Quantum Models for Psychological Measurements: An Unsolved Problem

Abstract

There has been a strong recent interest in applying quantum theory (QT) outside physics, including in cognitive science. We analyze the applicability of QT to two basic properties in opinion polling. The first property (response replicability) is that, for a large class of questions, a response to a given question is expected to be repeated if the question is posed again, irrespective of whether another question is asked and answered in between. The second property (question order effect) is that the response probabilities frequently depend on the order in which the questions are asked. Whenever these two properties occur together, it poses a problem for QT. The conventional QT with Hermitian operators can handle response replicability, but only in the way incompatible with the question order effect. In the generalization of QT known as theory of positive-operator-valued measures (POVMs), in order to account for response replicability, the POVMs involved must be conventional operators. Although these problems are not unique to QT and also challenge conventional cognitive theories, they stand out as important unresolved problems for the application of QT to cognition. Either some new principles are needed to determine the bounds of applicability of QT to cognition, or quantum formalisms more general than POVMs are needed.

Quantum-like modeling of cognition

“…In this section we repeat the discussion [36] on similarity between the schemes of representation of measurements in quantum physics and cognition.…”

“…As well as in quantum physics, one have to use generalized quantum observables represented by positive operator valued measures (POVMs) with corresponding state transformers [4,36]. In quantum physics POVM-type observables naturally arise in the framework of theory of open quantum systems describing interaction of a quantum system with an environment; especially useful is the Markovian approximation in the form of the Gorini-KossakowskiSudarshan-Lindblad equation.…”

“…For illustration we use two concrete applications [12][13][14][15][16][17][18]: the order and disjunction effects. The paper is concluded with a short review of recent research in quantum(-like) cognition, in particular, cognitive applications of the theory of open quantum systems [23,24,30,31] and positive operator valued measures [4,7,36].…”

Quantum Probabilistic Models Revisited: The Case of Disjunction Effects in Cognition

“…Although the Quantum-Like Approach provides great possibilities compared with the classical one, it appears that it cannot completely cover data from psychology and that a quantum formalism was not enough to explain some paradoxical findings (see [50]), so hyperbolic spaces were proposed [51][52][53]. From Equation (14), if Pr(b = β) − α∈a Pr(a = α|C)Pr(b = β|a = α, C) is different from zero, then various interference effects occur.…”

Numerical Simulation of Asano-Khrennikov-Ohya Quantum-like Decision Making Model

“…However, recently the mathematical formalism of QM started to be applied in cognitive science (Acacio de Barros and Suppes, 2009;Accardi et al, 2008Accardi et al, , 2009Asano et al, 2011aAsano et al, , 2011bAsano et al, , 2012aAsano et al, , 2012bBasieva et al, 2010;Busemeyer et al, 2006aBusemeyer et al, , 2006bBusemeyer et al, , 2008Busemeyer et al, , 2009Busemeyer et al, , 2012Takahashi, 2006, 2010;Conte et al, 2006Conte et al, , 2008Conte et al, , 2009Dzhafarov and Kujala, 2012;Khrennikov, 2009, 2013;Khrennikov et al, 2003Khrennikov et al, , 2004aKhrennikov et al, , 2004bKhrennikov et al, , 2006Khrennikov et al, , 2009aKhrennikov et al, , 2009bKhrennikov et al, , 2011aKhrennikov et al, , 2011bKhrennikov et al, , 2014Ohya and Volovich, 2011;Pothos and Busemeyer, 2009). …”