Complex Projective Line

Question

How can I go about showing that a collection of all states is the complex projective line $CP^1$?

All I understand at the moment is that an element in $CP^1$ is of the form $\lambda(\alpha|0\rangle+\beta|1\rangle) \: for \: \lambda \in C$.

Answer 1

The states, if I understand you correctly, are parametrized by the pairs of complex numbers $(\alpha, \beta)$ up to non-zero scalar. But that is exactly $\mathbb{C}P^1$ by definition. (see wiki)