While I am familiar with the isomorphism between finite-dimensional vector spaces with the same dimension and isometric isomorphism between Hilbert spaces, I realize I am getting confused by $n$-th tensor products of an infinite-dimensional separable Hilbert space $H$ ($H^{\otimes n} = \underbrace{H \otimes \dots \otimes H }_{\text{$n$ times}}$).
Is $H^{\otimes n}$ isomorph to $H^{\otimes m}$ for $ m \ne n$?
Since every separable infinite dimensional Hilbert space is isomorph to $\ell_2$, they should be isomorph. However, I have not been able to identify a bijection between $H^{\otimes n}$ and $H^{\otimes m}$ which preserves the algebraic structure. Pretty sure that I am missing something obvious.