On the Consistency of the Quantum-Like Representation Algorithm for Hyperbolic Interference |
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Authors: | Peter Nyman |
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Institution: | (1) International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, 35195 V?xj?, Sweden;(2) Department of Information Sciences, Tokyo University of Science, Yamasaki 2641, Noda-shi, Chiba 278-8510, Japan;(3) Department of Biological Science and Technology, Tokyo University of Science, Yamasaki 2641, Noda-shi, Chiba 278-8510, Japan;; |
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Abstract: | Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov 20]–28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or
hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations.
The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric,
hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric
interference was recently proved 29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric
interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning.
For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H
b|a
and H
a|b
(in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary
equivalent (in the sense of hyperbolic Hilbert space). |
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Keywords: | |
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