Linear rank preservers of tensor products of rank one matrices

Let $n_1,\dots ,n_k$ be integers larger than or equal to 2. We characterize linear maps $\varphi :M_{n_1\cdots n_k}\to M_{n_1\cdots n_k}$ such that $$\mathrm{r}ank(\varphi (A_1\otimes \cdots \otimes A_k))=1\text{whenever}\mathrm{r}ank(A_1\otimes \cdots \otimes A_k)=1\text{for all}{A}_i\in M_{n_i},i=1,\dots ,k\mathrm{.}$$ Applying this result, we extend two recent results on linear maps that preserving the rank of special classes of matrices.