Conditions for being able to transform a matrix into a Kronecker sum

Question

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.

Answer 1

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.