I was reading through the book "Quantum Computation and Quantum Information for Computer Scientists", and I got up to a problem about rotation matrices on the block sphere and I can't figure it out at all.
It first starts off by defining some matrices $X = \left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right], Y = \left[\begin{array}{cc}0 & -i \\ i & 0\end{array}\right], Z = \left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$. Then it went on to define
$ \begin{align*} R_x(\theta) = \cos\left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)X, \\ R_y(\theta) = \cos\left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)Y, \\ R_z(\theta) = \cos\left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)Z \end{align*}$
It claimed each of these represents a rotation of the bloch sphere on the specified axis. I don't know how to go about proving this at all.
It also said if you define $R_\hat{n}(\theta) = \cos\left(\frac{\theta}{2}\right)I - i\sin\left(\frac{\theta}{2}\right)(n_xX + n_yY + n_zZ)$
Then this represents a rotation of the bloch sphere about the axis specified by unit vector $\hat{n}$. I also don't know how to prove this.
Can someone explain how this works?
Start with the z rotation. Take your definition of the Bloch sphere where: $$ |\psi\rangle = \cos\left(\frac{\theta}{2}\right) \left[\begin{array}{c}1\\0\end{array}\right] + e^{i\phi}\sin\left(\frac{\theta}{2}\right) \left[\begin{array}{c}0\\1\end{array}\right], $$
and calculate $R_z(\Delta\phi)|\psi\rangle$. What do you get? Remember to factor out and ignore any overall phase: ie, if you get the result: $$R_z(\Delta\phi)|\psi\rangle = \left[\begin{array}{c}e^{ia}x\\z\end{array}\right]$$ for some $a,x \in \mathbb{R}$ and $z \in \mathbb{C}$, it'd be better to factor out the $e^{ia}$ to obtain: $$R_z(\Delta\phi)|\psi\rangle = e^{ia}\left[\begin{array}{c}x\\e^{-ia}z\end{array}\right]$$ so that you can compare your result more easily in terms of the original definition of $|\psi\rangle$.
Express your result in Bloch coordinates and see what happened to the values of $\theta$ and $\phi$. Then repeat for $R_x$ and $R_y$.
The second part is trickier, but more in the realm of ordinary geometry once you get the first part down.