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A purification of a density operator ${\rho }_A$ is a pure state $|\Psi {⟩}_{AE}$ on a larger system such that ${\mathrm{Tr}}_E|\Psi ⟩⟨\Psi |={\rho }_A$. A spectral decomposition ${\rho }_A={\sum }_j{p}_j|j⟩⟨j|$ gives the canonical construction $|\Psi ⟩={\sum }_j\sqrt{p_j}|j{⟩}_A|j{⟩}_E$.

For a finite irreducible Markov chain satisfying detailed balance with $\pi$, its transition matrix is a self-adjoint operator for $⟨f,g{⟩}_{\pi }={\sum }_x\pi (x)f(x)g(x)$. For a real orthonormal eigenbasis $(f_j)$ with eigenvalues $({\lambda }_j)$,Indeed, apply the spectral decomposition of $P^t$ to the function $1_{\{y\}}/\pi (y)$, whose coefficient against $f_j$ is $f_j(y)$. For $t=0$, every eigenvalue multiplier is interpreted as one.