Quantum-Like Models of Cognition

• Quantum-like modelling of cognition is an approach that uses Hilbert-space formalism and quantum probabilistic methods to represent complex mental processes.
• It employs quantum instruments, interference effects, and non-classical probability rules to explain contextuality, order effects, and paradoxes in decision making.
• This paradigm bridges classical statistical models with quantum-inspired computation, offering novel insights for neural architectures and artificial intelligence.

Quantum-like modelling of cognition is a research paradigm in which cognitive states, processes, and phenomena are formalized using mathematical methods originally developed for quantum theory, particularly the Hilbert-space formalism, projective measurement calculus, and open-system dynamics. Unlike reductionist quantum brain theories, quantum-like models do not posit genuine microscopic quantum processes in neural substrates but exploit the quantum-theoretic formalism as a generalized information-theoretic and probabilistic framework for understanding non-classical cognitive phenomena—contextuality, order effects, interference, superposition, and entanglement—observed in human decision-making, concept combination, and perception. This approach has demonstrated explanatory and predictive power for a range of paradoxical patterns that classical probability and logic cannot account for (Moreira et al., 2019, Bruza et al., 2013, Aerts et al., 2013, Khrennikov et al., 27 May 2025, Khrennikov, 2023, Fuyama et al., 7 Mar 2025, Maksymov, 5 Apr 2026, Khrennikov et al., 17 Sep 2025, Uprety et al., 2020, Basieva et al., 2016).

1. Hilbert Space Formalism and Representation of Mental States

In quantum-like cognitive models, each agent’s “mind” or conceptual entity is represented as a (typically finite-dimensional) complex Hilbert space $\mathcal{H} \cong \mathbb{C}^N$ (Moreira et al., 2019). The mental state is modelled by either:

• Pure state: A normalized vector $|\psi\rangle \in \mathcal{H}$, $\langle\psi|\psi\rangle=1$, expressing a definite but in general indeterminate cognitive state; e.g.,

$$|\psi\rangle = \sum_{i=1}^N \alpha_i |e_i\rangle,\quad \sum_i |\alpha_i|^2 = 1$$

where $\{|e_i\rangle\}$ corresponds to basic mental propositions or conceptual features.

• Mixed state (Density operator): A statistical mixture or uncertainty over pure states,

$$\rho = \sum_k p_k |\psi_k\rangle\langle\psi_k|, \quad \rho = \rho^\dagger, \quad \rho \succeq 0, \quad \operatorname{Tr} \rho = 1$$

which is essential for modeling populations, incomplete information, or decohered cognitive processes (Bruza et al., 2013, Khrennikov, 2023).

This state space enables representation of mental superposition (simultaneous potentialities), statistical uncertainty, and, via tensor-product structures, combinatorial conceptual spaces and multi-agent systems (Aerts et al., 2013).

2. Observables, Measurements, and State Updates

Decision questions, judgments, and cognitive observables (e.g., "Is Clinton honest?") are represented by Hermitian operators $\hat{A}$ on $\mathcal{H}$, typically via spectral decompositions

$$\hat{A} = \sum_i a_i P_i,\quad P_i^2 = P_i = P_i^\dagger,\quad \sum_i P_i = I$$

where each $P_i$ is a projection operator onto the subspace associated with outcome $|\psi\rangle \in \mathcal{H}$0 (Moreira et al., 2019).

Measurement (decision/action) is formalized via the Born rule:

$|\psi\rangle \in \mathcal{H}$1

Post-measurement, the state updates according to the "collapse" or Lüders rule:

$|\psi\rangle \in \mathcal{H}$2

When modeling sequential decisions or continuous beliefs, the mental state can evolve unitarily:

$|\psi\rangle \in \mathcal{H}$3

or, more generally, via a Lindblad-type open-system master equation for density matrices:

$|\psi\rangle \in \mathcal{H}$4

where $|\psi\rangle \in \mathcal{H}$5 captures internal deliberation, $|\psi\rangle \in \mathcal{H}$6 encode environmental/cognitive noise or context (Moreira et al., 2019, Asano et al., 19 Apr 2026, Khrennikov, 2023).

3. Emergence of Cognitive Biases and Quantum-Like Effects

Quantum-like models naturally account for key empirical violations of classical probability and logic in cognition:

• Violation of the law of total probability: In dichotomic scenarios, an interference term automatically appears,

$|\psi\rangle \in \mathcal{H}$7

explaining the disjunction effect (Moreira et al., 2019, Bruza et al., 2013, Basieva et al., 2016).

• Conjunction fallacy: For non-commuting projections, quantum probability allows,

$|\psi\rangle \in \mathcal{H}$8

which can exceed $|\psi\rangle \in \mathcal{H}$9, matching “Linda problem” patterns (Moreira et al., 2019, Bruza et al., 2013).

• Order effects: When the projection operators for two questions do not commute, the probability for sequential answers depends on order,

$\langle\psi|\psi\rangle=1$0

• Contextuality and non-existence of joint distributions: The requirement that measurement operators (questions) cannot all be jointly assigned probabilities consistent with all marginals and conditionals, as per the contextuality-by-default framework (Moreira et al., 2019, Uprety et al., 2020).
• Entanglement in concept combination: Composite or holistically-bonded concepts (e.g., "pet-fish" and the “guppy effect”) and joint decisions are modeled in tensor-product Hilbert spaces, supporting entangled states not decomposable into independent marginals (Aerts et al., 2013, 0805.3850).
• Interference and non-classical combination rules: The quantum formalism provides interference terms allowing overextension and underextension in concept conjunctions/disjunctions, immediately fitting empirical deviations from classical min/max rules (0805.3850, Aerts et al., 2013).

4. Measurement Theory, Instruments, and Non-Projective Updates

Quantum-like cognition demands a generalized theory of measurement beyond projective (Lüders) measurement.

• Quantum instruments (POVMs): Modern approaches encode cognitive measurements using Positive Operator-Valued Measures and associated state-update maps (instruments), with Kraus decompositions:

$\langle\psi|\psi\rangle=1$1

This enables modeling of “fuzzy” or probabilistic answer sets, allowing the simultaneous empirical fit of both order effects and response replicability—beyond the reach of projective measurement-only models (Fuyama et al., 7 Mar 2025, Khrennikov, 2023).

• Sharp Repeatable Non-Projective Measurements ($\langle\psi|\psi\rangle=1$2): Instruments of the form $\langle\psi|\psi\rangle=1$3, with partial isometries $\langle\psi|\psi\rangle=1$4 satisfying $\langle\psi|\psi\rangle=1$5, allow simultaneous sharpness (POVMs are projectors), repeatability, and non-projectivity (state-update is not simple projection). Noncommutativity of the state-update maps (i.e., the instrument) is critical for reproducing cognitive order effects and response replicability (Fuyama et al., 7 Mar 2025).
• Observable vs. update noncommutativity: The distinction between noncommuting observables (traditional quantum incompatibility) and noncommuting state-update maps (instrument calculus) is central. The latter enables cognitive models to produce order effects even when observables commute, a property not possible in standard (physical) quantum measurement (Fuyama et al., 7 Mar 2025).

5. Quantum-Like Models in Neural and Information Processing Architectures

Modern quantum-like modeling aims to bridge cognitive phenomena with neurodynamics and artificial systems via several mathematical constructions:

• Oscillatory neuronal networks and Prequantum Classical Statistical Field Theory (PCSFT): Macroscopic neuronal assemblies are modeled as systems of classical oscillators with random amplitudes $\langle\psi|\psi\rangle=1$6; covariance operators $\langle\psi|\psi\rangle=1$7 of these random fields are mapped to density matrices $\langle\psi|\psi\rangle=1$8 (Khrennikov et al., 27 May 2025, Khrennikov et al., 17 Sep 2025). Tensor-product decompositions via local operator algebras yield notions of “mental entanglement” at a coarse-grained, population-code level, detectable via partial transpose (PPT) criteria applied to EEG/MEG-derived covariance matrices.
• Quantum-tunnelling oscillator models: Perceptual ambiguity, bistable perception (e.g., Necker cube), and collective decision-making are framed as Schrödinger-like quantum tunnelling in engineered double-well (or multi-well) potentials, where cognitive states oscillate and tunnel between alternatives; networked versions explain social “bubbles” and polarization (Maksymov, 5 Apr 2026, Maksymov, 11 Aug 2025).
• Quantum-inspired computation and QL AI: Quantum-like algorithms using Hilbert-space–based representations of mental states and measurements, or networks of classical oscillators engineered to perform “quantum-inspired computation,” are proposed as efficient substrates for modeling and implementing humanlike and context-sensitive inference in AI, with measurable computational advantages (e.g., superpositional parallelism, interference-driven learning) (Khrennikov et al., 27 May 2025, Wang et al., 2019, Maksymov, 11 Aug 2025).

6. Key Theorems, Evaluation, and Empirical Validation

Quantum-like cognition provides rigorous, testable generalizations of classical probabilistic modelling. Notable results include:

Classical Constraint	Quantum-Like Generalization	Empirical/Mathematical Consequence
Law of total probability	Interference-amended total probability (Born rule + cross-term)	Disjunction effect, order effects
Sure-thing principle (Savage)	Born-rule formula with phase-dependent interference term	Violations in decision tasks
Existence of joint distributions	Contextuality-by-default; non-existence of global joint measures	Contextuality in sequential tasks
Classical logical bounds (min/max)	Interference and emergent, Fock-space combination formulas	Overextension/underextension
Sequential/parallel Markov chains	Quantum/non-commuting instrument process theories	CHSH/Bell inequality violations

Quantum-like models outperform Bayesian or Markovian models in predicting and fitting empirical order effects, conjunction/disjunction fallacies, context effects in multidimensional relevance judgment, and non-factorizable correlations in collective decisions (Moreira et al., 2019, Uprety et al., 2020, Basieva et al., 2016, Maksymov, 11 Aug 2025). However, Theorems in process-theoretic analysis (Tull et al., 8 Apr 2026) demonstrate that:

• Sequential data can always be modeled by sufficiently general (possibly non-measurement) classical Markov instruments.
• Order effects and contextuality only force genuinely non-classical (quantum) models if one observes Bell/CHSH-type violations in parallel/joint decision tasks.

Laboratory and online experiments using Stern–Gerlach–inspired designs, cognitive triple-slit analogs, and EEG/MEG entanglement measures are being developed as empirical tests for higher-order quantum-like phenomena (Uprety et al., 2020, Basieva et al., 2016, Khrennikov et al., 17 Sep 2025).

7. Extensions, Open Problems, and Future Directions

Quantum-like cognitive modeling remains a rapidly developing interdisciplinary field. Open questions and frontiers include:

• Establishing normative and axiomatic justifications for Hilbert-space dimension, measurement design, and phase parameter interpretation in cognitive models (Moreira et al., 2019, Bruza et al., 2013).
• Integrating more general probabilistic frameworks beyond standard Hilbert-space quantum mechanics, as motivated by empirical phenomena unaccounted for by the Born rule (e.g., third-order interference, unpacking effects) (Aerts et al., 2016, Basieva et al., 2016).
• Unifying the emerging links between classical neural oscillatory architectures and the formal machinery of quantum-like information processing, with experimental work targeting the detection of “mental entanglement” and operational mapping of cognitive observables to neural data (Khrennikov et al., 27 May 2025, Khrennikov et al., 17 Sep 2025).
• Developing hybrid quantum–classical or neuromorphic computational architectures for AI that exploit quantum-like processing at the hardware or algorithmic level (Maksymov, 11 Aug 2025, Khrennikov et al., 27 May 2025).
• Refining open-systems and dynamical treatments of cognition, where the temporal evolution of mixed or undecided states is modeled by GKSL-type master equations, predicting phenomena such as “cognitive beats” indicative of internal deliberative conflict and escape from classical decision equilibria (Asano et al., 19 Apr 2026).

Quantum-like models, through their mathematically rigorous extension of classical probabilistic and logical frameworks, provide a rich, generative paradigm for cognitive science, producing explanatory and predictive advances in modeling both individual and collective aspects of human mental life (Moreira et al., 2019, Aerts et al., 2013, Khrennikov et al., 27 May 2025, Khrennikov, 2023, Asano et al., 19 Apr 2026).