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arXiv:2503.05859 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

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

Miho Fuyama, Andrei Yu. Khrennikov, Masanao Ozawa

Published Nov 27, 2025• 7 citations
arXivHTMLPDF

Abstract

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, sharp repeatable non-projective measurements — S R P ¯ . 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 S R P ¯ highlights the role of non-commutativity of the state-update maps generated by measurement back action . 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): observable non-commutativity versus state-update-non-commutativity . 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)’.

DOI
10.1098/rsta.2024.0372

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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, sharp repeatable non-projective measurements — S R P ¯ . 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 S R P ¯ highlights the role of non-commutativity of the state-update maps generated by measurement back action . 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): observable non-commutativity versus state-update-non-commutativity . 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)’.
There are two basic theories of probability and two scientific perspectives – classical and quantum. In this chapter, assumptions of both the classical and the quantum probability theories will be presented and confronted. The rules for quantum probabilities differ from those of classical probabilities. In this chapter, we will describe what is the difference between the quantum cognitive modelling different from traditional cognitive modelling. In quantum probability theory (QPT) states of individual quantum, systems are called pure states, which are represented as vectors in a vector space (the Hilbert space). Properties of the Hilbert space and vectors’ operations will be explained. Two aspects of quantum theory open the door to addressing problems facing cognition and decision in a totally new light. The first is known as “contextuality” of judgements and decisions: the context generated by making a first judgement or decision interferes with subsequent judgements or decisions to produce order effects, so that judgements and decisions are non-commutative. The second aspect relates to quantum entanglement – a phenomenon whereby making an observation on one part of the system instantaneously affects the state in another part of the system, even if the respective systems are separated by space-like distances. This opens the door to developing quantum-like models of cognitive phenomena which are not decompositional in nature. Examples of cognitive processes modelled using the quantum account and results on the previous studies will be presented. We will also discuss the quantum–classical boundary.
The authors present the theory of quantum measurements in a humanities friendly way. The most general process of decision-making is represented with the aid of the formalism of quantum apparatuses and instruments. This measurement formalism generalizes the standard one based on the von Neumann–Lüders projection postulate. Generalized quantum observables are mathematically represented as positive operator valued measures (POVMs) and state transformers resulting from the feedback of measurements to the states of systems that are given by quantum instruments. The quantum scheme of indirect measurements (a special realization of quantum instruments) is applied to model decision-making as resulting from the interaction between the belief and decision states. The authors analyze the specific features of quantum instruments which are important for cognitive and social applications. In particular, the state transformers given by quantum instruments are in general less invasive than the state projections. Thus quantum-like decision-making need not be viewed as a kind of state collapse. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
An important challenge for quantum theories of cognition and decision concerns the incorporation of memory for recently made judgements and their effects on later judgements. First, we review a general approach to measurement based on system plus environment representations of states and measurement instruments. These more general measurement models provide ways to incorporate the effects of recent judgements on later judgements. Then we compare three different measurement models that are based on these more general measurement operations to account for a puzzling collection of question order effect findings.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.
Various effects in human cognition, often considered `non-classical', have been argued to be most naturally modelled by quantum-like models of decision making. We extend this approach to describe models of cognition and decision-making in general probabilistic process theories, which include both classical probabilistic models and quantum instrument models as special cases. We show how many aspects of quantum-like cognition can be described diagrammatically in process theories, before using our approach to assess the arguments for quantum-like models. While standard Bayesian classical models are insufficient, we prove that any sequential decision data can in fact be given a more general form of classical instrument model, and see that even simple deterministic models can exhibit all cognitive effects. Restricting attention to instruments induced by measurements, such as classical Bayesian and quantum POVM models, rules out such a result, but is challenged by the fact that such instruments cannot account for certain effects. Finally, we argue that to strictly rule out classical instrument models one should make use of parallel composition in the modelling of joint decisions, and find real world cognitive data violating Bell inequalities.
This short chapter contains basics of the mathematical formalism for thequantum measurement theory. In this book we proceed mainly withthe von Neumann measurement theory in which observables are given byHermitian operators and the state update by projections. However, we alsomention the measurement formalism based on quantum instruments, sinceit gives the general framework for quantum measurements. This formalismis used only in Chapters 10 and 18. The latter chapter is devoted to quantum-likemodeling – the applications of the mathematical formalism and methodologyof quantum mechanics (QM) to cognition, psychology, and decision making.Surprisingly, in such applications even the simplest effects can’t be described bythe von Neumann theory. One should use quantum instruments (compare withquantum physics where the main body of theory can be presented solelywithin the von Neumann measurement theory).