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I am looking into 1 qubit pauli channels, e.g. the dephasing channel $$\mathcal{E}(\rho)=(1-p)\rho+p\sigma_z\rho\sigma_z.$$ I found out it could be represented as $$\mathcal{E}(\rho)=\int p(\lambda)\cdot U(\lambda)\rho U(\lambda)^{\dagger} \text{d}\lambda$$ with $U(\lambda)=e^{-i\sigma_z\lambda}$ and $p(\lambda)$ a probability distribution - and therefore interpreted as an average over random rotations in the bloch sphere (more specific, random rotations around the z axis).

Now I only could find this in a certain paper (https://doi.org/10.1038/s41467-019-11502-4) and only for dephasing channels. Is there a general (maybe even introduction) book that covers this integral representation also for depolarizing channels? Nakahara, Nielsen and Chuang or Marinescu (which I have used mainly so far) don't seem to cover this.

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  • $\begingroup$ related: quantumcomputing.stackexchange.com/a/30402/55, quantumcomputing.stackexchange.com/q/28615/55, and links therein $\endgroup$
    – glS
    Jun 2, 2023 at 19:22
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    $\begingroup$ It should not be a surprise that a discrete probability distribution can be represented by a continuous one - just use delta functions! Like $p(\lambda)=(1-p)\delta(\lambda) + p \delta(\lambda-\pi/2)$. But if you're wondering why you can use other probability distributions, that's a different question. For depolarization, for example, you want a probability distribution over SU(2) such that an average of unitaries over that is the same as some discrete sum (the sum is given in Nielsen and Chuang) - then you want to look into spherical designs, in addition to @glS's links $\endgroup$ Jun 2, 2023 at 20:42

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