Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory
Abstract
The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.
References
[1]
Bell, J. L., Boolean-valued models and independence proofs in set theory, 2nd ed., Oxford UP, Oxford, 1985.
[2]
Birkhoff, G. and von Neumann, J., The logic of quantum mechanics, Ann. Math. 37, pp. 823–843, 1936.
[3]
Bruns G. and Kalmbach, G., “Some remarks on free orthomodular lattices,” Proc. Lattice Theory Conf., Houston, U.S.A., (Schmidt, J. ed.), pp. 397–408, 1973.
[4]
Chevalier G.: “Commutators and decompositions of orthomodular lattices,”. Order 6, 181–194 (1989)
[5]
Cohen P. J.: “The independence of the continuum hypothesis I,”. Proc. Nat. Acad. Sci. U.S.A. 50, 1143–1148 (1963)
[6]
Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.
[7]
Davies, E. B., Quantum theory of open systems, Academic, London, 1976.
[8]
Gibbins, P., Particles and paradoxes: The limits of quantum logic, Cambridge UP, Cambridge, UK, 1987.
[9]
Halmos, P. R., Introduction to Hilbert space and the theory of spectral multiplicity, Chelsea, New York, 1951.
[10]
Halvorson H., Clifton R.: “Maximal beable subalgebras of quantum mechanical observables,”. Int. J. Theor. Phys. 38, 2441–2484 (1999)
[11]
Holland, Jr., S. S., “Orthomodularity in infinite dimensions; A theorem of M. Solèr,” Bull. Amer. Math. Soc. 32, pp. 205–234, 1995.
[12]
Kalmbach, G., Orthomodular lattices, Academic, London, 1983.
[13]
Kochen S., Specker E. P.: “The problem of hidden variables in quantum mechanics,”. J. Math. Mech. 17, 59–87 (1967)
[14]
Marsden E. L.: “The commutator and solvability in a generalized orthomodular lattice,”. Pacific J. Math. 33, 357–361 (1970)
[15]
Okamura, K. and Ozawa, M., “Measurement theory in local quantum physics,” J. Math. Phys. 57, pp. 015209/1–015209/29, 2016.
[16]
Ozawa M.: “Boolean valued analysis and type I $${{\rm AW}^*}$$-algebras,”. Proc. Japan Acad. A 59, 368–371 (1983)
[17]
Ozawa M.: “Boolean valued interpretation of Hilbert space theory,”. J. Math. Soc. Japan 35, 609–627 (1983)
[18]
Ozawa M.: “A classification of type I $${{\rm AW}^*}$$-algebras and Boolean valued analysis,”. J. Math. Soc. Japan 36, 589–608 (1984)
[19]
Ozawa M.: “Quantum measuring processes of continuous observables,”. J. Math. Phys. 25, 79–87 (1984)
[20]
Ozawa M.: “Nonuniqueness of the cardinality attached to homogeneous $${{\rm AW}^*}$$-algebras,”. Proc. Amer. Math. Soc. 93, 681–684 (1985)
[21]
Ozawa M.: “Uncertainty relations for noise and disturbance in generalized quantum measurements,”. Ann. Physics 311, 350–416 (2004)
[22]
Ozawa M.: “Quantum perfect correlations,”. Ann. Physics 321, 744–769 (2006)
[23]
Ozawa M.: “Transfer principle in quantum set theory,”. J. Symbolic Logic 72, 625–648 (2007)
[24]
Ozawa M.: “Quantum reality and measurement: A quantum logical approach,”. Found. Phys. 41, 592–607 (2011)
[25]
Ozawa, M., “Universal uncertainty principle, simultaneous measurability, and weak values,” AIP Conf. Proc. 1363, pp. 53–62, 2011. arXiv:1106.5083 [quant-ph].
[26]
Ozawa, M., “Quantum set theory extending the standard probabilistic interpretation of quantum theory (extended abstract),” Electronic Proc. in Theoretical Computer Science (EPTCS) 172, pp. 15–26, 2014. arXiv:1412.8540 [quant-ph].
[27]
Pulmannová S.: “Commutators in orthomodular lattices,”. Demonstratio Math. 18, 187–208 (1985)
[28]
Redhead, M., Incompleteness, nonlocality, and realism: A prolegomenon to the philosophy of quantum mechanics, Oxford UP, Oxford, 1987.
[29]
Scott, D. and Solovay, R., “Boolean-valued models for set theory,” unpublished manuscript for Proc. AMS Summer Institute on Set Theory, Los Angeles: Univ. Cal., 1967.
[30]
Takeuti, G., Two applications of logic to mathematics, Princeton UP, Princeton, 1978.
[31]
Takeuti, G., “Quantum set theory,” Current Issues in Quantum Logic (Beltrametti, E. G. and van Fraassen, B. C., eds.), Plenum, New York, pp. 303–322, 1981.
[32]
Takeuti G.: “$${{\rm C}^*}$$-Algebras and Boolean valued analysis,”. Japan. J. Math. 9, 207–245 (1983)
[33]
Takeuti G.: “Von Neumann algebras and Boolean valued analysis,”. J. Math. Soc. Japan 35, 1–21 (1983)
[34]
Varadarajan, V. S., Geometry of quantum theory, Springer, New York, 1985.
[35]
von Neumann, J., Mathematical foundations of quantum mechanics, Princeton UP, Princeton, NJ, 1955, [Originally published: Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932)].
Index Terms
- Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory
Index terms have been assigned to the content through auto-classification.
Recommendations
Exact master equation and non-markovian decoherence for quantum dot quantum computing
In this article, we report the recent progress on decoherence dynamics of electrons in quantum dot quantum computing systems using the exact master equation we derived recently based on the Feynman---Vernon influence functional approach. The exact ...
Qubitless Quantum Logic
We discuss the implementation of quantum logic in a system of strongly interacting particles. The implementation is qubitless since “logical qubits” do not correspond to any well-defined physical substates. As an illustration, we present the results of ...
Deriving the correctness of quantum protocols in the probabilistic logic for quantum programs
This paper presents a sound axiomatization for a probabilistic modal dynamic logic of quantum programs. The logic can express whether a state is separable or entangled, information that is local to a subsystem of the whole quantum system, and the ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Copyright © 2016 Ohmsha and Springer Japan.
Publisher
Ohmsha
Japan
Publication History
Published: 01 March 2016
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 30 Nov 2024
Other Metrics
Citations
View Options
View options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in