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

If subsystem $S$ has pure state $|u⟩⟨u|$ in a normalized joint pure vector $|\Omega {⟩}_{SE}$, then $|\Omega {⟩}_{SE}=|u{⟩}_S\otimes |e{⟩}_E$. Indeed, for an orthonormal basis $\{v_j\}$ of $u^{\perp }$, each squared norm $\Vert (⟨v_j|\otimes I)|\Omega ⟩{\Vert }^2=⟨v_j|{\rho }_S|v_j⟩$ is zero. Therefore the joint vector has no component outside $\mathbb{C}u\otimes {\mathcal{H}}_E$. In particular, if Alice and Bob share a known pure entangled state, an adversary holding a purifying system has no correlations with it. This does not make every pure bipartite state useful for key distribution: a pure product state has no shared random correlations.

For $\rho ={\sum }_x{p}_x{\rho }_x$, choose component purifications $|u_x{⟩}_{AR}$ and an orthonormal flag register. The vector $|U⟩={\sum }_x\sqrt{p_x}|u_x⟩|x⟩$ is normalized and reduces to $\rho$ after tracing out $R$ and the flag. Orthogonal labels remove the cross terms. This construction turns mixture identities into pure-state overlap arguments and proves joint concavity of quantum fidelity.

Any two purifications of the same density operator on sufficiently large reference spaces differ by an isometry on the reference system, and by a unitary when the reference spaces have the same dimension.

The Hughston--Jozsa--Wootters theorem says that two pure-state ensembles represent the same density operator exactly when their subnormalized state vectors are related by an isometry, or by a unitary after padding the shorter ensemble with zero vectors.

For the nonzero spectral decomposition $\rho ={\sum }_{a=1}^r{\lambda }_a|e_a⟩⟨e_a|$, every pure-state quantum state ensemble of $\rho$ is specified by the displayed column linear isometry. To prove this, write $w_j=\sqrt{p_j}{\varphi }_j$. Positivity forces each $w_j$ into the support of $\rho$, and the coefficients $U_{ja}=⟨e_a|w_j⟩/\sqrt{{\lambda }_a}$ have orthonormal columns because ${\sum }_j|w_j⟩⟨w_j|=\rho$. Conversely this column identity reconstructs $\rho$; the squared vector norms are the probabilities and sum to one. Complete the columns to unitary bases after adding zero ensemble vectors to obtain the usual unitary-mixing formulation.

Uhlmann's theorem states that the fidelity of two density operators is the maximum absolute overlap between their purifications on a common reference space.