Obviously, the matrices: <math> L_{ij}=|i\rangle \langle j | </math> formsform a basis set in the operator space. Any quantum operator defined in this Hilbert can be expended by <math> \lbrace L_{ij} \rbrace </math> operators. In the following, let's call these matrices as a super basis to distinguish the eigen basis of quantum states. More speificallyspecifically the above super basis <math> \lbrace L_{ij} \rbrace </math> can be called a transition super basis because it describes the transition between states <math> |i\rangle </math> and <math> |j\rangle </math>. In fact, this is not the only super basis that does the trick. We can also use Pauli matrices plusand the identity matrix to form a super basis
