Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much known about maps that preserve this tensor rank?
Also an easier (more specific) question: suppose I'm working in $U(4)$. Might there exist a $U \in U(4)$ such that for any pair of matrices $A,B \in U(2)$, $U(A \otimes B)U^{-1}$ is another simple tensor in $U(4)$, without $U$ itself being the tensor product of two matrices?
Morally I feel like this wouldn't happen and that the set of maps that preserve simple tensors in this way are themselves simple tensors, but I'm unsure. Thanks!