Abstract
The combination of the question order and response replicability effects (QOE+RRE) cannot be modeled by von Neumann observables (with projection state update). In this chapter, we overcome this difficulty within quantum instrument theory. The mathematical construction is based on the indirect measurement scheme. The unitary operators describing the interaction of a system and an observable (question) are directly written via their actions. In this model, the systems are humans and the observables are questions asked to them. Our model covers not only of QOE + RRE, but even QOE +RRE + QQE (the later abbreviation is for QQ-equality derived by Wang and Busemeyer). However, we will be concentrated on the combination QOE+RRE. This chapter contains rather long calculations in Hilbert space. They are not complicated, but might be boring for inexperienced reader. One may be satisfied by the statement that, for QOE+RRE, it is possible to construct quantum instruments (by using the indirect measurement scheme). For more experienced reader, the calculations can serve as the basis example of construction of instruments for various combinations of psychological or social effects.
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Notes
- 1.
The construction based on the indirect measurement scheme was suggested by Masanao Ozawa in the series of discussions during the visit of Andrei Khrennikov to the Nagoya University in 2018.
- 2.
This terminology is from quantum physics and may be it is not so natural for cognitive modeling.
- 3.
It is easy to see that there exists at least one unitary operator \(U_a\) on \({\mathcal H}\otimes {\mathcal K}\) satisfying the given relations, since the relations define a one-to-one correspondence between two subsets of the orthonormal basis \(\{|\alpha ,\beta ,\gamma ,\delta \rangle \mid \alpha ,\beta ,\gamma ,\delta =0,1\}\) of \({\mathcal H}\otimes {\mathcal K}\).
- 4.
This is the good place to mention the p-adic and, in particular, 2-adic models of brain’s functioning [127, 229, 243].
- 5.
The easiest way to model this situation is to consider the brain as self-observer: consciousness \(\mathcal{C}\) performs observations on unconsciousness \(\mathcal{C}\) [268].
- 6.
This is the good place to make the following remark. One can be curious whether borrowing the quantum probability formalism from physics would not lead to paradoxes similar to the paradoxes generated by using the classical probability formalism in decision-making.
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Khrennikov, A.Y. (2023). Question Order and Response Replicability Effects. In: Open Quantum Systems in Biology, Cognitive and Social Sciences. Springer, Cham. https://doi.org/10.1007/978-3-031-29024-4_9
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DOI: https://doi.org/10.1007/978-3-031-29024-4_9
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