Density operator ρ: tr ρ = 1, ρ ≽ 0; pure ⇔ ρ² = ρ ⇔ tr ρ² = 1

Density operator ρ on a finite-dimensional Hilbert space H encodes the complete quantum-statistical state of the system: ρ is Hermitian (ρ† = ρ), positive semidefinite (⟨ψ|ρ|ψ⟩ ≥ 0 for all |ψ⟩), and has unit trace (tr ρ = 1). Pure vs…