 Square-root amplitude of a density operator

The unique positive square root of a density operator is a Hilbert-Schmidt operator $W$ with $\Vert W{\Vert }_{\mathrm{HS}}^2=\mathrm{tr}\rho =1$. It reproduces expectations by $\mathrm{tr}(A\rho )=\mathrm{tr}(W^{\ast }AW)$. In chosen orthonormal bases, vectorizing its entries gives a normalized bipartite vector $|W⟩⟩={\sum }_{r,s}{W}_{rs}|r⟩|s⟩$ whose reduced density operator is $W{W}^{\ast }=\rho$. Thus the positive operator amplitude and the vector purification are related but distinct objects.

More generally a coefficient matrix $C$ purifies $\rho$ when $C{C}^{\ast }=\rho$. For square nonsingular $C$, the matrix $U=C^{-1}\sqrt{\rho }$ obeys $U{U}^{\ast }=I$ and is unitary, so $\sqrt{\rho }=CU$. In an eigenbasis of $\rho$, $C=\mathrm{diag}(\sqrt{p_r})U^{\ast }$; its conjugated unitary columns give the second-factor Schmidt decomposition vectors. This is the matrix form of unitary freedom of purification.