An Operational Characterization for Optimal Observation of Potential Properties in Quantum Theory and Signal Analysis
- 37 Downloads
Quantum logic introduced a paradigm shift in the axiomatization of quantum theory by looking directly at the structural relations between the closed subspaces of the Hilbert space of a system. The one dimensional closed subspaces correspond to testable properties of the system, forging an operational link between theory and experiment. Thus a property is called actual, if the corresponding test yields “yes” with certainty. We argue a truly operational definition should include a quantitative criterion that tells us when we ought to be satisfied that the test yields “yes” with certainty. This question becomes particularly pressing when we inquire how the usual definition can be extended to cover potential, rather than actual properties. We present a statistically operational candidate for such an extension and show that its representation automatically captures some essential Hilbert space structure. If it is the nature of observation that is responsible for the Hilbert space structure, then we should be able to give examples of theories with scope outside the domain of quantum theory, that employ its basic structure, and that describe the optimal extraction of information. We argue signal analysis is such an example.
KeywordsPotential property Quantum logic Quantum theory Signal analysis Optimal observation
Unable to display preview. Download preview PDF.
- 1.Aerts, D.: The One and the Many: towards a unification of the quantum and the classical description of one and many physical entities. Ph.D. thesis, Vrije Universiteit Brussel (VUB) (1981) Google Scholar
- 2.Aerts, S., Aerts, D., Schroeck, F.E.: The necessity of combining mutually incompatible perspectives in the construction of a global view: quantum probability and signal analysis. In: Aerts, D., D’Hooghe, B., Note, N. (eds.) Worldviews, Science and Us: Redemarcating Knowledge and Its Social and Ethical Implications. World Scientific, Singapore (2005) Google Scholar
- 3.Aerts, S.: Quantum and classical probability as Bayes-optimal observation. quant-ph/0601138 (2006) Google Scholar
- 5.Cohen, L.: Time–Frequency Analysis. Prentice Hall, New York (1995) Google Scholar
- 7.Gabor, D.: Theory of communications. J. Inst. Elect. Eng. 93, 429–457 (1943) Google Scholar
- 9.Helmholtz, H.: Handbuch der Physiologischen Optik. Voss, Leipzig (1867) Google Scholar
- 15.Ville, J.: Theorié et applications de la notion de signal analytique. Cables Transm. 2A(1), 61–74 (1948) Google Scholar
- 17.Wootters, W.K.: The acquisition of information from quantum measurements. Ph.D. dissertation, University of Texas at Austin (1980) Google Scholar