Quantum-like Modeling in Cognition

• Quantum-like Modeling is a mathematical framework that employs quantum mechanics concepts, such as Hilbert spaces and density operators, to represent abstract cognitive processes.
• It transforms high-dimensional, concrete electromagnetic signals from neuronal activity into compressed, operator-based states for efficient abstraction and decision-making.
• The framework enables dynamic interplay between sensory data and abstract concepts, offering insights into memory recall, perceptual binding, and the nuances of brain information processing.

Quantum-like Modeling (QLM) refers to a class of mathematical frameworks that employ the structural apparatus of quantum theory—including Hilbert spaces, density operators, noncommuting observables, and linear operator dynamics—to model complex information processing outside quantum physics proper. QLM distinguishes itself from reductionist models aiming to ground cognitive or biological function in microphysical quantum processes; instead, it provides a rigorous means to represent and analyze nonclassical features (e.g., interference, contextuality, entanglement analogues) in macroscopic systems such as the brain, biosystems, or artificial intelligence. The formalism is fundamentally grounded in classical signal theory, with macroscopic brain dynamics modeled via the statistical structure of classical electromagnetic fields, where the transition from concrete, high-dimensional sensory data to abstract, compressed mental representations is achieved through the construction of quantum-like (QL) states.

1. Dual Information Coding: Classical and Quantum-like Regimes

QLM postulates that information in the brain is processed simultaneously in two regimes:

• Classical Regime: Concrete sensory and perceptual data are encoded by classical electromagnetic signals generated by collective neuronal activity. These are mathematically represented in the form of (Gaussian) random fields $\phi(x)$, each characterized by a covariance operator $D$ in a Hilbert space $H$ (typically $H = L^2(O)$ over a spatial domain $O$).
• Quantum-like Regime: For abstract cognitive operations—decision making, abstract thinking, conceptualization—the statistical properties of many such classical signals are compressed into a density operator $p$:

$p = \frac{D}{\operatorname{Tr} D}$

Here, $p$ encapsulates the higher-order statistical correlations (covariances) relevant to the concept, thus realizing a compressed, quantum-like representation (QLR) in the Hilbert space.

A key principle of QLM is active bidirectional interaction between these regimes. Concrete images can be “abstracted” into QL states for fast, symbolic processing, and QL concepts can be “decoded” into possible realizations (Gaussian field samples) in the classical regime, enabling the dynamic switching between perception and imagination, or abstraction and recall.

2. Mathematical Structure: Covariance and Density Operators

The technical core of QLM lies in reinterpreting the formalism of quantum mechanics through classical signal theory constructs:

• Covariance Operator ($D$): For a Gaussian random field $\phi$ with zero mean, the covariance operator is defined as

$$(Du, v) = E\left[ \left( \int \phi(x)u(x)\, dx \right) \left( \int \phi(x)v(x)\, dx \right) \right]$$

for test functions $u, v \in H$. Empirically, this operator captures the full information about signal correlations, and therefore about the underlying mental image.

• Density Operator ($p$): Normalization of $D$ yields the QL state:

$$p = \frac{D}{\operatorname{Tr} D}, \quad \operatorname{Tr} p = 1$$

This operator serves as the compressed code for the abstract concept. In the QLM model, all QL processing is encoded in operator dynamics, expectation values, and unitary or dissipative evolution, e.g., via the von Neumann equation:

$\frac{dp(t)}{dt} = -i[H, p(t)]$

where $H$ is a “mental Hamiltonian” governing the evolution of QL states.

• Decoding Abstract Concepts: To reconstruct concrete realizations (e.g., possible sensory mental images), the density operator $p$ is used as the covariance matrix for generating new samples of Gaussian random fields, ensuring each realization is consistent with the abstract concept but retains concrete variability.

3. Quantum-like Observables and Averages

QLM assigns abstract mental qualities (“mental qualia”) to self-adjoint operators $A$ acting on $H$. The subjective value (“feeling,” “feature,” or statistical relevance) assigned to $A$ for a given mental image/state $p$ is computed via the quantum trace formula:

$\langle A \rangle_p = \operatorname{Tr}(p A)$

This mechanism generalizes the classical averaging procedure for observables, allowing for linear superpositions and operator manipulations that underlie nonclassical cognitive effects—such as the context dependence of decision making, probabilistic interference, and order effects.

