Lindblad Master Equation: Qubit Decoherence

Lindblad Master Equation: dρ/dt = −i[H,ρ] + Σ_k γ_k(L_k ρ L_k† − ½{L_k†L_k, ρ})

For a qubit: Amplitude damping (L=σ₋, rate γ₁=1/T₁): energy relaxation |1⟩→|0⟩. Pure dephasing (L=σ_z/2, rate γ_φ): destroys off-diagonal coherences. Together: T₂ = (2T₁)⁻¹ + T_φ⁻¹ so T₂ ≤ 2T₁.

The Bloch sphere shows qubit state as (⟨X⟩, ⟨Y⟩, ⟨Z⟩). Pure states lie on the surface; mixed states inside. Decoherence shrinks the sphere into the Z-axis (amplitude damping) or the equatorial plane (dephasing). The trajectory spirals inward to the ground state.