Qubit, one or two complex numbers? I'm currently reading up on quantum computing and it seems like I have found some contradiction about how to represent qubits.
It is often stated that a qubit is represented as $a|0\rangle + b|1\rangle = (a, b)$ with both a and b being complex numbers.
However, it is stated just as often, that there is only one complex number needed, namely b, since to ignore the global phase shift means, that a becomes real while b stays complex. See for example: this and this.
What is it now? What don't I get here?
 A: In the representation $|\psi\rangle = a|0\rangle + b|1\rangle $ we must have $|a|^2+|b|^2=1$, so that gives us one constraint.  The second is that an overall global phase doesn't make any difference.  We can use these two freedoms to chose $a$ and $b$ in a specific way.  Traditionally we choose them such that $$|\psi\rangle = \cos \theta|0\rangle+\exp(i\phi)\sin \theta|1\rangle $$
See this wiki article on the Bloch sphere for details.
A: You are spot on. Ignoring the global phase does mean that $a$ is real while $b$ remains complex. As explained in the post above $\uparrow$. This forces the qubit to be confined to the Bloch sphere (AKA the 2-sphere). However the general description of the qubit is not the 2-sphere, it is the 3-sphere.
When the global phase is included (as it should be) the qubit is a 2-dimensional complex number $|\psi\rangle\in\mathbb{C}^2$. This means the qubit is isomorphic to vectors in $\mathbb{R}^4$. When the complex coefficients have a unit norm, as $|a|^2+|b|^2=1$, the qubit describes a vector in $\mathbb{R}^4$ extending to a point on the surface of the 3-sphere.
