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How to Measure the Quantum Measure

In memory of David Ritz Finkelstein

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

  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.

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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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Correspondence to Rafael Dolnick Sorkin.

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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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Keywords

  • Quantum measure theory
  • Coupling ancillas