Quantum model for psychological measurements: from the projection postulate to interference of mental observables represented as positive operator valued measures

TL;DR

This paper explores a quantum-like framework for psychological measurements, emphasizing the use of positive operator-valued measures (POVMs) to model mental observables and their interference effects, advancing understanding of cognitive phenomena.

Contribution

It introduces a POVM-based quantum model for cognitive measurements, extending the traditional projection postulate and providing a psychological interpretation of interference terms.

Findings

POVMs offer a more general framework for modeling mental observables.

Interference terms in POVMs have complex structures with psychological significance.

The model explains phenomena like ambiguous figure recognition and disjunction effects.

Abstract

Recently foundational issues of applicability of the formalism of quantum mechanics (QM) to cognitive psychology, decision making, and psychophysics attracted a lot of interest. In particular, in \cite{DKBB} the possibility to use of the projection postulate and representation of "mental observables" by Hermitian operators was discussed in very detail. The main conclusion of the recent discussions on the foundations of "quantum(-like) cognitive psychology" is that one has to be careful in determination of conditions of applicability of the projection postulate as a mathematical tool for description of measurements of observables represented by Hermitian operators. To represent some statistical experimental data (both physical and mental) in the quantum(-like) way, one has to use generalized quantum observables given by positive operator-valued measures (POVMs). This paper contains a brief review on POVMs which can be useful for newcomers to the field of quantum(-like) studies. Especially interesting for cognitive psychology is a variant of the formula of total probability (FTP) with the interference term derived for incompatible observables given by POVMs. We present an interpretation of the interference term from the psychological viewpoint. As was shown before, the appearance of such a term (perturbing classical FTP) plays the important role in cognitive psychology, e.g., recognition of ambiguous figures and the disjunction effect. The interference term for observables given by POVMs has much more complicated structure than the corresponding term for observables given by Hermitian operators. We elaborate cognitive interpretations of different components of the POVMs-interference term and apply our analysis to a quantum(-like) model of decision making.