here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: \begin{equation} L = (L_0\otimes \mathbb{I} + \mathbb{I} \otimes L_0). \end{equation} Suppose now that your numerical capabilities allow you to compute the singular value decomposition of $L_0=U_0S_0V_0^\dagger$.
Is it possible to write the SVD of the operator $L=USV^\dagger$ using just tensor product or compositions of the matrices $U_0$, $S_0$ and $V_0$?
My intuition says that is should be possible as it is possible to compute the spectrum and the inverse of $L$ just knowing the full spectrum of $L_0$. From a more physical point of view this seems to me reasonable since the properties of composite systems with no interactions between the subsystems can be always deduced from the local properties (I mean, the spectrum of an ensamble of non-interacting spins or electrons can be deduced knowing the spectrum of a single particle).
Thank you very much in advance, best regards
Alberto
Let us put your intuition to the test. Let $A=B=\mathbb C^2$, and $$ L_0=\begin{bmatrix}\sqrt2&3/\sqrt2\\ \sqrt2&-3/\sqrt2 \end{bmatrix} =\begin{bmatrix}1/\sqrt2&1/\sqrt2\\1/\sqrt2&-1/\sqrt2\end{bmatrix}\begin{bmatrix}2&0\\0&3\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}. $$ Now $$ L=L_0\otimes I+I\otimes L_0=\begin{bmatrix}\sqrt2&3/\sqrt2&0&0\\ \sqrt2&-3/\sqrt2&0&0\\0&0&\sqrt2&3/\sqrt2\\0&0&\sqrt2&-3/\sqrt2\end{bmatrix}+ \begin{bmatrix}\sqrt2&0&3/\sqrt2&0\\0&\sqrt2&0&3/\sqrt2\\ \sqrt2&0&-3/\sqrt2&0\\0&\sqrt2&0&-3/\sqrt2\end{bmatrix} =\begin{bmatrix}2\sqrt2&3/\sqrt2&3/\sqrt2&0\\ \sqrt2&-1/\sqrt2&0&3/\sqrt2\\ \sqrt2&0&-1/\sqrt2&3/\sqrt2\\0&\sqrt2&\sqrt2&-3\sqrt2\end{bmatrix}, $$ and (according to Wolfram Alpha) $$ S=\begin{bmatrix}5.746&0&0&0\\ 0&4.36177&0&0\\ 0&0&1/\sqrt2&0\\0&0&0&0.677124\end{bmatrix}, $$ and $$ U = \begin{bmatrix}-0.181482 & 0.912785 & 0 & 0.365907 \\ 0.392625 & 0.284406 & 0.707107 & -0.514741 \\ 0.392625 & 0.284406 & -0.707107 & -0.514741 \\ -0.811637 & 0.0710617 & 0& -0.579823 \end{bmatrix}, $$ $$ V=\begin{bmatrix}0.103934 & 0.776329 & 0 & -0.6217\\ -0.315078 & 0.420862 & -0.707107 & 0.472865\\ -0.315078 & 0.420862 & 0.707107 & 0.472865\\ 0.889185 &0.207517 & 0 & 0.407783 \end{bmatrix}. $$
Or consider $$ L_0=\begin{bmatrix}5/2&-1/2\\-1/2&5/2\end{bmatrix} =\begin{bmatrix}1/\sqrt2&1/\sqrt2\\1/\sqrt2.&-1/\sqrt2\end{bmatrix} \begin{bmatrix}2&0\\0&3\end{bmatrix} \begin{bmatrix}1/\sqrt2&1/\sqrt2\\1/\sqrt2.&-1/\sqrt2\end{bmatrix}^* $$ (same $U_0$ and $L_0$). Note that $S_0$ is also the same as the previous example. Now $$ L=L_0\otimes I+I\otimes L_0=\begin{bmatrix}5/2&-1/2&0&0\\-1/2&5/2&0&0\\0&0&5/2&-1/2\\0&0&-1/2&5/2\end{bmatrix}+ \begin{bmatrix}5/2&0&-1/2&0\\0&5/2&0&-1/2\\-1/2&0&5/2&0\\0&-1/2&0&5/2\end{bmatrix} =\begin{bmatrix}5&-1/2&-1/2&0\\-1/2&5&0&-1/2\\-1/2&0&5&-1/2\\0&-1/2&-1/2&5\end{bmatrix}, $$ and $$ S=\begin{bmatrix}4&0&0&0\\ 0&5&0&0\\ 0&0&5&0\\ 0&0&0&6\end{bmatrix}. $$
Of course this doesn't show that there is no relation between $U,S,V$ and $U_0,S_0,V_0$; but if there is one, it's unlikely that it is simple.