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When learning the basics of quantum computing, the Bloch sphere comes early on as a visualization technique of quantum states. It shows the state of a single qubit as a point on this sphere:

Had a and b been real numbers, it would have been easy since there would only be two dimensions (degrees of freedom). However, in reality a,b\in\mathbb{Z}, making our visualization task much more challenging because there are now 4 dimensions (two for each complex number). The Bloch sphere is a clever mapping from this 4D reality into something we can visualize.

We start by representing each of the complex coefficients using their polar representation, where the magnitudes and angles are real numbers:

Since a global phase doesn't affect the observable properties of a qubit (see the appendix for more on this), we can multiply this state by the global state e^{-i\phi_a} to get [1]:

There's only a single angle in this equation: \phi_b-\phi_a; this is the relative phase between the two components of the state. Let's call it just \phi, and then:

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