Quantum Density Matrix

Bloch sphere visualization — pure vs mixed states, decoherence, purity

θ (polar): 1.05

φ (azimuth): 0.79

Mixedness: 0.00

Decoherence γ: 0.05

ρ = (I + r·σ)/2 where r = Bloch vector. Pure: |r|=1, Tr(ρ²)=1. Mixed: |r|<1, Tr(ρ²)<1. Decoherence: off-diagonal ρ_{01}→0 (dephasing). Bloch vector shrinks toward z-axis then origin.