= Pure-target upper bound on trace distance
{title2=$D(\rho,|v\rangle\langle v|)\leq\sqrt{1-\langle v|\rho|v\rangle}$}
For two normalized <pure states>, the difference of their projectors has two <eigenvalues> $\pm\sqrt{1-|\langle u|v\rangle|^2}$, giving that <trace distance>. Write a mixed <density operator> as $\rho=\sum_jq_j|u_j\rangle\langle u_j|$. Convexity of the <trace norm> and weighted <Cauchy-Schwarz inequality> give $D(\rho,|v\rangle\langle v|)\leq\sum_jq_j\sqrt{1-|\langle u_j|v\rangle|^2}\leq\sqrt{1-\langle v|\rho|v\rangle}$. Thus a small <probability> of failing the pure-state projector test forces trace-distance proximity, including for correlated multipartite outputs.
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