For example, say $|\phi_1\rangle$ and $|\phi_2\rangle$ are elements in $\mathcal H_{2^n}$, Hilbert space of dimension $2^n$(or to physics people, states of $n$ 2 state particles). Can you define "addition" or "multiplication" of the two elements?
If we restrict it to 0,1 strings (e.g., 10010), we can define a group by just using the group structure of $\mathbb F_2^n$.
Can we do the similar thing on the Hilbert space?
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Sorry, I guess I should have clarified. I mean "projective Hilbert Space" not Hilbert space. In physics we just call it Hilbert space...
Complex projective space $\mathbb{CP}^n$ (the projective space of an $n+1$-dimensional complex Hilbert space) does not have a (topological) group structure for any value of $n$. This can be deduced from the fact that its Euler characteristic is $n+1$, and various results can be used to show that if $\mathbb{CP}^n$ had a group structure its Euler characteristic would have to be $0$.