Resource for geometric representation of quantum channels

Question

I was wondering if anyone knows about any good resources on representing unital/quantum channels by using rotations/pauli matrices. It is mentioned in Nielsen&Chuang on p774, but i feel it is still unclear. When asking my professor he reffers me to this paper: https://arxiv.org/pdf/1902.00906. But is is still unclear. Especially how we get $\rho=\frac{1}{2}(I+w*\sigma)$ and similary $$ P(\frac{1}{2}(I+w*\sigma))=\frac{1}{2}(I+(Mw)*\sigma) $$ for $M_{ij}=\frac{1}{2}Tr[\sigma_{j}P(\sigma_{i})]$, where $\sigma_{i}$ are the Pauili matrices for $i\in \{1,2,3\}$.

Answer 1

Chapters 5.2 and 10.7 in the book "Geometry of Quantum States" by Bengtsson & Zyczkowski (alt link) may be of interest to you. Alternatively, you could have a look at Chapters 2.1.3 and 4.6 in "The Mathematical Language of Quantum Theory" by Heinosaari & Ziman.