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Is the function PU(2) and SO(3) induced by the Bloch sphere bijective?

The element $U \in PU(2)$ is not uniquely determined by the pair $|u_0\rangle = U|0\rangle$, $|u_1\rangle = U|1\rangle \in \mathbb{CP}^1$. There is a freedom, $U = e^{i\alpha}|u_0\rangle\langle 0| + e^...
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Clarification defining/finding the relative phase of a qubit

TL;DR: It doesn't matter. Both $e^{i(\theta_0-\theta_1)}$ and $e^{i(\theta_1-\theta_0)}$ are valid. The textbook is correct and you have made no mistake. Either of the two expressions is appropriate. ...

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