy, Phys. Rev. Lett. 68, 2981 (1992) MathSciNetADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Y. Aharonov, A. Botero, S. Popescu, B. Reznik, J. Tollaksen, Phys. Lett. A 301, 130 (2002) MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    D. Rohrlich, Y. Aharonov, Phys. Rev. A 66, 042102 (2002) ADSCrossRefGoogle Scholar
  20. 20.
    J. Tollaksen, Pre- and post-selection, weak values, and contextuality. J. Phys. A, Math. Gen. 40, 9033–9066 (2007). quant-ph/0602226 MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    J. Tollaksen, Probing contextuality with pre-and-post-selection. J. Phys. Conf. Ser. 70, 012014 (2007) ADSCrossRefGoogle Scholar
  22. 22.
    J. Tollaksen, Non-statistical weak measurements, in Quantum Information and Computation V, ed. by E. Donkor, A. Pirich, H. Brandt. Proc of SPIE, vol. 6573 (SPIE, Bellingham, 2007). CID 6573-33 Google Scholar
  23. 23.
    J.S. Lundeen, K.J. Resch, A.M. Steinberg, Phys. Rev. A 72(1), 016101 (2005) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    S.E. Ahnert, M.C. Payne, Phys. Rev. A 70, 042102 (2004) MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    K. Yokota, T. Yamamoto, M. Koashi, N. Imoto, New J. Phys. 11, 033011 (2009) ADSCrossRefGoogle Scholar
  26. 26.
    N.D. Mermin, Phys. Rev. Lett. 74, 831 (1995) MathSciNetADSCrossRefzbMATHGoogle Scholar
  27. 27.
    N.D. Mermin, in Potentiality, Entanglement and Passion-at-a-Distance, ed. by R.S. Cohen et al. (Kluwer Academic, Norwell, 1997), pp. 149–157 CrossRefGoogle Scholar
  28. 28.
    J. Dressel, A.N. Jordan, Phys. Rev. A 85, 012107 (2012) ADSCrossRefGoogle Scholar
  29. 29.
    N.D. Mermin, Rev. Mod. Phys. 65, 803 (1993) MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Y. Aharonov, S. Popescu, J. Tollaksen, L. Vaidman, Phys. Rev. A 79, 052110 (2009) MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    N.W.M. Ritchie, J.G. Story, R.G. Hulet, Phys. Rev. Lett. 66, 1107 (1991) ADSCrossRefGoogle Scholar
  32. 32.
    G.J. Pryde, J.L. O’Brien, A.G. White, T.C. Ralph, H.M. Wiseman, Phys. Rev. Lett. 94, 220405 (2005) ADSCrossRefGoogle Scholar
  33. 33.
    O. Hosten, P. Kwiat, Science 319, 787 (2008) ADSCrossRefGoogle Scholar
  34. 34.
    P.B. Dixon, D.J. Starling, A.N. Jordan, J.C. Howell, Phys. Rev. Lett. 102, 173601 (2009) ADSCrossRefGoogle Scholar
  35. 35.
    P.B. Dixon, D.J. Starling, A.N. Jordan, J.C. Howell, Phys. Rev. A 80, 041803(R) (2009) ADSCrossRefGoogle Scholar
  36. 36.
    Y. Kedem, Phys. Rev. A 85, 060102 (2012) ADSCrossRefGoogle Scholar
  37. 37.
    A. Feizpour, X. Xingxing, A.M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011) ADSCrossRefGoogle Scholar
  38. 38.
    D.J. Starling, P.B. Dixon, N.S. Williams, A.N. Jordan, J.C. Howell, Phys. Rev. A 82, 011802(R) (2010). arXiv:0912.2357 ADSCrossRefGoogle Scholar
  39. 39.
    A.K. Pan, A. Matzkin, Phys. Rev. A 85, 022122 (2012) ADSCrossRefGoogle Scholar
  40. 40.
    A.G. Kofman, S. Ashhab, F. Nori, Phys. Rep. 520 (2012) Google Scholar
  41. 41.
    P. Kwiat et al., Increase of signal-to-noise ratio in weak value measurements, in Quantum Theory: A Two-Time Success Story. Yakir Aharonov Festschrift, ed. by D. Struppa, J. Tollaksen (Springer, Milan, 2013). doi: 10.1007/978-88-470-5217-8_26 Google Scholar
  42. 42.
    J. Dressel et al., Unpublished Google Scholar
  43. 43.
    N. Brunner, A. Acin, D. Collins, N. Gisin, V. Scarani, Phys. Rev. Lett. 91, 180402 (2003) ADSCrossRefGoogle Scholar
  44. 44.
    D.J. Starling, P.B. Dixon, A.N. Jordan, J.C. Howell, Phys. Rev. A 82, 063822 (2010) ADSCrossRefGoogle Scholar
  45. 45.
    J. Tollaksen, Robust weak measurements on finite samples. J. Phys. Conf. Ser. 70, 012015 (2007). quant-ph/0703038 ADSCrossRefGoogle Scholar
  46. 46.
    H.F. Hofmann, Phys. Rev. A 83, 022106 (2011) ADSCrossRefGoogle Scholar
  47. 47.
    E. Madelung, Z. Phys. 40, 322–326 (1926) ADSCrossRefzbMATHGoogle Scholar
  48. 48.
    E. Nelson, Phys. Rev. 150, 1079 (1966) ADSCrossRefGoogle Scholar
  49. 49.
    J. Dressel, A.N. Jordan, Phys. Rev. Lett. 109, 230402 (2012) ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2014

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of RochesterRochesterUSA
  2. 2.Institute for Quantum StudiesChapman UniversityOrangeUSA
  3. 3.Institute for Quantum Studies, Schmid College of Science and TechnologyChapman UniversityOrangeUSA

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