I'm studying Nielsen&Chuang Book and need some clarification on mapping arbitrary rotation operator onto geometric tranformation of vector on Bloch sphere. I thought that $R_{\vec{n}}(\theta)$ means rotation of point on Bloch sphere defined by our state vector $\psi$ around $\vec{n}$ axis for angle $\theta$. So far so good, but then I tried to geometrically "cheat" on exercise 4.40 in "universal gates" section, which includes this equation:
For arbitrary $\alpha$, $\beta$,
$E(R_{\vec{n}}(\alpha), R_{\vec{n}}(\alpha + \beta)) = |1 - exp(i\beta/2)|$, where $E(U, V) = max_{|\psi\rangle} || (U-V)|\psi\rangle||$
Okay, let's say our operators correspond to rotations of point on circle made by intersection of Bloch sphere and plane orthogonal to our rotation axis $\vec{n}$, for $\alpha$ and $\alpha + \beta$ angles, respectively. Then we need to find maximum length of difference between two resulting vectors for all $\psi$ and $\alpha$. This length is equal to length of chord $\alpha + \beta - \alpha = \beta$ angle in our circle, which radius is of course is maximum(and equal to Bloch sphere radius 1) for $\psi$ orthogonal for $\vec{n}$ i.e. our plane contains Bloch sphere center. Then our maximum chord length(simple geometry for isosceles triangle with two sides equal to 1 and angle $\beta$ between them) will be $2sin(\beta/2)$.
But this is not we see in our exercise. Actually, correct answer with some easy trigonometry transforms to $2sin(\beta/4)$, just a half of angle we found geometrically. The second strange thing is that if we substitute $\beta$ for $2\pi$ than error value in exercise answer will be equal to $|1 - exp(i\pi)| = |1-(-1)| = 2$. But how it's possible if rotations for $\alpha$ and $\alpha+2\pi$ are essentially the same? I definitely miss something, but I can't understand what...
