 Purification of a density operator

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 $|\Psi ⟩={\sum }_j\sqrt{p_j}|j{⟩}_A|j{⟩}_E$.