I know the definition of monopartite, bipartite quantum system and the fact that they are defined as Hilbert space. And I also find the theorem that any two Hilbert spaces which have the same dimensions are isomorphic, cf. here.
My question: If there are isomorphism between monopartite system $H^1=H_{n_a\times n_b}$ and bipartite system $H^2=H_{n_a}\otimes H_{n_b}$ since $H^1$ and $H^2$ have the same dimensions, how do we distinguish monopartite system and bipartite system? ($H_n$ means n-dimensional Hilbert space.) Or one system can be monopartite or bipartite system depending on my perspective?
My think: I'm not sure but it may be distinguished by partial trace. For example, let bipartite system $H_2\otimes H_2$ has orthonormal basis $\{|0\rangle|0\rangle, |0\rangle|1\rangle, |1\rangle0\rangle, |1\rangle|1\rangle\}$ and monopartite system $H_4$ has orthonormal basis $\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\}$. We can operate partial trace to $|0\rangle|1\rangle\langle 0|\langle1|$ but it's impossible for $|3\rangle\langle 3|$.
