LINDBLAD MASTER EQUATION

Open quantum system dynamics and decoherence on the Bloch sphere

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The Lindblad master equation dρ/dt = −i[H,ρ] + Σ_k γ_k(L_k ρ L_k† − ½{L_k†L_k, ρ}) is the most general Markovian, completely positive evolution of an open quantum system. The jump operators L_k model environment interactions: amplitude decay L=σ₋ shrinks the Bloch vector toward |0⟩, while pure dephasing L=σ_z collapses coherences without affecting populations. Steady states arise from the balance between coherent driving (Rabi oscillations) and dissipation, tracing spiral trajectories on the Bloch sphere. This framework underlies all of quantum optics and quantum computing error models.