International Journal of Theoretical Physics

, Volume 56, Issue 1, pp 232–258 | Cite as

How to Measure the Quantum Measure

In memory of David Ritz Finkelstein


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.


Quantum measure theory Coupling ancillas 



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.


  1. 1.
    Aharonov, Y., Zubairy, M. S.: Time and the quantum: Erasing the past and impacting the future. Science 307 (5711) 875–879 (2005).
  2. 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. 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. 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 ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 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 ADSMathSciNetCrossRefGoogle Scholar
  6. 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 ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 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 ADSMathSciNetCrossRefGoogle Scholar
  8. 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 ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 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. 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)Google Scholar
  11. 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 ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 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 ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 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 ADSCrossRefGoogle Scholar
  14. 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)Google Scholar
  15. 15.
    Lee, C. M., Selby, J.H.: Higher-order interference in extensions of quantum theory (2015)Google Scholar
  16. 16.
    Lee, C. M., Selby, J.H.: Higher-order interference doesn’t help in searching for a needle in a haystack (2016)Google Scholar
  17. 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. 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. 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. 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 ADSCrossRefGoogle Scholar
  21. 21.
    Sinha, A., Vijay, A. H., Sinha, U.: On the superposition principle in interference experiments (2014)Google Scholar
  22. 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 MATHGoogle Scholar
  23. 23.
    Sorkin, R.D.: How Interconnected is the Quantum World? Workshop on Free Will and Retrocausality in the Quantum World. Trinity College, CambridgeGoogle Scholar
  24. 24.
    Sorkin, R. D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9(1994), 3119–3128 (1994). doi: 10.1142/S021773239400294X ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 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)Google Scholar
  26. 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)Google Scholar
  27. 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 CrossRefGoogle Scholar
  28. 28.
    Surya, S.: Directions in causal set quantum gravity. In: Dasgupta, A. (ed.) Recent Research in Quantum Gravity. Nova Science Publishers, NY (2013) (2011)Google Scholar
  29. 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. 30.
    Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87(052), 104 (2013). doi: 10.1103/PhysRevA.87.052104 Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Álvaro Mozota Frauca
    • 1
  • Rafael Dolnick Sorkin
    • 1
    • 2
  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of PhysicsSyracuse UniversitySyracuseUSA

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