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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.

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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 $$ R_n(\theta)|\psi\rangle\langle\psi|R_n(-\theta). $$

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

You're supposed to show that the operation is the same in both representations. That applying a rotation matrix to the state vector gives the same result as converting the state to a bloch vector, rotating the bloch vector, then converting back.

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