Suppose that $A$ and $B$ are complex $d\times d$ matrices. Suppose moreover that they are Hermitian positive semidefinite and that for some $k>0$ $$ A^{\otimes k} = B^{\otimes k},$$ where $\otimes$ denotes the Kronecker (i.e., tensor) product. Is it then true that $A=B$? Or can we at least relate $A$ and $B$ somehow?
Thank you in advance for your help.