SYMPLECTIC SPACE OR ORTHOGONAL SPACE OF n QUBITS

Abstract

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