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arXiv:2607.11968 · Journal of Mathematical Physics

Double Covariance Model for Entangled Quantum States: Gaussian Reduction to Second Order Covariances

Andrei Khrennikov

quant-phSubmitted Jul 12, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

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.

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Invited talk at QIP26 conference, Vaxjoe, June 2026

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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.
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.
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 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 (ρ\rhoρ). 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".
The characterisation of quantum correlations, also known as entanglement, is central to quantum information theory. Operationally, entanglement gives rise to the notion of quantum non-locality, which challenges the ‘reality’ of quantum mechanics (adopting the terminology of Einstein, Podolski and Rosen) and arguably represents its most distinctive feature, in the sense that it calls for a departure from certain foundational tenets of classical physics. It should thus not come as a surprise that entanglement turns out to be a key resource for the implementation of a number of quantum communication protocols and also necessary to achieve computational speed-ups with quantum hardware, in the sense that accessing regions of the Hilbert space with entangled states allows one to shorten the computational depth of certain algorithms. Not surprisingly, generating and maintaining entanglement is generally difficult in practice, as it requires strong interactions between selected components of a system that must at the same time be isolated from the environment. Thanks to the techniques developed in quantum optical set-ups to achieve squeezed light, entanglement is, however, comparatively accessible with optical quantum continuous variables. As we shall see, in fact, continuous variable entanglement is closely related to the notion of squeezing. It is therefore of the utmost importance to be able to qualify and quantify the entanglement of quantum continuous variables, which is the subject matter of the present chapter. The first part of the chapter is devoted to techniques and criteria based on second moments to detect the entanglement of Gaussian states and also to the description of possible ways to quantify such quantum correlations, bearing in mind that the determination of entanglement monotones of clear operational significance is still an open problem, even in the restricted arena of Gaussian states. Although the criteria based on second moments are sufficient, but not necessary, for the detection of non-Gaussian entanglement too, we will push our analysis further by introducing more general criteria based on higher order statistical moments, which will be put to use to detect, on paper, the quantum correlations of an entangled cat-like state. We will also address the issues of entanglement distillation and quantum non-locality tests, in whose contexts we shall encounter other non-Gaussian operations and states, such as photon-subtracted states.
The characterisation of quantum correlations, also known as entanglement, is central to quantum information theory. This chapter devotes to techniques and criteria based on second moments to detect the entanglement of Gaussian states. It also devotes to the description of possible ways to quantify such quantum correlations, bearing in mind that the determination of entanglement monotones of clear operational significance is still an open problem, even in the restricted arena of Gaussian states. The chapter addresses the issues of entanglement distillation and quantum non-locality tests, in whose contexts we shall confront other non-Gaussian operations and states, such as photon subtracted states. It introduces a general criteria based on higher order statistical moments, which will be put to use to detect the quantum correlations of an entangled cat-like state and addresses the issues of entanglement distillation and quantum non-locality tests.