Symplectic space and orthogonal space of n qubits

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In the Hilbert space of n qubits, we introduce the symplectic space (n odd) and the orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping the local operations of n qubits into the symplectic group or orthogonal group, and prove that the generalized ``magic basis'' is just the bi-orthonormal basis (that is, the orthonormal basis of both Hilbert space and the orthogonal space). Finally, an example is given to discuss the application in physics of this mathematical structure.