The density matrix ρ generalizes quantum state descriptions to mixed states: ρ = Σ p_i |ψ_i⟩⟨ψ_i|. It encodes both diagonal populations (probabilities) and off-diagonal coherences (quantum interference). For a qubit, ρ = (I + r·σ)/2 where r is the Bloch vector — pure states touch the sphere surface (|r|=1), mixed states reside inside. The purity Tr(ρ²) ranges from 1/d (maximally mixed) to 1 (pure), and coherences decay under decoherence, shrinking r toward the origin.