Fiber products of measures and quantum foundations

doi:10.1017/CBO9781139519687.006

4 - Fiber products of measures and quantum foundations

Published online by Cambridge University Press: 05 June 2016

Adam Brandenburger and

Edited by

Ali Eskandarian and

Adam Brandenburger: Affiliation:
New York University
H. Jerome Keisler: Affiliation:
University of Wisconsin-Madison
Jennifer Chubb: Affiliation:
University of San Francisco
Ali Eskandarian: Affiliation:
George Washington University, Washington DC
Valentina Harizanov: Affiliation:
George Washington University, Washington DC

Book contents

Get access

Summary

Abstract. With a view to quantum foundations, we define the concepts of an empirical model (a probabilistic model describing measurements and outcomes), a hidden-variable model (an empirical model augmented by unobserved variables), and various properties of hidden-variable models, for the case of infinite measurement spaces and finite outcome spaces. Thus, our framework is general enough to include, for example, quantum experiments that involve spin measurements at arbitrary relative angles. Within this framework, we use the concept of the fiber product of measures to prove general versions of two determinization results about hidden-variable models. Specifically, we prove that: (i) every empirical model can be realized by a deterministic hidden-variable model; (ii) for every hidden-variable model satisfying locality and λ-independence, there is a realizationequivalent hidden-variable model satisfying determinism and λ-independence.

Introduction. Hidden variables are extra variables added to the model of an experiment to explain correlations in the outcomes. Here is a simple example. Alice's and Bob's computers have been prepared with the same password. We know that the password is either p2s4w6r8 or 1a3s5o7d, but we do not know which it is. If Alice now types in p2s4w6r8 and this unlocks her computer, we immediately know what will happen when Bob types in one or other of the two passwords. The two outcomes—when Alice types a password and Bob types a password—are perfectly correlated. Clearly, it would be wrong to conclude that, when Alice types a password on her machine, this somehow causes Bob's machine to acquire the same password. The correlation is purely informational: It is our state of knowledge that changes, not Bob's computer. Formally, we can consider an r.v. (random variable) X for Alice's password, an r.v. Y for Bob's password, and an extra r.v. Z. The r.v. Z takes the value z1 or z2 according as the two machines were prepared with the first or the second password. Then, even though X and Y will be perfectly correlated, they will also be independent (trivially so), conditional on the value of Z. In this sense, the extra r.v. Z explains the correlation.

Type: Chapter
Information: Logic and Algebraic Structures in Quantum Computing , pp. 71 - 87

DOI: https://doi.org/10.1017/CBO9781139519687.006 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] S., Abramsky and A., Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, New Journal of Physics, vol. 13 (2011), p. 113036.Google Scholar

[2] H., Adler, A Geometric Introduction to Forking and Thorn-Forking, Journal of Mathematical Logic, vol. 9 (2009), pp. 1–21.Google Scholar

[3] J., Bell, On the Einstein-Podolsky-Rosen Paradox, Physics, vol. 1 (1964), pp. 195–200.Google Scholar

[4] J., Bell, Introduction to the hidden-variable question, Foundations of Quantum Mechanics, Proceedings of the International School of Physics ‘Enrico Fermi,’ Course IL (NewYork), Academic Press, 1971, (Reprinted in [6, 1987, pp. 29–39],), pp. 171–181.

[5] J., Bell, An exchange on local beables, Dialectica, vol. 39 (1985), pp. 85–96.Google Scholar

[6] J., Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 1987.

[7] I., Ben Yaacov and H. J., Keisler, Randomizations of models as metric structures, Confluentes Mathematici, vol. 1 (2009), pp. 197–223, also available at http://www.math.wisc.edu/keisler.Google Scholar

[8] P., Billingsley, Probability and measure, 3rd ed., Wiley, 1995.

[9] A., Brandenburger and H. J., Keisler, A canonical hidden-variable space, (2012), available at http://www.adambrandenburger.com and http://www.math.wisc.edu/keisler.

[10] A., Brandenburger and N., Yanofsky, A classification of hidden-variable properties, Journal of Physics A: Mathematical and Theoretical, vol. 41 (2008), p. 425302.Google Scholar

[11] A. P., Dawid and M., Studený, An Alternative Approach to Conditional Independence, Artificial Intelligence and Statistics 99, Proceedings of the 7th Workshop (D., Heckerman and J., Whittaker, editors), Morgan Kaufmann, 1999, pp. 32–40.

[12] W. M., Dickson, Quantum Chance and Non-Locality: Probability and Non-Locality in the Interpretations of Quantum Mechanics, Cambridge University Press, 2005.

[13] A., Einstein, B., Podolsky, and N., Rosen, Can quantum-mechanical description of physical reality be considered complete?, Physical Review, vol. 47 (1935), pp. 777–780.Google Scholar

[14] A., Fine, Hidden variables, joint probability and the Bell inequalities, Physical Review Letters, vol. 48 (1982), pp. 291–295.Google Scholar

[15] C., Flori and T., Fritz, Compositories and Gleaves, (2013), available at http://arxiv.org/ abs/1308.6548.

[16] D. M., Greenberger, M. A., Horne, and A., Zeilinger, Going beyond Bell's theorem, Bell's Theorem, Quantum Theory and Conceptions of the Universe (M., Kafatos, editor), Kluwer, 1989, pp. 69–72.

[17] L., Hardy, Nonlocality for two particles without inequalities for almost all entangled states, Physical Review Letters, vol. 71 (1993), pp. 1665–1668.Google Scholar

[18] J., Jarrett, On the physical significance of the locality conditions in the Bell arguments, Noûs, vol. 18 (1984), pp. 569–589.Google Scholar

[19] S., Kochen and E., Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics, vol. 17 (1967), pp. 59–87.Google Scholar

[20] A., Shimony, Events and processes in the quantum world, Quantum Concepts in Space and Time (R., Penrose and C., Isham, editors), Oxford University Press, 1986, pp. 182–203.

[21] R., Shortt, Universally measurable spaces: An invariance theorem and diverse characterizations, Fundamenta Mathematicae, vol. 121 (1984), pp. 169–176.Google Scholar

[22] J., Swart, A conditional product measure theorem, Statistics & Probability Letters, vol. 28 (1996), pp. 131–135.Google Scholar

[23] J., von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, 1932, (Translated as Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.).

Book contents

4 - Fiber products of measures and quantum foundations

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive