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.
Similar content being viewed by others
Notes
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.
We can also express this by saying that every history which is inside of O but outside of E has measure zero.
References
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
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)
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
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
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
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
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
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
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
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)
Gudder, S.: Quantum measures and the coevent interpretation. Rep. Math. Phys. 67(1), 137–156 (2011). doi:10.1016/s0034-4877(11)80019-4
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
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
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)
Lee, C. M., Selby, J.H.: Higher-order interference in extensions of quantum theory (2015)
Lee, C. M., Selby, J.H.: Higher-order interference doesn’t help in searching for a needle in a haystack (2016)
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
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
Sawant, R., Samuel, J., Sinha, A., Sinha, S., Sinha, U.: Non-classical paths in interference experiments (2013). doi:10.1103/PhysRevLett.113.120406
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
Sinha, A., Vijay, A. H., Sinha, U.: On the superposition principle in interference experiments (2014)
Sinha, U., Couteau, C., Jennewein, T., Laflamme, R., Weihs, G.: Ruling out multi-order interference in quantum mechanics (2010). doi:10.1126/science.1190545
Sorkin, R.D.: How Interconnected is the Quantum World? Workshop on Free Will and Retrocausality in the Quantum World. Trinity College, Cambridge
Sorkin, R. D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9(1994), 3119–3128 (1994). doi:10.1142/S021773239400294X
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)
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)
Sorkin, R. D.: An exercise in “anhomomorphic logic”. J. Phys. Conf. Ser. 67, 012018,2007 (2007). doi:10.1088/1742-6596/67/1/012018
Surya, S.: Directions in causal set quantum gravity. In: Dasgupta, A. (ed.) Recent Research in Quantum Gravity. Nova Science Publishers, NY (2013) (2011)
Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. doi:10.1007/s10701-010-9429-z (2009)
Vaidman, L.: Past of a quantum particle. Phys. Rev. A 87(052), 104 (2013). doi:10.1103/PhysRevA.87.052104
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
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-016-3181-x