# Exercise 4.6 in Quantum Computing and Quantum Information Nielsen and Chuang

Question 4.6: One reason why the $$R_\hat{n}(θ)$$ operators are referred to as rotation operators is the following fact, which you are to prove. Suppose a single qubit has a state represented by the Bloch vector $$λ$$. Then the effect of the rotation$$R_\hat{n}(θ)$$ on the state is to rotate it by an angle $$θ$$ 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.

In this question $$\hat{n} = (n_x, n_y, n_z)$$ is a unit vector in three dimension and we have $$R_\hat{n}(θ) = cos(\theta/2) I - isin(\theta/2)(n_xX + n_yY + n_zZ)$$

I don't understand the idea of this question? I need some explanation of this question.

Imagine you have a state $$|\psi\rangle$$ with Bloch vector $$\vec{m}$$. Now apply this rotation $$R_n(\theta)$$ to $$|\psi\rangle$$. Can you show how the vector $$\vec{m}$$ is changed by this unitary?
Without wanting to do too much of the calculation for you, I find it instructive to write $$|\psi\rangle\langle\psi|=\frac{1}{2}(I+\vec{m}\cdot\vec{\sigma})$$ and, letting $$\vec{n}^\perp$$ be perpendicular to $$\vec{n}$$, we have $$|\psi\rangle\langle\psi|=\frac{1}{2}(I+(\vec{n}\cdot\vec{m})\vec{n}\cdot\vec{\sigma}+(\vec{n}^\perp\cdot\vec{m})\vec{n}^\perp\cdot\vec{\sigma}).$$ The unitary evolution is described by $$R_n(\theta)|\psi\rangle\langle\psi|R_n(-\theta).$$