Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 3 / a

The partial trace removes the other orthonormal factor in the Schmidt decomposition:For instance, tracing $|a_k{b}_k⟩⟨a_{\ell }{b}_{\ell }|$ over $B$ gives $⟨b_{\ell }|b_k⟩|a_k⟩⟨a_{\ell }|$, which vanishes unless $k=\ell$. Consequently the nonzero eigenvalues of the two reduced density operators agree, with multiplicity. If the subsystem dimensions differ, the orthonormal Schmidt families can be completed to full bases and the extra eigenvalues are zero.