Quantum-Like Modelling of Cognition

• Quantum-like modelling of cognition is an interdisciplinary framework that uses quantum mathematical tools like Hilbert space and operator theory to represent complex cognitive states.
• It explains empirical phenomena such as order effects, interference, and contextuality, challenging classical probability with non-commutative, superposed mental processes.
• The approach extends to applications in decision-making, information retrieval, and neural dynamics, offering practical insights for analyzing and simulating human cognition.

Quantum-like Modelling of Cognition refers to the application of mathematical structures and probabilistic tools from quantum theory—not quantum physics per se—to model human cognitive processes, judgment, decision-making, information retrieval, and concept formation. This interdisciplinary paradigm captures systematic violations of classical (Kolmogorovian) probability theory, such as contextuality, interference, order effects, and non-Boolean logic, observed in empirical studies of cognition and behavior. Quantum-like approaches employ Hilbert space formalism, operator theory, and non-commutative probability, generalizing or replacing the joint-distribution assumptions of classical frameworks to account for the intrinsically contextual, superposed, and sometimes entangled nature of mental states.

1. Foundations: Hilbert Space, Non-Commutativity, and Cognitive States

Central to quantum-like modelling is the representation of cognitive states as vectors $|\psi\rangle$ or density matrices $\rho$ in a complex Hilbert space $\mathcal{H}$ (Khrennikov, 2023). Questions, judgments, and mental properties are modeled as Hermitian operators $A$, with outcomes corresponding to spectral projectors $\{\Pi_a\}$. The Born rule computes probabilities of responses as $P(a)\!=\!\mathrm{Tr}(\rho \Pi_a)$, allowing for contextual influences and dependencies beyond classical conditioning.

Non-commutativity is key: pairs of cognitive observables $A$ and $B$ generally do not commute ($[A, B]\neq0$), which induces order effects—i.e., the sequence in which questions are posed affects joint probabilities $P_{AB}(a,b)\neq P_{BA}(b,a)$ (Uprety et al., 2020). Measurements are treated not as mere queries but as operators that actively update the cognitive state, introducing state-dependence and “collapse” phenomena analogous to quantum measurement (Moreira et al., 2019).

2. Interference, Superposition, and Quantum Logic

Quantum-like models naturally encode superposition: prior to judgment, mental states can exist in linear combinations of alternatives, $|\psi\rangle = \alpha |A\rangle + \beta |B\rangle$, with ambiguity or indecision modeled as nontrivial coefficients (Maksymov, 11 Aug 2025). Measurement (a decision or answer) collapses the state, enforcing selection and altering the probabilities of subsequent judgments (“order effects”).

Interference is an empirical signature: in decision scenarios involving conjunctions or disjunctions (e.g., Tversky & Shafir’s “disjunction effect,” Kahneman’s “conjunction fallacy”), observed probabilities systematically violate the classical formula of total probability, but fit quantum-like models with nonzero interference terms:

$$P(B) = P(A)P(B|A) + P(\neg A)P(B|\neg A) + 2\sqrt{P(A)P(B|A)P(\neg A)P(B|\neg A)}\cos\phi$$

for an appropriate phase $\phi$ (Moreira et al., 2019, Widdows et al., 2023, Uprety et al., 2020).

Underlying these features is a shift from Boolean logic (distributive lattice of events) to quantum logic (orthomodular lattices), where distributivity fails and context-sensitivity emerges from noncommuting projectors (Gunji et al., 2014, Khrennikov, 2023).

3. Measurement Theory, Instruments, and Generalizations

Standard quantum-like models use projective measurements (Lüders rule) for state updates. However, this framework cannot simultaneously accommodate question-order effects (QOE) and response-replicability effects (RRE) observed in cognition (e.g., subjects repeat previous answers, but answer order affects outcomes) (Fuyama et al., 7 Mar 2025). The class of sharp, repeatable, non-projective (SR̄P) instruments was introduced to resolve this: these are measurement maps $\{\mathcal{I}(x)\}$ such that each is sharp (associated with a projective effect), repeatable (immediate repetitions yield the same outcome with certainty), but not Lüders/projective in update. Instrument noncommutativity—even for commuting observables—is essential for modeling joint QOE/RRE regimes.

Comparison of instrument classes:

Model Class	Allows QOE	Allows RRE	QQ-equality
Projective	Yes	No	Yes
SR̄P	Yes	Yes	Yes/No

Traditional projective models are thus special cases within a more general instrument calculus.

4. Conceptual Spaces, Fock Space, and Quantum Combination of Concepts

Quantum concept theory treats individual concepts as vectors in Hilbert space, and their combinations (conjunction/disjunctions) via superpositions and tensor products. The Brussels Fock space model $\mathcal{F}(H) = H_1 \oplus H_2$ encodes logical combinations in $H_2$ (tensor) and emergent novel concepts in $H_1$ (superposition) (Aerts et al., 2013, 0805.3850). Membership probabilities for combined concepts are computed as mixtures of classical (product) and quantum (interference) terms, fitting anomalous empirical data (Guppy effect, over- and underextension) that violate fuzzy-min/max rules, but arise naturally in models with interference and contextuality.

