# Do Bloch sphere rotations span $SU(2)$ (up to a global phase)? [duplicate]

It is well known that the Pauli group $$\{I,X,Y,Z\}$$ spans the group of $$2\times2$$ unitary matrices, $$SU(2)$$, for example see this link.

A general Bloch sphere rotation by an angle $$\alpha$$ about an arbitrary axis, $$\hat{n}$$ is given by:

$$R_{\hat{n}}\left(\alpha\right)=\cos\left(\frac{\alpha}{2}\right)I-i\sin\left(\frac{\alpha}{2}\right)\left(n_x X+n_y Y+n_z Z\right)$$

Is it true, that up to a global phase, Bloch sphere rotations span $$SU(2)$$?

If so, can it be proved without using group theory and abstract algebra?

• actually technically the Pauli group has $8$ elements: $\pm \mathbb{I},\pm X,\pm Y,\pm Z$. See doi.org/10.1063/1.528006 Commented Jul 21, 2022 at 21:45
• All you need to do is to show that any SU(2) matrix has the form $R_{\hat n}$ so write the most general SU(2) matrix and find the coefficients $n_i$ and $\alpha$ using trace orthogonality of the Pauli matrices… Commented Jul 21, 2022 at 21:48
• $SU(2)$ is not a vector space, so it isn't the span of anything. The four matrices you've given span all of $\mathbb C^{2\times2},$ which includes $SU(2)$ but also a lot of other junk. $\mathfrak{su}(2)$ is a vector space and is spanned by just $X,Y,Z.$ The map $\exp:\mathfrak{su}(2)\to SU(2)$ surjects, and your $R$ should be explainable as interpreting $\hat n,a$ as a element of $\mathfrak{su}(2)$ and then exponentiating.
– HTNW
Commented Jul 21, 2022 at 21:56
• Commented Jul 22, 2022 at 3:29
• Another possible duplicate: physics.stackexchange.com/q/174562/2451 Commented Jul 22, 2022 at 6:56