Rotation operators

The rotation operators are defined as:
 Rx(θ)=(cos(θ2)isin(θ2)isin(θ2)cos(θ2))\ R_x\left ( \theta \right ) = \begin{pmatrix} \cos \left ( \frac{\theta}{2} \right ) & -i \sin \left ( \frac{\theta}{2} \right) \ -i \sin \left ( \frac{\theta}{2} \right) & \cos \left ( \frac{\theta}{2} \right ) \end{pmatrix} \  Ry(θ)=(cos(θ2)sin(θ2)sin(θ2)cos(θ2))\ R_y\left ( \theta \right ) = \begin{pmatrix} \cos \left ( \frac{\theta}{2} \right ) & - \sin \left ( \frac{\theta}{2} \right) \ \sin \left ( \frac{\theta}{2} \right) & \cos \left ( \frac{\theta}{2} \right ) \end{pmatrix} \  Rz(θ)=(eiθ200eiθ2)\ R_z\left ( \theta \right ) = \begin{pmatrix} e^{-i \frac{\theta}{2}} & 0 \ 0 & e^{i \frac{\theta}{2}} \end{pmatrix} \

The rotation operators are generated by exponentiation of the Pauli matrices according to
 exp(iAx)=cos(x)I+isin(x)A\ exp{(i A x)} = \cos\left ( x \right )I+i\sin\left ( x \right )A \
where P is one of the three Pauli Matrices.

Note that the Rz rotation operator can also be expressed as
(eiθ200eiθ2)(eiθ200eiθ2)=(100eiθ)\begin{pmatrix} e^{i \frac{\theta}{2}} & 0 \ 0 & e^{i \frac{\theta}{2}} \end{pmatrix} \begin{pmatrix} e^{-i \frac{\theta}{2}} & 0 \ 0 & e^{i \frac{\theta}{2}} \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & e^{i \theta} \end{pmatrix} \
which differs from the definition above by a global phase only.