Quantum-like approaches have been extended to high-dimensional formal models of conceptual spaces, combining geometric and operational perspectives (e.g., Gärdenfors convex regions with vector superpositions) and algorithms for conceptual scaling in cognitive learning tasks (S et al., 2018).

5. Neurobiological Substrates and Quantum-Like Information Processing

Despite the macroscopic scale of neurons and synapses, quantum-like information processing can be realized by neuronal network activity. Models built from oscillatory neural circuits employ classical electromagnetic signals as basis for abstract cognition, with covariance operators normalized to produce quantum-like density matrices $\rho$ for mental states (Khrennikov et al., 27 May 2025, Khrennikov, 2010).

PCSFT (prequantum classical statistical field theory) formalizes this bridge: neuronal oscillations yield covariance matrices $C$, which, after normalization, become cognitive states $\rho=C/\mathrm{Tr}C$; observables are quadratic forms, and classic interference, superposition, and entanglement become emergent correlates of structured oscillatory dynamics (Khrennikov et al., 17 Sep 2025). Experimental predictions involve EEG/MEG observable signatures, Bell-type inequalities for mental entanglement, and operational quantum-like effects detected in cognitive and neurophysiological tests.

Posner models posit biological realizations of quantum-protected information storage and computation via phosphorus nuclear spins in Ca₉(PO₄)₆ molecules, enabling theoretical quantum error correction, communication, and computation (Halpern et al., 2017).

6. Generalized Probability Theory and Operational Embedding

Recent neural-cognition models extend quantum-like formalisms beyond standard Hilbert space, embedding neuronal networks in ordered-linear-state spaces (GPT framework). Here, the states are normalized nonnegative matrices (e.g., synaptic weights), and cognitive observables/effects are positive linear functionals (Khrennikov et al., 29 Oct 2024). Instrument noncommutativity (of measurement maps) substitutes for operator noncommutativity, achieving quantum-like order and interference effects in settings not strictly quantum. GPT-based models accommodate order effects, non-repeatability, and interference phenomena, and can be applied to medical diagnostics, social networks, and biological systems.

7. Algorithms, Applications, and Empirical Outcomes

Quantum-like cognitive architectures can be instantiated both in simulation and on quantum circuit hardware. Mental states and inquiries are encoded via qubit registers and gate operations; non-commutativity, entanglement, and collapse are mapped to specific circuit elements (Widdows et al., 2023, Wang et al., 2019). Quantum-inspired computation exploits oscillatory neural networks for collective inference and decision, demonstrating scalability and robustness in AI, financial, defense, and social systems (Maksymov, 11 Aug 2025).

Case studies include quantum probability ranking in IR (density operator similarity), contextual information retrieval using superposition, and modeling band formation, polarization, and consensus dynamics in social networks via tensor-product constructions and open-system decoherence (Aerts et al., 2013, Maksymov, 11 Aug 2025).

8. Representative Cognitive Phenomena and Experimental Evidence

Quantum-like models have successfully addressed:

• Order effects: Noncommuting measurements reproduce response alteration due to question sequence (Uprety et al., 2020, Khrennikov, 2023).
• Conjunction/disjunction fallacies: Interference terms boost combined probabilities above classical limits (Moreira et al., 2019).
• Contextuality: Failure of classical joint distributions, explained via quantum-like contextuality measures (Khrennikov, 2023).
• Replicability and memory effects: Specialized instrument models achieve simultaneous order effects and response replicability (Fuyama et al., 7 Mar 2025, Franco, 2016).
• Medical diagnostics: EEG-derived GPT representations offer improved discriminative power between neuro-psychiatric conditions (Khrennikov et al., 29 Oct 2024).
• Conceptual emergence and IR/NLP: Quantum contextual and meaning-based approaches outperform classical vector-space models in representing semantic and typicality data (Aerts et al., 2013, 0805.3850).

9. Open Questions, Extensions, and Future Directions

Key research directions include development and testing of cognitive triple-slit experiments to probe limits of quantum probabilistic formalism and search for higher-order interference; operational axiomatizations of quantum-like probability for cognition; integration of GPT and Hilbert-space models; and empirical validation of mental entanglement and open-system cognitive dynamics using neuroimaging techniques (Basieva et al., 2016, Khrennikov et al., 17 Sep 2025).

Quantum-like modelling of cognition provides a mathematically principled, empirically validated framework capturing the contextual, nonclassical, and dynamic properties of human thought. It continues to expand across cognitive psychology, neuroscience, artificial intelligence, information retrieval, and collective social behaviour, driven by increasingly sophisticated theoretical, computational, and experimental methodologies.