Conditions for being able to transform a matrix into a Kronecker sum Does anyone know any conditions under which a Hermitian matrix $H$ can be transformed by a unitary matrix $U$ into a Kronecker product?
That is,
$H'=UHU^\dagger=A\oplus B\equiv A\otimes I+I\otimes B$
Thank you in advance for your help.
 A: A preliminary attempt: I'll stick to the case where both $A$ and $B$ are necessarily of size $n$.
Two Hermitian matrices are similar if and only if they share eigenvalues.  Note that if $A,B$ have eigenvalues $\lambda_1,\dots,\lambda_n$ and $\mu_1,\dots,\mu_n$ respectively, then the eigenvalues of $A \oplus B$ are given by $\lambda_i + \mu_j$ for all $1 \leq i,j \leq n$. Note in particular that arranging these eigenvalues in a matrix leaves you with the rank-$2$ matrix
$$
M = \pmatrix{
\lambda _1 + \mu_1 & \cdots & \lambda_1 + \mu_n\\
\vdots & \ddots & \vdots\\
\lambda_n + \mu_1 & \cdots & \lambda_n + \mu_n} 
\\= 
\pmatrix{\lambda_1\\ \vdots \\ \lambda_n} \pmatrix{1&\cdots & 1} + 
\pmatrix{1\\ \vdots \\ 1}\pmatrix{\mu_1 & \cdots & \mu_n}
\\= 
\pmatrix{\lambda_1 & 1\\ \vdots & \vdots\\ \lambda_n & 1}
\pmatrix{
1 & \cdots & 1\\
\mu_1 & \cdots & \mu_n}.
$$
So, a size $n^2 \times n^2$ Hermitian matrix $H$ can be transformed into a matrix of the form $A \oplus B$ if and only if its eigenvalues can be arranged into a matrix of the above form.