4. Temporal Multiplexing: Scale Separation in Information Integration

A crucial feature of QLM is the exploitation of multiple temporal scales in neuronal signal processing:

• Physical (Fine) Time Scale ($t$): The electromagnetic field varies on fast timescales ($\sim 1$ ms), reflecting rapid neural oscillations and raw sensory input.
• Mental (Coarse) Time Scale ($T$): On much longer timescales ($T \sim 80$ ms, a typical mental integration window), the brain performs ergodic averaging:

$$(f) = \lim_{T \to \infty} \frac{1}{T} \int_0^T f(\phi(s)) ds$$

This procedure “filters” the high-frequency noise and produces stable covariance statistics suitable for encoding as a QL state.

The hierarchy of time scales enables the system to abstract robust, noise-immune representations of perceptual input and to convert noisy, concrete signals into crisp, operator-defined, fast-to-process QL representations.

5. Implications and Applications in Cognitive Neuroscience

The QLM framework provides the following implications for cognitive science and neuroscience:

• Efficient Abstract Processing: The use of operator-based QL states allows the brain to quickly process, manipulate, and combine abstract concepts without retaining every concrete detail, aligning with the observed speed and generality of high-level cognitive operations.
• Decision Making with Incomplete Information: Because QL averages entail the suppression of fine-grained details in favor of dominant statistical correlations, QLM naturally explains how decisions can be made under uncertainty and incomplete input.
• Binding Problem and Nonlocality: The model posits that high-level correlations encoded in the covariance operator—potentially driven by background electromagnetic field fluctuations—could mediate “super strong” statistical (not quantum-physical) correlations between spatially separated neuronal circuits, offering an explanation for the integration of distributed brain activity into unified percepts (the binding problem).
• Memory Storage and Recall: Mapping from concrete, time-dependent images (covariance statistics of Gaussian fields) to abstract QL states (density operators) provides a viable mechanism by which memories can be stored in compressed, conceptual form and later regenerated as specific images or scenarios.
• Concept Formation: By clustering across many instances (e.g., individual images of “houses”), the brain can construct low-dimensional QL representations (via spectral projection onto subspaces), thus efficiently building concepts from high-dimensional, noisy input streams.

6. Theoretical Distinction from Quantum Physical Models

A defining tenet of this QLM approach is the lack of reliance on microscopic quantum physical processes in the brain. Instead, the formalism is imported as an information-theoretic structure—a spectral coding strategy over classical electromagnetic field statistics. This positions QLM as a bridge between classical neural computation and the formal tools of quantum theory, suggesting that some features previously considered haLLMarks of quantum physics (e.g., operator noncommutativity, interference, linear trace averages) can “emerge” on the macroscopic scale from appropriate statistical signal encoding.

QLM therefore challenges strict divisions between classical and quantum paradigms within the neuroscience of cognition. It suggests that the brain’s information processing is fundamentally classical at the substrate level but quantum-like at the level of data compression, abstraction, and symbolic manipulation.

7. Tabular Summary: Structure–Function Map in the QLM Brain Model

QLM Entity	Classical Analogue	Cognitive Function
Gaussian field $\phi$	Electromagnetic brain signal	Concrete mental image
Covariance $D$	Statistical correlations	Encoded sensory correlation
Density $p$	Normalized covariance	Abstract concept/QL state
Observable $A$	Linear operator	Mental quality/decision probe
$\langle A \rangle_p$	Trace average	Qualia value/decision output
$T >\!> t$ averaging	Ergodic integration	Abstraction, noise filtering

This structure-function map illustrates the direct correspondence between formal entities in QLM and their roles in a statistical model of cognition based on classical brain dynamics.

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In sum, Quantum-like Modeling, as developed in this framework, provides a mathematically rigorous, operator-centric theory that allows macroscopic systems—specifically, brains characterized by classical electromagnetic activity—to exploit the compression and computational benefits of quantum information processing without invoking actual microphysical quantum effects (Khrennikov, 2010). This establishes a robust theoretical foundation for operator-based models of cognition and opens new avenues for experimental and computational exploration of abstraction, learning, binding, and decision making in neural systems.