 Solution

A density operator is a positive self-adjoint trace-class operator with trace one. In finite dimensions this is a positive semidefinite Hermitian matrix with $\mathrm{tr}\rho =1$. Its expectation of an observable is $\boxed{⟨A⟩=\mathrm{tr}(A\rho )}$ whenever the expectation is defined.

In the source's square-root notation, take the unique positive square root $W=\sqrt{\rho }$. It has Hilbert-Schmidt norm one and satisfies $W{W}^{\ast }=\rho$, soThe standard vector purification is the associated vector $|W⟩⟩={\sum }_{r,s}{W}_{rs}|{\xi }_r⟩|{\eta }_s⟩$ on a system and a chosen copy. Tracing out the copy gives $\rho$; measuring $A\otimes I$ gives the same trace formula. Thus the operator square root and the purified vector are two related objects, distinguished by this vectorization.

For a general coefficient matrix $C$, expansion of the expectation givesThus $\boxed{{\rho }_1=C{C}^{\ast }}$. Positivity is immediate from $v^{\ast }C{C}^{\ast }v=\Vert C^{\ast }v{\Vert }^2$, and its trace is one by normalization of $\Psi$.

If $C$ is nonsingular, it is square and ${\rho }_1$ is positive definite. DefineThen $U{U}^{\ast }=C^{-1}{\rho }_1(C^{\ast }{)}^{-1}=I$, which for a square matrix also implies $U^{\ast }U=I$. Thus $U$ is unitary, $\sqrt{{\rho }_1}=CU$, and $C=\sqrt{{\rho }_1}{U}^{\ast }$.

In an eigenbasis of ${\rho }_1$, its square root is $\mathrm{diag}(\sqrt{p_r})$. Hence $c_{rs}=\sqrt{p_r}\overline{U_{sr}}$. DefineUnitarity gives $⟨{\zeta }_r|{\zeta }_t⟩={\delta }_{rt}$, and substitution gives the Schmidt decompositionFor singular or rectangular $C$, the same conclusion follows on its support by a singular value decomposition; the stated nonsingular case needs only the unitary calculation above.

For the supplied Bell state, the requested matrices areThe Schmidt vectors can be chosen as ${\zeta }_1={\eta }_2$, ${\zeta }_2=-{\eta }_1$, and the two coefficients are $1/\sqrt{2}$. The square root's canonical vector purification is $(|{\xi }_1{\eta }_1⟩+|{\xi }_2{\eta }_2⟩)/\sqrt{2}$; the original Bell vector is a different purification of the same reduced state, related by a unitary on the second factor. This illustrates the unitary freedom of purification.