= Rank bound for a weighted operator trace
{title2=$|\operatorname{Tr}(\rho T)|^2\leq(\sum_{j<k}\lambda_j)\operatorname{Tr}(\rho T^\dagger T)$}
If $\rho$ is a <density operator> with decreasing <eigenvalues> $\lambda_j$ and $T$ has <matrix rank> at most $k$, let $\Pi$ project onto the image of $T$. The <Cauchy-Schwarz inequality> for the <Hilbert-Schmidt inner product> gives
$$
|\operatorname{Tr}(\rho T)|^2\leq\operatorname{Tr}(\rho\Pi)\operatorname{Tr}(\rho T^\dagger T).
$$
The <Hermitian effect variational principle> bounds $\operatorname{Tr}(\rho\Pi)$ by $\sum_{j<k}\lambda_j$, proving the claimed estimate without assuming $\rho$ is invertible.
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