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.

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