Where does this solid angle come from?

[Heidi]

TL;DR

interferometry prblem (with mixed states)

I found this paper
https://arxiv.org/abs/quant-ph/0412216
We have an interferometer with to arms. The firsr has a couple of HWP's inclened by an angle theta
and the second has the crossed couple. A mixed state is in input.
i look to the figure withe the Bloch sphere. i see 2 paths on it. one down and one up. i suppose
that it corresponds to the arms and that that at the end of the two curves they meet when the arms merge. Is it really the Bloch spere in the figure ? i do not see the r vector into it. or is it a sphere into it with a radius length r?
thanks

 

[Heidi]

I am looking to the picture of the interferometer. There are two HWPs on each arm. on the upper arm the first is labelled by theta1 and the 2nd by theta2. what are these angles? what happens when they are equal?