Let $A$ and $B$ be two (finite-dimensional) Hermitian matrices and $n$ be a positive integer. We define the matrix $$ L_i = A\otimes \dots\otimes A\otimes B\otimes A\otimes \dots\otimes A~, $$ where there are $n$ factors and $B$ is the $i$-th factor (with $1 \leq i \leq n$). We then define the "symmetric sum" $$ L = \sum_{i=1}^n L_i~. $$ Question: Can we say anything about the eigenvectors of $L$ knowing the eigenvectors of $A$ and $B$?
This question seems simple (and maybe it is) but it has resisted my best attempts. If $A$ and $B$ commute, this is trivial so I am interested in any case where they don't commute. I am not necessarily asking for a general solution of the problem and would be interested to see any (non-commuting) example. Even the case where $A= \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} $ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ was too complicated (the constraint satisfied by the eigenvectors is hard to solve). I feel like some form of this problem should have appeared before and that's why I am asking here!
For some background, this problem arises in quantum information theory, when trying to determine the optimal quantum measurement that distinguishes two states, involving $n$ copies of the system. Note that for this application, we have $\mathrm{Tr}\,B = 0$.
Thanks a lot for your help!
Edit: I don't have enough reputation to comment but wanted to mention that I am interested in the limit where $n$ becomes large.
