Quantum-like cognition and decision-making in the light of quantum measurement theory

We characterize the class of quantum measurements that matches the applications of quantum theory to cognition (and decision-making)-quantum-like modelling. Projective measurements describe the canonical measurements of the basic observables of quantum physics. However, the combinations of the basic cognitive effects, such as the question order and response replicability effects (RREs), cannot be described by projective measurements. We motivate the use of the special class of quantum measurements, namely, <i>sharp repeatable non-projective measurements</i>-[Formula: see text] This class is practically unused in quantum physics. Thus, physics and cognition explore different parts of quantum measurement theory. Quantum-like modelling is not the automatic borrowing of the quantum formalism. Exploring the class [Formula: see text] highlights the role of <i>non-commutativity of the state-update maps generated by measurement back action</i>. Thus, 'non-classicality' in quantum physics as well as quantum-like modelling for cognition is based on two different types of non-commutativity, of operators (observables) and instruments (state-update maps): <i>observable non-commutativity</i> versus <i>state-update-non-commutativity</i>. We speculate that distinguishing quantum-like properties of the cognitive effects is the expression of the latter, or possibly both.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.