Skip to main content
Log in

How to Measure the Quantum Measure

In memory of David Ritz Finkelstein

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

The histories-based framework of Quantum Measure Theory assigns a generalized probability or measure μ(E) to every (suitably regular) set E of histories. Even though μ(E) cannot in general be interpreted as the expectation value of a selfadjoint operator (or POVM), we describe an arrangement which makes it possible to determine μ(E) experimentally for any desired E. Taking, for simplicity, the system in question to be a particle passing through a series of Stern-Gerlach devices or beam-splitters, we show how to couple a set of ancillas to it, and then to perform on them a suitable unitary transformation followed by a final measurement, such that the probability of a final outcome of “yes” is related to μ(E) by a known factor of proportionality. Finally, we discuss in what sense a positive outcome of the final measurement should count as a minimally disturbing verification that the microscopic event E actually happened.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. If we wish to be more cautious, we can say only that the event that the particle travelled some other path than what our measurement indicated did not happen. For example if we measured (0,0) then this, complementary event would comprise the last 7 histories in the Table 3.

  2. We can also express this by saying that every history which is inside of O but outside of E has measure zero.

References

  1. Aharonov, Y., Zubairy, M. S.: Time and the quantum: Erasing the past and impacting the future. Science 307 (5711) 875–879 (2005). http://science.sciencemag.org/content/307/5711/875

  2. Barnum, H., Mueller, M. P., Ududec, C.: Higher-order interference and single-system postulates characterizing quantum theory. doi:10.1088/1367-2630/16/12/123029 (2014)

  3. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. i. Phys. Rev. 85, 166–179 (1952). doi:10.1103/PhysRev.85.166

  4. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. ii. Phys. Rev. 85, 180–193 (1952). doi:10.1103/PhysRev.85.180

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bombelli, L., Lee, J., Meyer, D., Sorkin, R. D.: Space-time as a causal set. Phys. Rev. Lett. 59, 521–524 (1987). doi:10.1103/PhysRevLett.59.521

    Article  ADS  MathSciNet  Google Scholar 

  6. Brown, H. R., Wallace, D.: Solving the measurement problem: De Broglie–Bohm loses out to Everett. Found. Phys. 35(4), 517–540 (2005). doi:10.1007/s10701-004-2009-3

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Caves, C. M., Milburn, G. J.: Quantum-mechanical model for continuous position measurements. Phys. Rev. A 36, 5543–5555 (1987). doi:10.1103/PhysRevA.36.5543

    Article  ADS  MathSciNet  Google Scholar 

  8. Craig, D., Dowker, F., Henson, J., Major, S., Rideout, D., Sorkin, R. D.: A bell inequality analog in quantum measure theory. J. Phys. A 40, 501–523,2007 (2006). doi:10.1088/1751-8113/40/3/010

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Danan, A., Farfurnik, D., Bar-Ad, S., Vaidman, L.: Asking photons where they have been. Phys. Rev. Lett. 111 (240), 402 (2013). doi:10.1103/PhysRevLett.111.240402

    Google Scholar 

  10. Dowker, F.: Causal sets and the deep structure of spacetime. In: Ashtekar, A (ed.) 100 Years of Relativity - Space-time Structure: Einstein and Beyond World Scientific (2005) (2005)

  11. Gudder, S.: Quantum measures and the coevent interpretation. Rep. Math. Phys. 67(1), 137–156 (2011). doi:10.1016/s0034-4877(11)80019-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Gudder, S. P., Sorkin, R. D.: Two-site quantum random walk. Gen. Relativ. Gravit. 43(12), 3451–3475 (2011). doi:10.1007/s10714-011-1245-z

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Herzog, T. J., Kwiat, P. G., Weinfurter, H., Zeilinger, A.: Complementarity and the quantum eraser. Phys. Rev. Lett. 75, 3034–3037 (1995). doi:10.1103/PhysRevLett.75.3034

    Article  ADS  Google Scholar 

  14. Kauten, T., Keil, R., Kaufmann, T., Pressl, B., Brukner, A., Weihs, G.: Obtaining tight bounds on higher-order interferences with a 5-path interferometer (2015)

  15. Lee, C. M., Selby, J.H.: Higher-order interference in extensions of quantum theory (2015)

  16. Lee, C. M., Selby, J.H.: Higher-order interference doesn’t help in searching for a needle in a haystack (2016)

  17. Martin, X., O’Connor, D., Sorkin, R. D.: The random walk in generalized quantum theory. Phys. Rev. D 71(2005), 024029 (2004). doi:10.1103/PhysRevD.71.024029

  18. Rideout, D. P., Sorkin, R. D.: A classical sequential growth dynamics for causal sets. Phys. Rev. D 61, 024002,2000 (1999). doi:10.1103/PhysRevD.61.024002

  19. Sawant, R., Samuel, J., Sinha, A., Sinha, S., Sinha, U.: Non-classical paths in interference experiments (2013). doi:10.1103/PhysRevLett.113.120406

    Google Scholar 

  20. Scully, M. O., Drühl, K.: Quantum eraser: a proposed photon correlation experiment concerning observation and “delayed choice” in quantum mechanics. Phys. Rev. A 25, 2208–2213 (1982). doi:10.1103/PhysRevA.25.2208

    Article  ADS  Google Scholar 

  21. Sinha, A., Vijay, A. H., Sinha, U.: On the superposition principle in interference experiments (2014)

  22. Sinha, U., Couteau, C., Jennewein, T., Laflamme, R., Weihs, G.: Ruling out multi-order interference in quantum mechanics (2010). doi:10.1126/science.1190545

    MATH  Google Scholar 

  23. Sorkin, R.D.: How Interconnected is the Quantum World? Workshop on Free Will and Retrocausality in the Quantum World. Trinity College, Cambridge

  24. Sorkin, R. D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9(1994), 3119–3128 (1994). doi:10.1142/S021773239400294X

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Sorkin, R. D.: Quantum measure theory and its interpretation. In: Feng, D.H., Hu, B.-L. (eds.) Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, held Philadelphia, September 8–11, 1994. (International Press, Cambridge Mass. 1997), pp 229–251 (1995)

  26. Sorkin, R. D.: Causal sets: discrete gravity (notes for the valdivia summer school). In: Gomberoff, A., Marolf, D. (eds.) Lectures on Quantum Gravity (Series of the Centro De Estudios científicos), Proceedings of the Valdivia Summer School, held January 2002 in Valdivia, Chile. (Springer 2005), pp 305–328 (2003)

  27. Sorkin, R. D.: An exercise in “anhomomorphic logic”. J. Phys. Conf. Ser. 67, 012018,2007 (2007). doi:10.1088/1742-6596/67/1/012018

    Article  Google Scholar 

  28. Surya, S.: Directions in causal set quantum gravity. In: Dasgupta, A. (ed.) Recent Research in Quantum Gravity. Nova Science Publishers, NY (2013) (2011)

  29. Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. doi:10.1007/s10701-010-9429-z (2009)

  30. Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87(052), 104 (2013). doi:10.1103/PhysRevA.87.052104

    Google Scholar 

Download references

Acknowledgments

AMF would like to thank his PSI partners for their useful discussions and the long hours working together. This research was supported in part by NSERC through grant RGPIN-418709-2012. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rafael Dolnick Sorkin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frauca, Á.M., Sorkin, R.D. How to Measure the Quantum Measure. Int J Theor Phys 56, 232–258 (2017). https://doi.org/10.1007/s10773-016-3181-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-016-3181-x

Keywords

Navigation