Quantum Computing

Hilbert space: Banach space with implied norm. All Cauchy sequences converge.

$$\mathrm{|\psi ⟩}=\alpha \cdot \mathrm{|0⟩}+\beta \cdot \mathrm{|1⟩}$$

Statistical (incoherent) mixtures of pure states cannot be described by state vectors.

Density operator ρ

Properties

• ρ is Hermitean
• ρ >= 0
• Tr(ρ) = 1

$$\rho =\alpha \cdot \mathrm{|0⟩}+\beta \cdot \mathrm{|1⟩}$$

Interesting things

Purity: Tr(ρ²)

Linear entropy: 1 - Tr(ρ²)

Canonical Ensemble

$$\rho =\frac{\exp (\beta H)}{\mathrm{Tr}\left(\exp \right(-\beta H\left)\right)}$$

Author: Danny (remove the ".nospam" to send)

Last modification on: Wed, 15 Jul 2026 .