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.

In a Hilbert-Schmidt orthonormal operator basis, the generator $-i[H,\cdot ]$ is represented on the traceless Hermitian subspace by a real skew-symmetric matrix. Its propagator is therefore in the special orthogonal group and preserves the purity of a density operator. For $N>2$, these rotations form the adjoint image of the accessible unitary group, generally a proper subgroup of the full Euclidean rotation group; preservation of the full spectrum of the density operator imposes further constraints.

For a Hilbert-Schmidt orthonormal traceless Hermitian basis, the Generalized Bloch representation gives the displayed identity by orthogonality. Nonnegative eigenvalues of a density operator sum to one, so their squared sum is at most one, with equality precisely for a rank-one projector. For $N>2$, the radius bound alone does not guarantee positivity: $(2/N)I-P$ for a rank-one projector $P$ has trace one and squared norm one, but one negative eigenvalue.

A time-independent Lindblad equation becomes an affine linear equation for the Generalized Bloch representation. If $\mathcal{L}(I)={\sum }_d[V_d,V_d^{†}]$, then $A_{mn}=\mathrm{Tr}({\sigma }_m\mathcal{L}({\sigma }_n))$ and $c_m=\mathrm{Tr}({\sigma }_m\mathcal{L}(I))/N$ for traceless basis indices. The offset vanishes exactly when the dynamics are unital. Equilibria solve $A{s}_{\ast }=-c$, and deviations evolve as $s(t)-s_{\ast }=e^{At}(s(0)-s_{\ast })$.

Decay of the off-diagonal coherence of a two-level density operator. Population relaxation contributes half its total rate; additional dephasing contributes ${\gamma }_{\varphi }\ge 0$ in the usual completely positive two-level model. Thus transverse relaxation should not automatically be identified with pure dephasing alone.

For a two-level density operator, rates ${\gamma }_{12}$ from level one to two and ${\gamma }_{21}$ in the reverse direction give ${\dot{p}}_1=-{\gamma }_{12}{p}_1+{\gamma }_{21}{p}_2$. Their sum sets the longitudinal population relaxation rate, and their ratio fixes the stationary population imbalance. If both rates vanish, populations are conserved.

An arbitrary real affine equation has steady states exactly when $\mathrm{rank}A=\mathrm{rank}[A\mid -c]$. Its solution set is $s_{\ast }+\ker A$. For a finite-dimensional time-independent Lindblad equation, physical stationary states always exist by time averaging trajectories. If $A$ is invertible the steady state is unique. Conversely, a nonzero traceless Hermitian stationary direction can be decomposed into the difference of two equally weighted density operators; time averaging their trajectories yields two distinct stationary density operators. Thus singularity of the physical Bloch matrix also obstructs physical uniqueness.