Embeddings of spaces of quregisters into special linear groups

Abstract: We study embeddings of the unit sphere of complex Hilbert spaces of dimension a power $2^n$ into the corresponding groups of non-singular linear transformations. For the case of $n=1$, the sphere $S_2$ of qubits is identified with $\mbox{SU}(2)$ and the algebraic structure of this last group is carried into $S_2$. Hence it is natural to analyse whether is it possible, for $n\geq 2$, to carry the structure of the symmetry group $\mbox{SU}(2^n)$ into the unit sphere $S_{2^n}$. For $n=2$ the embeddings of $S_{2^2}$ into $\mbox{GL}(2^2)$, obtained as tensor products of the above embedding, fails to determine a bijection between $S_{2^2}$ and $\mbox{SU}(2^2)$, but they determine entanglement measures consistent with von Neumann entropy.