Beyond Separability: Quantum Extension of Lyapunov Convergence via Entangled Steady States

Extends the classical Lyapunov convergence result D(p_t || π_α) → 0 to the quantum regime via quantum relative entropy under CPTP dynamics satisfying detailed balance. Constructs a projector interaction Hamiltonian H_α = H_0 − αP whose unique ground state is maximally entangled for α > Λ (Proposition 3.1). Invokes Spohn's theorem (1978) for monotonic decrease of quantum relative entropy. Defines the classical-quantum boundary operationally via entanglement negativity. Derives a quantitative negativity lower bound as an explicit function of (T, α, d) with linear onset scaling near the entanglement threshold. Provides an exactly solvable minimal model yielding a necessary-and-sufficient phase boundary α_quant(T) = kT ln(d+1) with exact negativity and universal temperature collapse. Appendix D derives quantum detailed balance from the Davies weak-coupling limit. Fifth paper in a series unifying thermodynamic laws under a single variational equation.