Generalized Bloch representation ($\rho =I/N+{\sum }_{k=1}^{N^2-1}{s}_k{\sigma }_k$)

Choose a Hilbert-Schmidt inner product orthonormal Hermitian operator basis with final element $I/\sqrt{N}$. A density operator has real coefficients $r_k=\mathrm{Tr}({\sigma }_k\rho )$ and fixed final coefficient $1/\sqrt{N}$. The remaining coefficients form its generalized Bloch vector. Positivity restricts these real vectors to a compact convex body; for $N>2$ this body is not the entire Euclidean ball allowed by a purity bound.