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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 $i\in \{1,2,3\}$.

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  • $\begingroup$ Here are posts explaining the Bloch vector as well as the Pauli transfer matrix (the $\Lambda$ therein is your $M$ & for unital channels the $v_j$ are zero). Is this what you are looking for? $\endgroup$ Commented May 21 at 13:14
  • $\begingroup$ @FrederikvomEnde no not exactly. I was wondering where i could read more about it, and get a better geometric intuition. Or read the proofs for the equalities mentioned above. $\endgroup$ Commented May 21 at 15:06

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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.

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