§IV.A, p. 4 — one qubit: eigenvalues 0, −2g, −g, −g of L_{σ₋} on |0⟩⟨0|, σ_z, σ_x, σ_y

Consider a single qubit with Hamiltonian $0$ and one jump operator $\sigma_- = |0\rangle\langle 1| = \begin{pmatrix}0&1\\0&0\end{pmatrix}$ with rate $g>0$, so that $\mathcal L(\rho) = g\,(2\sigma_-\rho\sigma_-^\dagger - \sigma_-^\dagger\sigma_-\rho - \rho\sigma_-^\dagger\sigma_-)$. Then

where $\sigma_z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$, $\sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix}$, $\sigma_y = \begin{pmatrix}0&-i\\i&0\end{pmatrix}$ (basis order $|0\rangle,|1\rangle$).

These four eigenvectors form a basis of the $2\times2$ matrices, so they give the full spectrum $\lambda_0 = 0$, $\lambda_3 = -2g$, $\lambda_1 = \lambda_2 = -g$ of the single-qubit Liouvillian; the $n$-qubit spectrum is built from them.

Formalization Note The page says "$\mathcal L(\sigma_i) = \lambda_i\sigma_i$ iff $\sigma_i\in\Sigma_k$"; read literally the "iff" is false (e.g. $2\sigma_z$ is an eigenvector outside $\Sigma_k$), so the four eigen-equations are stated.