Weak Values and Quantum Properties

Abstract

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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Note that this reasoning assumes the pointer states \(\left| \varphi _{x_{0} -ga_{k}}\right\rangle \) of Eq. (7) are orthogonal. Otherwise a pointer state does not correlate with a single eigenstate and no element of reality can be defined.

  2. 2.

    This implementation of the three-box paradox [2] has been described in details elsewhere [23, 24].

  3. 3.

    The estimate minimizes a specific distance d in Hilbert space, namely \(d=\mathrm {Tr}\left[ \left| \psi (t_{i} )\right\rangle \left\langle \psi (t_{i})\right| \left( A-A_{\mathrm {est} }\right) ^{2}\right] ,\) and the resulting best estimate is [40] \(A_{\mathrm {est} }=\sum _{f}{\text {Re}}A_{f}^{w}\left| b_{f}\right\rangle \left\langle b_{f}\right| {\text {Re}}A_{f}^{w}\) where \(A_{f}^{w}\) is the weak value (19).

  4. 4.

    We have already noticed above that even in the absence of postselection, no element of reality can be defined if the pointer states are not orthogonal.

  5. 5.

    Unsurprisingly, the current density appears in the numerator of the following weak value of the momentum, \(\mathrm {Re}\frac{\left\langle x\right| P\left| \psi (t)\right\rangle }{\left\langle x\right| \left. \psi (t)\right\rangle }=\frac{mj(x,t)}{\rho (x,t)}\).

  6. 6.

    The corresponding classical expression is

    (28)

    where \(\rho (q)\) is the configuration space classical distribution. The integral is taken over \(\mathcal {B}_{f}\) which is the set of all q’s taken at \(t_{w}\) such that at the final time \(t_{f}\) we have \(B(q,t_{f})=b_{f}\). In a classical setting, the filtering needs to be done before the weak interaction takes place. Note that the denominator is simply the normalization constant for the density due to the filtering (see [46] for details).

References

  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)

    ADS  Article  Google Scholar 

  2. 2.

    Aharonov, Y., Vaidman, L.: Complete description of a quantum system at a given time. J. Phys. A 24, 2315 (1991)

    ADS  MathSciNet  Article  Google 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)

  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)

    ADS  Article  MATH  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  8. 8.

    Aharonov, Y., Botero, A.: Quantum averages of weak values. Phys. Rev. A 72, 052111 (2005)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  12. 12.

    Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1958)

    MATH  Google Scholar 

  13. 13.

    Gilton, M.J.R.: Whence the eigenstate–eigen value link? Stud. Hist. Phil. Sci. B 55, 92 (2016)

    MathSciNet  Google Scholar 

  14. 14.

    Bohm, D.: Quantum Mechanics. Prentice-Hall, Englewood Cliffs, NJ (1951)

    MATH  Google 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)

    MATH  Google 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.), https://plato.stanford.edu/archives/spr2016/entries/qm-collapse

  17. 17.

    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)

    ADS  Article  MATH  Google Scholar 

  18. 18.

    Redhead, M.: Incompleteness, Nonlocality, and Realism. Clarendon Press, Oxford (1987)

    MATH  Google Scholar 

  19. 19.

    Ballentine, L.E.: Quantum Mechanics. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  20. 20.

    Vaidman, L.: Time-symmetrized counterfactuals in quantum theory. Found. Phys. 29, 755 (1999)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–935 (1935)

    ADS  Article  MATH  Google Scholar 

  22. 22.

    Aharonov, Y., Bergmann, P.G., Lebowitz, J.L.: Time symmetry in the quantum process of measurement. Phys. Rev. 134, B1410 (1964)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Duprey, Q., Matzkin, A.: Null weak values and the past of a quantum particle. Phys. Rev. A 95, 032110 (2017)

    ADS  Article  Google Scholar 

  25. 25.

    Sokolovski, D.: Comment on “Null weak values and the past of a quantum particle”. Phys. Rev. A 97, 046102 (2018)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Kirkpatrick, K.A.: Classical three-box ‘paradox’. J. Phys. A 36, 4891 (2003)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Mohrhoff, U.: Objective probabilities, quantum counterfactuals, and the ABL rule. Am. J. Phys. 69, 864 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Vaidman, L.: Defending time-symmetrised quantum counterfactuals. Stud. Hist. Phil. Mod. Phys. 30, 373 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Matzkin, A.: Observing trajectories with weak measurements in quantum systems in the semiclassical regime. Phys. Rev. Lett 109, 150407 (2012)

    ADS  Article  Google 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)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Aharonov, Y., Vaidman, L.: The two-state vector formalism: an updated review. Lect. Notes Phys. 734, 399 (2008)

    ADS  Article  Google Scholar 

  35. 35.

    Aharonov, Y., Cohen, E., Landsberger, T.: The two-time interpretation and macroscopic time-reversibility. Entropy 19, 111 (2017)

    ADS  Article  Google Scholar 

  36. 36.

    Sokolovski, D., Akhmatskaya, E.: An even simpler understanding of quantum weak values. Ann. Phys. 388, 382 (2018)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Svensson, B.E.Y.: What is a quantum-mechanical weak value the value of? Found. Phys. 43, 1193 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Dressel, J., Jordan, A.N.: Contextual-value approach to the generalized measurement of observables. Phys. Rev. A 85, 022123 (2012)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  40. 40.

    Dressel, J.: Weak values as interference phenomena. Phys. Rev. A 91, 032116 (2015)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

  42. 42.

    Kastner, R.E.: Demystifying weak measurements. Found. Phys. 47, 697 (2017)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Sokolovski, D.: Weak measurements measure probability amplitudes (and very little else). Phys. Lett. A 380, 1593 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Cohen, E.: What weak measurements and weak values really mean. Found. Phys. 47, 1261 (2017)

    ADS  Article  MATH  Google 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 preparation

  47. 47.

    Aharonov, Y., Rohrlich, D., Popescu, S., Skrzypczyk, P.: Quantum Cheshire cats. New J. Phys. 15, 113015 (2013)

    ADS  Article  Google Scholar 

  48. 48.

    Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87, 052104 (2013)

    ADS  Article  Google Scholar 

  49. 49.

    Danan, A., Farfurnik, D., Bar-Ad, S., Vaidman, L.: Asking photons where they have been. Phys. Rev. Lett. 111, 240402 (2013)

    ADS  Article  Google 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)

    ADS  Article  Google 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)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Matzkin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Matzkin, A. Weak Values and Quantum Properties. Found Phys 49, 298–316 (2019). https://doi.org/10.1007/s10701-019-00245-3

Download citation

Keywords

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