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Quantum-Like Representation of Dynamic Functional Connectivity: Double Covariance Formalism

Andrei Yu. Khrennikov, Makio Yamada, Irina Basieva

Published Jun 23, 2026• 0 citations
Publisher

Abstract

We introduce a neurophysiological adaptation of the Double Covariance Model (DCM) to provide a generative, fourth-order statistical framework for brain dynamics. Moving beyond descriptive sliding-window methods, the proposed theory implements a two-scale temporal scheme that explicitly separates fast micro-time stochastic neural fluctuations from the macro-time scale of emergent cognitive states. By treating the variance of covariance as a first-class mathematical object, the model computes the fourth-order moment of localized micro-signals to directly reconstruct a complex-valued network density matrix ($\rho$). This architecture transitions classical neurophysiological data into a quantum-like state representation, enabling the direct application of quantum information measures—such as concurrence and von Neumann entropy—to model macroscopic integration. Ultimately, this framework establishes a principled, non-phenomenological link between low-level stochastic network dynamics and the unified macroscopic phenomenon of "mental entanglement".

DOI
10.20944/preprints202606.1616.v1

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This article presents a concrete mathematical framework for the generation of entangled quantum states from classical stochastic processes. We demonstrate that any density operator ρABρ_{AB}ρAB​ of a composite system can be derived from the correlations between two underlying stochastic processes, X(t)X(t)X(t) and Y(t)Y(t)Y(t), representing the random fluctuations of its subsystems. This construction utilizes a two-scale temporal scheme - micro and macro time - where quantum correlations emerge as macro-correlations derived from underlying micro-correlations. We propose the Double Covariance Model (DCM), which reproduces the fundamental properties of quantum theory by treating the quantum state as the fourth-order moment structure of an underlying classical probability space.
We develop an interacting extension of the Double Covariance Model (DCM), a stochastic subquantum framework in which macroscopic quantum dynamics emerge through coarse-graining of correlated microscopic fluctuations. Starting from local stochastic differential equations on subsystem Hilbert spaces, we derive a closed evolution equation for a coarse-grained double covariance operator using multi-scale Itô calculus and sliding-window averaging. The construction explicitly incorporates two separated temporal scales: a fast microscopic fluctuation scale governing subquantum stochastic processes and a slower macroscopic observation scale associated with coarse-grained dynamics. Within the hydrodynamic limit, where the ratio between microscopic correlation time and averaging-window scale vanishes, rapidly fluctuating corrections disappear and the effective dynamics converges to a deterministic macroscopic transport equation. We show that the emergent macroscopic dynamics has the exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form: coherent Hamiltonian evolution arises from deterministic subquantum flow, while dissipative channels emerge from quadratic noise correlations. The framework further demonstrates how non-separable interaction Hamiltonians can arise from strictly local, state-dependent stochastic feedback fields. In the fluctuation-free limit, the model reduces naturally to the standard von Neumann equation, providing a unified stochastic foundation for both open and closed quantum dynamics.
This work introduces a rigorous mathematical approach for producing entangled quantum states from classical stochastic dynamics. We show that any density matrix ρAB describing a composite quantum system can be reconstructed from the correlations of two foundational stochastic processes, X(t) and Y(t), which model the random behavior of the individual subsystems. The framework employs a dual temporal scale—micro and macro time—where quantum correlations naturally arise as emergent macro-level correlations derived from fine-grained micro-level interactions. We formulate the Double Covariance Model (DCM), which captures the essential features of quantum mechanics by interpreting the quantum state as a fourth-order statistical structure within an underlying classical probabilistic model.
We propose a novel approach to the problem of interconnecting the probabilistic formalisms of classical and quantum physics, focusing on its most challenging aspect: the classical probabilistic generation of entangled states. We show that the statistics encoded in the density operators of composite quantum systems correspond to fourth-order classical statistics. Specifically, to generate a density operator, one must consider the covariance of a random covariance operator. We term this framework the Double Covariance Model (DCM). This double covariance possesses a non-trivial internal structure that arises from the interplay between two distinct time scales, combining temporal and statistical covariances. In this article, we exploit a well-known property of Gaussian processes: the second-order moment determines the moments of higher orders, specifically the fourth-order moment. This Gaussian reduction simplifies the DCM by reducing it to second-order statistics. Utilizing (circular) Gaussian processes simplifies and generalizes the DCM construction for entangled states, rendering it mathematically rigorous. Furthermore, it clarifies the classical probabilistic meaning of concurrence, a foundational quantitative measure of entanglement.
We propose a novel approach to the problem of interconnecting the probabilistic formalisms of classical and quantum physics, focusing on its most challenging aspect: the classical probabilistic generation of entangled states. We show that the statistics encoded in the density operators of composite quantum systems correspond to fourth-order classical statistics. Specifically, to generate a density operator, one must consider the covariance of a random covariance operator. We term this framework the Double Covariance Model (DCM). This double covariance possesses a non-trivial internal structure that arises from the interplay between two distinct time scales, combining temporal and statistical covariances. In this article, we exploit a well-known property of Gaussian processes: the second-order moment determines the moments of higher orders, specifically the fourth-order moment. This Gaussian reduction simplifies the DCM by reducing it to second-order statistics. Utilizing (circular) Gaussian processes simplifies and generalizes the DCM construction for entangled states, rendering it mathematically rigorous. Furthermore, it clarifies the classical probabilistic meaning of concurrence, a foundational quantitative measure of entanglement.
Quantum-like modeling (QLM) – quantum theory applications outside of physics – are intensively developed with applications in biology, cognition, psychology, and decision-making. For cognition, QLM should be distinguished from quantum reductionist models in the spirit of Hameroff and Penrose and well as Umezawa and Vitiello. QLM is not concerned with just quantum physical processes in the brain but also QL information processing by macroscopic neuronal structures. Although QLM of cognition and decision-making has seen some success, it suffers from a knowledge gap that exists between oscillatory neuronal network functioning in the brain and QL behavioral patterns. Recently, steps toward closing this gap have been taken using the generalized probability theory and prequantum classical statistical field theory (PCSFT) – a random field model beyond the complex Hilbert space formalism. PCSFT is used to move from the classical ``oscillatory cognition'' of the neuronal networks to QLM for decision-making. In this study, we addressed the most difficult problem within this construction: QLM for entanglement generation by classical networks, i.e., “mental entanglement.” We started with the observational approach to entanglement based on operator algebras describing “local observables” and bringing into being the tensor product structure in the space of QL states. Moreover, we applied the standard states entanglement approach: entanglement generation by spatially separated networks in the brain. Finally, we discussed possible future experiments on “mental entanglement” detection using the EEG/MEG technique.