Rank bound for a weighted operator trace ($|\mathrm{Tr}(\rho T){|}^2\le ({\sum }_{j<k}{\lambda }_j)\mathrm{Tr}(\rho T^{†}T)$)

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 givesThe Hermitian effect variational principle bounds $\mathrm{Tr}(\rho \Pi )$ by ${\sum }_{j<k}{\lambda }_j$, proving the claimed estimate without assuming $\rho$ is invertible.