Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Double Covariance Model: A Stochastic Reconstruction of Quantum Entangled States via Interplay of Micro-Macro Time Scales

Andrei Khrennikov

TL;DR

The Double Covariance Model (DCM) reinterprets the quantum density operator as a macro-covariance that emerges from the micro-scale cross-covariances of two classical stochastic processes across a macro time window. It provides concrete constructions to realize Bell states, arbitrary pure states, and mixed states from purely classical time series, and it derives subsystem states via a stochastic partial trace that connects marginal covariances to quantum reductions. By treating entanglement as a macro-time phenomenon driven by micro-time consistency, the DCM offers a relational, scale-aware foundation for quantum states and a stochastic realization of the partial trace. This framework bridges classical probability and quantum structure, with potential applications in quantum-like modeling across disciplines and a fresh perspective on the universality of relational fields.

Abstract

This article presents a concrete mathematical framework for the generation of entangled quantum states from classical stochastic processes. We demonstrate that any density operator $ρ_{AB}$ of a composite system can be derived from the correlations between two underlying stochastic processes, $X(t)$ and $Y(t)$, 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.

The Double Covariance Model: A Stochastic Reconstruction of Quantum Entangled States via Interplay of Micro-Macro Time Scales

TL;DR

The Double Covariance Model (DCM) reinterprets the quantum density operator as a macro-covariance that emerges from the micro-scale cross-covariances of two classical stochastic processes across a macro time window. It provides concrete constructions to realize Bell states, arbitrary pure states, and mixed states from purely classical time series, and it derives subsystem states via a stochastic partial trace that connects marginal covariances to quantum reductions. By treating entanglement as a macro-time phenomenon driven by micro-time consistency, the DCM offers a relational, scale-aware foundation for quantum states and a stochastic realization of the partial trace. This framework bridges classical probability and quantum structure, with potential applications in quantum-like modeling across disciplines and a fresh perspective on the universality of relational fields.

Abstract

This article presents a concrete mathematical framework for the generation of entangled quantum states from classical stochastic processes. We demonstrate that any density operator of a composite system can be derived from the correlations between two underlying stochastic processes, and , 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.
Paper Structure (26 sections, 2 theorems, 70 equations)

This paper contains 26 sections, 2 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Models in physics based on micro-macro time scale interplay
  3. Remarks on mathematics
  4. The density operator as a double covariance: from micro- to macro-scale correlations
  5. 1. The Micro-scale Cross-Covariance Operator
  6. 2. Vectorization and the Macro-level
  7. 3. The Macro-Covariance Operator
  8. Density operator from micro-scale time series
  9. Density operator as a covariance of random operator
  10. Construction of classical stochastic processes behind density operators
  11. Bell states from classical correlations
  12. Generation of an arbitrary pure state
  13. Generation of mixed states
  14. Generalization of micro-dynamics to jump processes
  15. Deriving subsystem states from composite systems
  16. ...and 11 more sections

Key Result

Lemma 1

Let $(\Omega,\mathcal{F},\mathbb P)$ be a probability space. Let $\{S_k\}_{k=1}^M$ be a family of random variables with values in a measurable space $(E,\mathcal{E})$, and let $\eta$ be a discrete random variable taking values in $\{1,\dots,M\}$ with Assume that $\eta$ is independent of the family $\{S_k\}_{k=1}^M$. Define the random variable Then, for any measurable function $f : E \to L,$ wher

Theorems & Definitions (3)

  • Lemma 1: Random selection of stochastic processes
  • Lemma 2: The Discrete Synchronization Condition
  • proof