Foundations of Physics

, Volume 49, Issue 3, pp 298–316 | Cite as

Weak Values and Quantum Properties

  • A. MatzkinEmail author


We investigate in this work the meaning of weak values through the prism of property ascription in quantum systems. Indeed, the weak measurements framework contains only ingredients of the standard quantum formalism, and as such weak measurements are from a technical point of view uncontroversial. However attempting to describe properties of quantum systems through weak values—the output of weak measurements—goes beyond the usual interpretation of quantum mechanics, that relies on eigenvalues. We first recall the usual form of property ascription, based on the eigenstate-eigenvalue link and the existence of “elements of reality”. We then describe against this backdrop the different meanings that have been given to weak values. We finally argue that weak values can be related to a specific form of property ascription, weaker than the eigenvalues case but still relevant to a partial description of a quantum system.


Measurement in quantum mechanics Properties of quantum systems Weak measurements Post-selected quantum systems 



Dipankar Home (Bose Institute, Kolkata) and Urbasi Sinha (Raman Research Institute, Bangalore) are thanked for useful discussions on an earlier version of the manuscript. Partial support from the Templeton Foundation (Project 57758) is gratefully acknowledged.


  1. 1.
    Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)ADSCrossRefGoogle Scholar
  2. 2.
    Aharonov, Y., Vaidman, L.: Complete description of a quantum system at a given time. J. Phys. A 24, 2315 (1991)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    See e.g. Hosten, O., Kwiat, P.: Observation of the spin hall effect of light via weak measurements, Science 319, 787 (2008)Google Scholar
  4. 4.
    Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm, L.K., Steinberg, A.M.: Observing the average trajectories of single photons in a two-slit interferometer. Science 332, 1170 (2011)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Goggin, M.E., Almeida, M.P., Barbieri, M., Lanyon, B.P., OBrien, J.L., White, A.G., Pryde, G.J.: Violation of the Leggett-Garg inequality with weak measurements of photons. PNAS 108, 1256 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Dixon, P.B., Starling, D.J., Jordan, A.N., Howell, J.C.: Ultrasensitive beam deflection measurement via interferometric weak value amplification. Phys. Rev. Lett. 102, 173601 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    Harris, J., Boyd, R.W., Lundeen, J.S.: Weak value amplification can outperform conventional measurement in the presence of detector saturation. Phys. Rev. Lett. 118, 070802 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    Aharonov, Y., Botero, A.: Quantum averages of weak values. Phys. Rev. A 72, 052111 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    Hofmann, H.F.: Derivation of quantum mechanics from a single fundamental modification of the relations between physical properties. Phys. Rev. A 89, 042115 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Leggett, A.J.: Comment on “How the result of a measurement of a component of the spin of a spin-(1/2 particle can turn out to be 100‘”. Phys. Rev. Lett. 62, 2325 (1989)ADSCrossRefGoogle Scholar
  11. 11.
    Peres, A.: Comment on “How the result of a measurement of a component of the spin of a spin-(1/2 particle can turn out to be 100‘”. Phys. Rev. Lett. 62, 2326 (1989)ADSCrossRefGoogle Scholar
  12. 12.
    Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1958)zbMATHGoogle Scholar
  13. 13.
    Gilton, M.J.R.: Whence the eigenstate–eigen value link? Stud. Hist. Phil. Sci. B 55, 92 (2016)MathSciNetGoogle Scholar
  14. 14.
    Bohm, D.: Quantum Mechanics. Prentice-Hall, Englewood Cliffs, NJ (1951)zbMATHGoogle Scholar
  15. 15.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press, Princeton, NJ (1955). (Originally published in German in 1932 by Springer)zbMATHGoogle Scholar
  16. 16.
    Norsen, T.: Foundations of Quantum Mechanics, Springer (Cham, Switzerland) 2017; G. Ghirardi, ”Collapse Theories, The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.),
  17. 17.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Redhead, M.: Incompleteness, Nonlocality, and Realism. Clarendon Press, Oxford (1987)zbMATHGoogle Scholar
  19. 19.
    Ballentine, L.E.: Quantum Mechanics. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  20. 20.
