2.11 Drawing points on the Bloch sphere

We know that the state |0\rangle corresponds to the north pole of the Bloch sphere, and the state |1\rangle to the south, but what about an arbitrary state |\psi\rangle=\alpha|0\rangle+\beta|1\rangle? By definition, we can find the parametrisation in terms of \theta and \varphi, but there is also a neat “trick” for finding the point on the Bloch sphere that corresponds to |\psi\rangle, which goes as follows.

1. Calculate \lambda=\beta/\alpha (assuming that \alpha\neq0, since otherwise |\psi\rangle=|1\rangle).
2. Write \lambda=\lambda_x+i\lambda_y and mark the point p=(\lambda_x,\lambda_y) in the xy-plane (i.e. the plane \{z=0\}).
3. Draw the line going through the south-pole (which corresponds to |1\rangle) and the point p. This will intersect the Bloch sphere in exactly one other point, and this is exactly the point corresponding to |\psi\rangle.

Note that this lets you draw the point on the sphere, but doesn’t (immediately) give you the coordinates for it. That is, this method is nice for geometric visualisation, but the parametrisation method is much better when it comes to actually doing calculations.