[Physics] Rotations in Bloch Sphere about an arbitrary axis

Question

I am trying to understand the following statement. "Suppose a single qubit has a state represented by the Bloch vector $\vec{\lambda}$. Then the effect of the rotation $R_{\hat{n}}(\theta)$ on the state is to rotate it by an angle $\theta$ about the $\hat{n}$ axis of the Bloch sphere. This fact explains the rather mysterious looking factor of two in the definition of the rotation matrices."
I could work out that the rotation operators $R_x(\theta)$, $R_y(\theta)$ and $R_z(\theta)$ are infact rotations about the $X,Y$ and $Z$ axis.
I know the rotation matrices in terms of the Pauli matrices, i.e $R_x(\theta) = e^{-i\sigma_x /2}$ and the rotation matrices for $\sigma_y$ and $\sigma_z$ follows in the same manner. I could also prove that $R_{\hat{n}}(\theta) = cos(\frac{\theta}{2})I – i sin(\frac{\theta}{2})(n_x\sigma_x+n_y\sigma_y+n_z\sigma_z)$ using the Taylor expansion. But the difficulties for me start from here. How do I show that $R_\hat{n}(\theta)$ is infact a rotation about $\hat{n}$ axis by $\theta$. How can I construct a concrete proof?

Answer 1

You can in fact construct a concrete proof by direct computation:

1. Take a mixed quantum state represented by a density operator, $\rho = \frac{1}{2}(I + \vec{r} \cdot \vec{\sigma})$, where $I$ is the identity operator on a Hilbert space of two dimensions (representing your quantum state). $\vec{r}$ is the Bloch vector representing the mixed state.
2. Conjugate this quantum state by your unitary operation $U = R_{\hat{n}}(\theta)$ to compute $\rho' = U \rho U^{\dagger}$.
3. You will find that $\rho' = \frac{1}{2}(I + \vec{r'} \cdot \vec{\sigma})$ for some $\vec{r'}$, and you will note that this $\vec{r'}$ is obtained by rotating $\vec{r}$ around $\hat{n}$ by $\theta$ .

This link should help: http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf