Quantum-like cognition#

Quantum-like (or quantum cognition) models use the mathematics of quantum probability to describe human judgement. Nothing in the brain is assumed to be quantum. The motivation is empirical: people’s answers show patterns that classical probability handles badly, such as question-order effects, the conjunction fallacy and violations of the law of total probability. Quantum probability represents beliefs as a vector and questions as projections, and projections need not commute, so the order of questions can matter.

        flowchart LR
    accTitle: Quantum-like cognition, diagram 1
    accDescr: A belief state is projected onto the yes subspace of question A, renormalised, then projected onto question B; the result is compared with the opposite order.
    B["belief state ψ<br/>unit vector"]:::hum --> PA["question A<br/>projector Pₐ"]:::op
    PA -->|"P(yes to A) = ‖Pₐψ‖²"| S1["state after answering A<br/>Pₐψ / ‖Pₐψ‖"]:::op
    S1 --> PB["question B<br/>projector P_b"]:::op --> R["P(yes A, then yes B)<br/>= ‖P_b Pₐ ψ‖²"]:::out
    R --> CMP{"compare with<br/>B-then-A order"}:::base
    classDef op fill:#2a1b3d,stroke:#d2a8ff,color:#e6edf3
    classDef out fill:#0f2e1a,stroke:#7ee787,color:#e6edf3
    classDef hum fill:#3d1414,stroke:#ff7b72,color:#e6edf3
    classDef base fill:#262626,stroke:#8b949e,color:#e6edf3,stroke-dasharray:4 3

From the basics#

Quantum-like versus quantum: Quantum-like models borrow the probability rules and run on any computer. Quantum models run as circuits on quantum hardware, and quantum-inspired algorithms borrow quantum ideas for classical heuristics; this library only covers the first. The concepts page explains the difference.
Projection and Lüders’ rule: Answering “yes” projects the belief onto the “yes” subspace and renormalises it. If projectors do not commute, \(\|P_b P_a \psi\|^2 \neq \|P_a P_b \psi\|^2\) in general: the order effect.
The QQ equality: Projective models make a parameter-free prediction: the total probability of giving the same answer to both questions does not depend on the order. Wang and Busemeyer tested it on dozens of national surveys. Quantum Mind provides the test (qq_test).
Interference: The probability of an outcome when an intermediate question is not asked can differ from the average over its answers. The difference is an interference term, as in the double-slit experiment.
Open-system dynamics: Beliefs that drift and lose coherence over time are modelled with density matrices and the Lindblad equation. Quantum Mind uses this for trust, which a robot can change just by asking about it.
Fair comparison: A quantum-like model is useful only if it predicts better than classical models fitted to the same data. Every family in the library ships with classical baselines and a distinctive test.

A small example: an order effect in three lines#

from quantum_mind.families.order_effects import QuantumOrderModel
m = QuantumOrderModel(a=2.35, b=0.97, g=0.58, ranks=(1, 2))
p = m.predict()
print(p['AB'].round(3), p['BA'].round(3))   # the joint answers depend on the order

Where Quantum Mind fits#

This is the library’s first pillar: twelve families of quantum-like models of judgement, decision, memory, concepts, trust and perception, each next to classical baselines, all fitted and compared the same way, with circuits for each.

References#

Busemeyer, J. R., & Bruza, P. D. (2024). Quantum Models of Cognition and Decision (2nd ed.). Cambridge University Press.
Pothos, E. M., & Busemeyer, J. R. (2013). Can quantum probability provide a new direction for cognitive modeling? Behavioral and Brain Sciences, 36(3), 255-274.
Wang, Z., & Busemeyer, J. R. (2013). A quantum question order model supported by empirical tests of an a priori and precise prediction. Topics in Cognitive Science, 5(4), 689-710.
Wang, Z., Solloway, T., Shiffrin, R. M., & Busemeyer, J. R. (2014). Context effects produced by question orders reveal quantum nature of human judgments. PNAS, 111(26), 9431-9436.
Pothos, E. M., & Busemeyer, J. R. (2022). Quantum cognition. Annual Review of Psychology, 73, 749-778.