On the Possibility to Combine the Order Effect with Sequential Reproducibility for Quantum Measurements

Abstract

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.

Quantum-like models cannot account for the conjunction fallacy

“…However, it seems to face some new challenges like response replicability (cf. Basieva and Khrennikov 2015).…”

The Ellsberg paradox: A challenge to quantum decision theory?

“…But they have no reason to believe that one color is more likely than another. Hence, by the heuristic of insu¢ cient reason (or equal a' priori probabilities) 3 , they should assign the same probability to each color in urn U . 4 Hence, they should have no reason to prefer K to U or U to K on probabilistic grounds.…”

The Ellsberg Paradox: A Challenge to Quantum Decision Theory?

“…But they have no reason to believe that one color is more likely than another. Hence, by the heuristic of insu¢ cient reason (or equal a' priori probabilities) 3 , they should assign the same probability to each color in urn U . 4 Hence, they should have no reason to prefer K to U or U to K on probabilistic grounds.…”

“…For example, order e¤ects, the Linda paradox, the disjunction fallacy and the conjunction fallacy (see Busemeyer and Bruza, 2012, for an introduction and review). For papers examining the limits of standard quantum theory when applied to cognitive psychology, see Khrennikov et al (2014) and Basieva and Khrennikov (2015).…”

“…This was the situation before the advent of the source method, see subsection 3.5 below 3. Insu¢ cient reason or equal a' priori probabilities in now commonly referred to as indi¤ erence.…”