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5

Quoting from the linked source: "thus SWAP has negative eigenvalues, which means that $T\otimes I$ is not positive and therefore $T$ is not completely positive", where $T$ is the transpose. So they are not saying that the SWAP is not a realisable operation; they are saying that $T$ isn't. As you note, the SWAP is a perfectly fine unitary gate. That ...

4

Assuming w.l.o.g. that $p\in\mathbb{R}$, the linear map in the question may be rewritten as $$\mathcal{E}(\rho) = p^2\rho+p^2X\rho X = 2p^2\left(\frac12\rho + \frac12 X\rho X\right)$$ where $X$ is the Pauli matrix. Thus, the action of $\mathcal{E}$ can be understood as the composition $\mathcal{E}=\mathcal{S}_{2p^2}\circ\mathcal{X}_{\frac12}$ of a scaling ...

3

Examples The condition $[A_a, B_b]=0$ is sufficient, but not necessary for $\Phi$ and $\Psi$ to commute. Indeed, the standard Kraus representations of many commuting pairs of channels have non-commuting Kraus operators, e.g. bit- and phase-flip channel, depolarizing and unitary channel, amplitude damping and phase-flip channel etc. However, Kraus operators ...

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