Hilbert-Schmidt inner product ($⟨A,B{⟩}_{\mathrm{HS}}$)

For Hilbert-Schmidt operators $A$ and $B$, the Hilbert-Schmidt inner product is $⟨A,B{⟩}_{\mathrm{HS}}=\mathrm{tr}(A^{\ast }B)$. Its induced norm is the Hilbert-Schmidt norm.

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.

The Hilbert-Schmidt distance is $d_{\mathrm{HS}}(A,B)=\Vert A-B{\Vert }_{\mathrm{HS}}$.