Almost All Quantum Channels Are Diagonalizable

We prove the statement “The collection of all elements of $\mathcal{S}$ which have only simple eigenvalues is dense in $\mathcal{S}$” for different sets $\mathcal{S}$, including: all quantum channels, the unital channels, the positive trace-preserving maps, all Lindbladians (GKSL-generators), and all time-dependent Markovian channels. Therefore any element from each of these sets can always be approximated by diagonalizable elements of the same set to arbitrary precision.