Thermo-field entanglement description of Markovian two-state relaxation

TL;DR

This work addresses how to reconcile classical symmetric two-state Markov relaxation with quantum entanglement dynamics using thermo-field dynamics (TFD). It embeds the Markov process into a dissipative two-level system via the von Neumann equation with relaxation, $i\hbar\frac{\partial \rho}{\partial t}=[H,\rho]-\lambda(\rho-\rho_{eq})$, with $H=\frac{\hbar\omega}{2}\sigma_z$, and analyzes the entanglement structure through the TFD extended density matrix. The key results are a closed-form intrinsic entanglement component $b_{qe}(t)=\frac{1}{4} e^{-\lambda t} \sin^2(\omega t)$ and a clean entropy decomposition $\hat{S}(t)=S_{\mathrm{cl}}(t)+S_{\mathrm{qe}}(t)$, where $S_{\mathrm{cl}}(t)$ is the Shannon-type classical contribution and $S_{\mathrm{qe}}(t)=-2k_B b_{qe}(t)\ln b_{qe}(t)$ is purely quantum. This establishes that a single classical timescale $\lambda^{-1}$ governs entanglement decay while coherent dynamics occur at $\omega$, providing an information-theoretic link between relaxation rates and entanglement lifetimes with potential extensions to asymmetric, non-Markovian, and multi-level settings.

Abstract

We present a unified description of symmetric two-state Markov relaxation and intrinsic entanglement dynamics based on thermo-field dynamics (TFD). A classical two-state Markov process is embedded into a dissipative two-level quantum system by identifying the Markov relaxation rate with the dissipation parameter in a von Neumann equation with a relaxation term. Using the reduced extended density matrix in the TFD formalism, we explicitly separate classical thermal mixing from intrinsic quantum entanglement. For a minimal exchange-like two-level subspace, we obtain a closed-form expression for the intrinsic entanglement component, $b_{qe}(t)=\frac{1}{4}e^{-λt}\sin^2(ωt)$, demonstrating that a single classical timescale controls the decay envelope of genuine entanglement. We further show that the extended entanglement entropy naturally decomposes into a classical Shannon-type contribution and a purely quantum entanglement contribution, clarifying how stochastic relaxation constrains entanglement loss in a minimal setting.

Figures (3)

• Figure 1: Classical symmetric two-state Markov relaxation $p_1(t)=\frac{1}{2}(1-e^{-\lambda t})$ plotted versus $\omega t$ for representative values of $\lambda/\omega$.
• Figure 2: Intrinsic quantum entanglement component extracted from the reduced extended density matrix, $b_{qe}(t)=\frac{1}{4} e^{-\lambda t}\sin^2(\omega t)$, for representative values of $\lambda/\omega$. The exponential decay envelope is entirely determined by the Markov relaxation rate $\lambda$.
• Figure 3: Entropy decomposition for a representative $\lambda/\omega$ (here $\lambda/\omega=0.5$). The classical entropy $S_{\mathrm{cl}}(t)$, extended entropy $\hat{S}(t)$, and intrinsic entanglement contribution $S_{\mathrm{qe}}(t)=\hat{S}-S_{\mathrm{cl}}$ are shown. While $S_{\mathrm{cl}}$ saturates at $k_B\ln 2$, the intrinsic contribution decays to zero with the Markov relaxation rate.