Bloch Sphere vs 2D hilbert Space

Question

For a 2 level quantum system we can represent it in two ways: in 2D Hilbert space and also in the form of a Bloch sphere

Now,

In what space does the Bloch sphere lie?

Are they equivalent? because what seems slighlty counterintuitive to me is that in a '2D' Hilbert space we are representing a point by two complex numbers which is 4 real parameters, whose magnitude I admit are related by (alpha)^2+(beta)^2=1, but in a Bloch sphere we represent it just by two real parameters, so are 2D Hilbert space and Bloch sphere equivalent representations of a two level system?

Why is one method of representation preferred over the other? in the context of quantum information processing

Answer 1

They are equivalent. Let me first remind you that a '2D' Hilbert space are described by only 2 independent real parameters. You though it would be 4, but the normalization constraint you mentioned minus 1 out of 4, and a global phase minus another 1, so it's 4-1-1=2.

FYI, the 'global phase' means for the following transformation $$\alpha\to\alpha e^{i\phi}, \beta\to\beta e^{i\phi} $$ your state in Hilbert space remains unchanged.

The reason people would like to use Bloch sphere is that it’s more straightforward to illustrate. Also it’s able to generalize to higher spin case.