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Section 5.6 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{5.6.1} \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{5.6.2} \end{equation}

thus defining the three matrices

\begin{align} \sigma_x \amp = \begin{pmatrix} 0\amp 1\\ 1\amp 0 \end{pmatrix} ,\tag{5.6.3}\\ \sigma_y \amp = \begin{pmatrix} 0\amp -i\\ i\amp 0 \end{pmatrix} ,\tag{5.6.4}\\ \sigma_z \amp = \begin{pmatrix} 1\amp 0\\ 0\amp -1 \end{pmatrix}\text{,}\tag{5.6.5} \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{5.6.6}\\ \tr(\sigma_m) \amp = 0 ,\tag{5.6.7}\\ \sigma_m^\dagger \amp = \sigma_m\text{.}\tag{5.6.8} \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{5.6.9} \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{5.6.10} \end{equation}

The normalization conditions

\begin{equation} |a|^2+|b|^2 = 1 = |c|^2+|d|^2\tag{5.6.11} \end{equation}

allow us to write

\begin{align} a \amp = e^{i\alpha}\cos\theta ,\tag{5.6.12}\\ b \amp = e^{i\beta}\sin\theta ,\tag{5.6.13}\\ c \amp = e^{i\gamma}\sin\phi ,\tag{5.6.14}\\ d \amp = e^{i\delta}\cos\phi\text{,}\tag{5.6.15} \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{5.6.16} \end{equation}

which forces in general

\begin{align} \tan\theta \amp = \tan\phi ,\tag{5.6.17}\\ e^{i(\alpha-\gamma)} \amp = -e^{i(\beta-\delta)}\tag{5.6.18} \end{align}

(with the second condition unnecessary if \(\theta=0\) or \(\theta=\frac{\pi}{2}\)), so that

\begin{align} \theta \amp = \phi ,\tag{5.6.19}\\ \alpha + \delta \amp = \beta + \gamma - \pi\text{.}\tag{5.6.20} \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{5.6.21} \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{5.6.22} \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{5.6.23} \end{equation}

where we have replaced \(\alpha\) by \(\theta\) for consistency.