I have been studying the tensor product of Hilbert spaces, and I know that if ${\psi_k}$ and ${\phi_l}$ are orthonormal basis of $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively, then ${\psi_k\otimes\phi_l}$ is an orthonormal basis of $\mathcal{H}_1\otimes\mathcal{H}_2$. My question is, is the converse statement true?. Is it true the following statement:
Let $\{\lambda_n\}$ be an orthonormal basis of $\mathcal{H}_1\otimes\mathcal{H}_2$. Then, there exist orthonormal basis ${\psi_k}$ and ${\phi_l}$ of $\mathcal{H}_1$ and $\mathcal{H}_2$ respectively, such that $\lambda_n=\psi_k\otimes\phi_l$ for every $\lambda_n$?