Quantum Decoherence & Einselection

Einselection (environment-induced superselection): The environment monitors the system continuously, selecting a preferred pointer basis — the only states that survive decoherence robustly.
Density matrix evolution: ρ(t) = Σᵢⱼ ρᵢⱼ(0)·exp(−Γ|i−j|²t)|i⟩⟨j|. Off-diagonal (coherence) elements decay exponentially; diagonal (classical probability) is preserved.
Purity: Tr(ρ²) starts at 1 (pure state), decays to 1/N (maximally mixed) at rate ~2Γ. Pointer states: eigenstates of the system-environment interaction Hamiltonian H_SE = λ·Ŝ⊗Ê — the stable basis. Superpositions of pointer states are fragile.