On the Possibility to Combine the Order Effect with Sequential Reproducibility for Quantum Measurements
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In this paper we study the problem of a possibility to use quantum observables to describe a possible combination of the order effect with sequential reproducibility for quantum measurements. By the order effect we mean a dependence of probability distributions (of measurement results) on the order of measurements. We consider two types of the sequential reproducibility: adjacent reproducibility (\(A-A\)) (the standard perfect repeatability) and separated reproducibility(\(A-B-A\)). The first one is reproducibility with probability 1 of a result of measurement of some observable A measured twice, one A measurement after the other. The second one, \(A-B-A\), is reproducibility with probability 1 of a result of A measurement when another quantum observable B is measured between two A’s. Heuristically, it is clear that the second type of reproducibility is complementary to the order effect. We show that, surprisingly, this may not be the case. The order effect can coexist with a separated reproducibility as well as adjacent reproducibility for both observables A and B. However, the additional constraint in the form of separated reproducibility of the \(B-A-B\) type makes this coexistence impossible. The problem under consideration was motivated by attempts to apply the quantum formalism outside of physics, especially, in cognitive psychology and psychophysics. However, it is also important for foundations of quantum physics as a part of the problem about the structure of sequential quantum measurements.
KeywordsOrder effect Repeatability Quantum-like models POVMs in cognitive science
The authors would like to thank J. Busemeyer and E. Dzhafarov for fruitful discussions and M. D’ Ariano, P. Lahti, W.M. de Muynck, and M. Ozawa for fruitful comments and advices. This project was supported by researcher-fellowship at Mathematical Institute of Linnaeus University (I. Basieva, 2014-15).
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