Section 5.10 Parameterization of Hermitian and Unitary Matrices
Any \(2\times2\) Hermitian matrix, \(M\text{,}\) can be written in the form
\begin{equation}
M = \begin{pmatrix} t+z\amp x-iy\\ x+iy\amp t-z \end{pmatrix}\tag{5.10.1}
\end{equation}
with \(x,y,z,t\in\RR\text{.}\) Looking at the matrix coefficients of these variables, we can write
\begin{equation}
M = t \,I + x \,\sigma_x + y \,\sigma_y + z \,\sigma_z\tag{5.10.2}
\end{equation}
thus defining the three matrices
\begin{align}
\sigma_x \amp = \begin{pmatrix}
0\amp 1\\
1\amp 0 \end{pmatrix} ,\tag{5.10.3}\\
\sigma_y \amp = \begin{pmatrix}
0\amp -i\\
i\amp 0 \end{pmatrix} ,\tag{5.10.4}\\
\sigma_z \amp = \begin{pmatrix}
1\amp 0\\
0\amp -1 \end{pmatrix}\text{,}\tag{5.10.5}
\end{align}
which are known as the Pauli matrices. The Pauli matrices \(\{\sigma_m\}\) have several interesting properties. First of all, each Pauli matrix squares to the identity matrix, is tracefree, and of course is Hermitian:
\begin{align}
\sigma_m^2 \amp = I ,\tag{5.10.6}\\
\tr(\sigma_m) \amp = 0 ,\tag{5.10.7}\\
\sigma_m^\dagger \amp = \sigma_m\text{.}\tag{5.10.8}
\end{align}
Also of interest is that these matrices anticommute, and that the product of any two is the third, for instance:
\begin{equation}
\sigma_x \sigma_y = i\,\sigma_z = -\sigma_y \sigma_x\text{.}\tag{5.10.9}
\end{equation}
Thus, any \(2\times2\) Hermitian matrix can be written as a real linear combination of the Pauli matrices and the identity matrix.
What about \(2\times2\) unitary matrices? Now we must have \(MM^\dagger=I\text{,}\) that is,
\begin{equation}
\begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix}
\begin{pmatrix} a^*\amp c^*\\ b^*\amp d^* \end{pmatrix}
=
\begin{pmatrix} |a|^2+|b|^2\amp ac^*+bd^*\\ ca^*+db^*\amp |c|^2+|d|^2 \end{pmatrix}
= I\text{.}\tag{5.10.10}
\end{equation}
The normalization conditions
\begin{equation}
|a|^2+|b|^2 = 1 = |c|^2+|d|^2\tag{5.10.11}
\end{equation}
allow us to write
\begin{align}
a \amp = e^{i\alpha}\cos\theta ,\tag{5.10.12}\\
b \amp = e^{i\beta}\sin\theta ,\tag{5.10.13}\\
c \amp = e^{i\gamma}\sin\phi ,\tag{5.10.14}\\
d \amp = e^{i\delta}\cos\phi\text{,}\tag{5.10.15}
\end{align}
where we can assume that \(\theta,\phi\in[0,\frac{\pi}{2}]\text{.}\) The remaining condition becomes
\begin{equation}
0
= ac^*+bd^*
= e^{i(\alpha-\gamma)}\cos\theta\sin\phi + e^{i(\beta-\delta)}\sin\theta\cos\phi\tag{5.10.16}
\end{equation}
which forces in general
\begin{align}
\tan\theta \amp = \tan\phi ,\tag{5.10.17}\\
e^{i(\alpha-\gamma)} \amp = -e^{i(\beta-\delta)}\tag{5.10.18}
\end{align}
(with the second condition unnecessary if \(\theta=0\) or \(\theta=\frac{\pi}{2}\)), so that
\begin{align}
\theta \amp = \phi ,\tag{5.10.19}\\
\alpha + \delta \amp = \beta + \gamma - \pi\text{.}\tag{5.10.20}
\end{align}
Some special cases are \(\alpha=0=\delta\text{,}\) \(\beta=\frac{\pi}{2}=\gamma\text{,}\) corresponding to
\begin{equation}
U_x =
\begin{pmatrix}
\cos\theta\amp i\sin\theta\\ i\sin\theta\amp \cos\theta
\end{pmatrix}\text{,}\tag{5.10.21}
\end{equation}
\(\alpha=0=\delta\text{,}\) \(\beta=0\text{,}\) \(\gamma=\pi\text{,}\) corresponding to
\begin{equation}
U_y =
\begin{pmatrix}
\cos\theta\amp \sin\theta\\ -\sin\theta\amp \cos\theta
\end{pmatrix}\text{,}\tag{5.10.22}
\end{equation}
and \(\theta=0\text{,}\) \(\delta=\alpha\text{,}\) corresponding to
\begin{equation}
U_z =
\begin{pmatrix}
e^{i\theta}\amp 0\\ 0\amp e^{-i\theta}
\end{pmatrix}\text{,}\tag{5.10.23}
\end{equation}
where we have replaced \(\alpha\) by \(\theta\) for consistency.
