= Flagged purification of a quantum ensemble
{title2=$|U\rangle=\sum_x\sqrt{p_x}|u_x\rangle|x\rangle$}
= Flagged purifications of a quantum ensemble
{synonym}
For $\rho=\sum_xp_x\rho_x$, choose component purifications $|u_x\rangle_{AR}$ and an orthonormal flag register. The vector $|U\rangle=\sum_x\sqrt{p_x}|u_x\rangle|x\rangle$ 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>.
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