Thermo-field entanglement description of Markovian two-state relaxation

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.