    Vaidman, L.: Time-symmetrized counterfactuals in quantum theory. Found. Phys. 29, 755 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–935 (1935)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Aharonov, Y., Bergmann, P.G., Lebowitz, J.L.: Time symmetry in the quantum process of measurement. Phys. Rev. 134, B1410 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Matzkin, A., Pan, A.K.: Three-box paradox and ’Cheshire cat grin’: the case of spin-1 atoms. J. Phys. A 46, 315307 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Duprey, Q., Matzkin, A.: Null weak values and the past of a quantum particle. Phys. Rev. A 95, 032110 (2017)ADSCrossRefGoogle Scholar
  25. 25.
    Sokolovski, D.: Comment on “Null weak values and the past of a quantum particle”. Phys. Rev. A 97, 046102 (2018)ADSCrossRefGoogle Scholar
  26. 26.
    Duprey, Q., Matzkin, A.: Reply to comment on null weak values and the past of a quantum particle. Phys. Rev. A 97, 046103 (2018)ADSCrossRefGoogle Scholar
  27. 27.
    Kastner, R.E.: The three-box paradox and other reasons to reject the counterfactual usage of the ABL rule. Found. Phys. 29, 851 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kirkpatrick, K.A.: Classical three-box ‘paradox’. J. Phys. A 36, 4891 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mohrhoff, U.: Objective probabilities, quantum counterfactuals, and the ABL rule. Am. J. Phys. 69, 864 (2001)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Vaidman, L.: Defending time-symmetrised quantum counterfactuals. Stud. Hist. Phil. Mod. Phys. 30, 373 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Matzkin, A.: Observing trajectories with weak measurements in quantum systems in the semiclassical regime. Phys. Rev. Lett 109, 150407 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Duprey, Q., Kanjilal, S., Sinha, Urbasi, Home, D., Matzkin, A.: The quantum Cheshire cat effect: theoretical basis and observational implications. Ann. Phys. 391, 1 (2018)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Price, H.: Does time-symmetry imply retrocausality? How the quantum world says maybe? Stud. Hist. Phil. Sci. Mod. Phys. 43, 75 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Aharonov, Y., Vaidman, L.: The two-state vector formalism: an updated review. Lect. Notes Phys. 734, 399 (2008)ADSCrossRefGoogle Scholar
  35. 35.
    Aharonov, Y., Cohen, E., Landsberger, T.: The two-time interpretation and macroscopic time-reversibility. Entropy 19, 111 (2017)ADSCrossRefGoogle Scholar
  36. 36.
    Sokolovski, D., Akhmatskaya, E.: An even simpler understanding of quantum weak values. Ann. Phys. 388, 382 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Svensson, B.E.Y.: What is a quantum-mechanical weak value the value of? Found. Phys. 43, 1193 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Dressel, J., Jordan, A.N.: Contextual-value approach to the generalized measurement of observables. Phys. Rev. A 85, 022123 (2012)ADSCrossRefGoogle Scholar
  39. 39.
    Ipsen, A.C.: Disturbance in weak measurements and the difference between quantum and classical weak values. Phys. Rev. A 91, 062120 (2015)ADSCrossRefGoogle Scholar
  40. 40.
    Dressel, J.: Weak values as interference phenomena. Phys. Rev. A 91, 032116 (2015)ADSCrossRefGoogle Scholar
  41. 41.
    Vaidman, L., Ben-Israel, A., Dziewior, J., Knips, L., Weissl, M., Meinecke, J., Schwemmer, C., Ber, R., Weinfurter, H.: Weak value beyond conditional expectation value of the pointer readings. Phys. Rev. A 96, 032114 (2017)ADSCrossRefGoogle Scholar
  42. 42.
    Kastner, R.E.: Demystifying weak measurements. Found. Phys. 47, 697 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sokolovski, D.: Weak measurements measure probability amplitudes (and very little else). Phys. Lett. A 380, 1593 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Cohen, E.: What weak measurements and weak values really mean. Found. Phys. 47, 1261 (2017)ADSCrossRefzbMATHGoogle Scholar
  45. 45.
    Cohen Tannoudju, C., Diu, B., Laloe, F.: Quantum Mechanics, pp. 238–239. Hermann-Wiley Interscience, Paris (1977)Google Scholar
  46. 46.
    Matzkin, A.: in preparationGoogle Scholar
  47. 47.
    Aharonov, Y., Rohrlich, D., Popescu, S., Skrzypczyk, P.: Quantum Cheshire cats. New J. Phys. 15, 113015 (2013)ADSCrossRefGoogle Scholar
  48. 48.
    Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87, 052104 (2013)ADSCrossRefGoogle Scholar
  49. 49.
    Danan, A., Farfurnik, D., Bar-Ad, S., Vaidman, L.: Asking photons where they have been. Phys. Rev. Lett. 111, 240402 (2013)ADSCrossRefGoogle Scholar
  50. 50.
    Englert, B.-G., Horia, K., Dai, J., Len, Y.L., Ng, H.K.: Past of a quantum particle revisited. Phys. Rev. A 96, 022126 (2017)ADSCrossRefGoogle Scholar
  51. 51.
    Zhou, Z.-Q., Liu, X., Kedem, Y., Cui, J.-M., Li, Z.-F., Hua, Y.-L., Li, C.-F., Guo, G.-C.: Phys. Rev. A 95, 042121 (2017)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Modélisation (CNRS Unité 8089)Université de Cergy-PontoiseCergy-Pontoise CedexFrance

Personalised recommendations