Purification of a density operator
= Purification of a density operator
A purification of a density operator $\rho_A$ is a pure state $|\Psi\rangle_{AE}$ on a larger system such that $\operatorname{Tr}_E|\Psi\rangle\langle\Psi|=\rho_A$. A <spectral decomposition> $\rho_A=\sum_jp_j|j\rangle\langle j|$ gives the canonical construction $|\Psi\rangle=\sum_j\sqrt{p_j}|j\rangle_A|j\rangle_E$.