## 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.

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## 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.85.166Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. ii. Phys. Rev.

**85**, 180–193 (1952). doi:10.1103/PhysRev.85.180Bombelli, 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.59.521Brown, 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/s10701-004-2009-3Caves, C. M., Milburn, G. J.: Quantum-mechanical model for continuous position measurements. Phys. Rev. A

**36**, 5543–5555 (1987). doi:10.1103/PhysRevA.36.5543Craig, 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:10.1088/1751-8113/40/3/010Danan, 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.111.240402Dowker, 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)80019-4Gudder, S. P., Sorkin, R. D.: Two-site quantum random walk. Gen. Relativ. Gravit.

**43**(12), 3451–3475 (2011). doi:10.1007/s10714-011-1245-zHerzog, 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.75.3034Kauten, 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.71.024029Rideout, D. P., Sorkin, R. D.: A classical sequential growth dynamics for causal sets. Phys. Rev. D

**61**, 024002,2000 (1999). doi:10.1103/PhysRevD.61.024002Sawant, 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.25.2208Sinha, 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/S021773239400294XSorkin, 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:10.1088/1742-6596/67/1/012018Surya, 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.

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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

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

### Keywords

- Quantum measure theory
- Coupling ancillas