Density Matrix Formalism | Pure States, Mixed States & Purity
Quick Technical Answer:
The Density Matrix (or density operator) ρ describes statistical ensembles of pure and mixed quantum states: ρ = ∑ p_i |ψ_i⟩⟨ψ_i|. A state is pure if and only if Tr(ρ²) = 1, and mixed if Tr(ρ²) < 1.
Formula / Unitary:
\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|, \quad \text{Tr}(\rho) = 1, \quad \rho \ge 0, \quad \gamma = \text{Tr}(\rho^2) \le 1
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Frequently Asked Questions
Why can statevectors not describe decoherence and mixed states?
A statevector |ψ⟩ assumes 100% coherence and zero classical uncertainty. When a qubit interacts with a noisy environment or is entangled with unmeasured degrees of freedom, only a density matrix can describe the statistical mixture.