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arXiv:2301.05735 · International Journal of Modern Physics A

Classical entanglement and entropy

Haowu Duan, Alex Kovner, Vladimir V. Skokov

Published Aug 30, 2024• 1 citation
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Abstract

Motivated by recent discussions of entanglement in the context of high energy scattering, we consider the relation between the entanglement entropy of a highly excited state of a quantum system and the classical entanglement entropy of the corresponding classical system. We show on the example of two weakly coupled harmonic oscillators that the two entropies are equal. Quantum mechanically, the reduced density matrix which yields this entropy is close to the maximally entangled state. We thus observe that the nature of entanglement in this type of state is purely classical.

DOI
10.1142/s0217751x24501045

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In this review article, we first introduce the density matrix, along with its properties and time evolution. We note that one of the principal uses of the density matrix is to calculate measurement probabilities in quantum mechanics. When only some of the variables can be measured, it is necessary to integrate out the variables that are inaccessible to measurement. The resulting subsystem is then analogous to an open quantum system, where the unmeasured variables belong to an unspecified environment. Coupled and Lorentz-covariant harmonic oscillators are described and their differences are explained. We introduce the density matrix for two harmonic oscillators and demonstrate how they become entangled as well as entangled excited states of the Lorentz-covariant harmonic oscillator. We elaborate on the concept of von Neumann entropy and show how entropy increases when not all the variables of the system can be measured. We use the timeseparation between quarks as an example of an unmeasurable quantity and then present a discussion of von Neumann entropy and entanglement. We include “boiling quarksgluons”, whose most interesting feature is the transition from the confinement phase to the plasma phase, to illustrate the application of the density function and entropy. In this context, we discuss the quark-parton puzzle and a possible explanation.