I am going over chapter 3 of Preskill's lecture notes regarding complete positivity. Specifically, on page 19, it is mentioned that since SWAP has eigenstates with eigenvalue -1, it is not positive, thus not a physically realizable operation.
This is quite strange to me since SWAP is quite common in quantum circuits. Also, I tried more general form of SWAP, e.g., adding a global phase, or adding a phase to $|01\rangle$ and $|10\rangle$ like iSWAP, but it seems like there will always be some eigenstates with eigenvalues not positive, or not even real.
So, from the point of quantum channels, why are SWAP gates possible? Is it because we have to involve a third qubit into consideration? But I don't think on those superconducting chips, we need to take over a third qubit when performing a SWAP gate.