edited title
glS
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edited title
glS
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# How to get Bloch sphere angles given arbitrary qbitqubit as linear combination of basis vectors?

Clarify that I want a general formula

# How to get Bloch sphere angles given arbitrary qbit as linear combination of basis vectors?

I understand that given a pure state $$|\psi\rangle$$, we can express it in terms of two angles $$\theta$$ and $$\varphi$$ such that $$|\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\varphi}\sin(\theta/2)|1\rangle$$, and this is derived by converting from $$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$ into their representations in terms of $$r\mathrm{e}^{i\theta}$$, and then factoring and rearranging that.

But how do I convert between the two representations given arbitrary states? I know that $$|0\rangle = (\theta,\varphi) = (0,0), |1\rangle = (\pi,0), |+\rangle = (\pi/2, 0)$$ etc, but how do I get it for an arbitrary state $$|\psi\rangle$$?

So far, I have:

1. If $$\alpha$$ is complex, shift the entire state by phase $$\bar{\alpha}$$, where $$\bar\alpha$$ is the complex conjugate of $$\alpha$$, to end up with $$\alpha\bar\alpha |0\rangle + \bar\alpha\beta|1\rangle$$
2. Use the formulas:

$$\theta = 2 \arccos(\alpha\bar\alpha) \\ \varphi = ???$$

Improved formating
Martin Vesely
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