Consider a Hermitian operator $H_\mathcal{AB}$ acting on the Hilbert space $ \mathcal{A}\otimes\mathcal{B}$. What is the smallest commuting set of separable operators in the form $A_k\otimes B_k$, such that $$ H_\mathcal{AB}=\sum_k A_k\otimes B_k? $$ As pointed out in the comments, a set like this might not even exist.
It would also be interesting to constrain $A_k$ and $B_k$ to be Hermitian themselves. Note that if we didn't require commutativity, we could use the operator singular value decomposition.