A system in quantum mechanics is modeled by a state, i.e. a $N$ dimensional with trace $1$ positive semidefinite matrix where $N$ is the number of possible values for an observable (e.g. momentum, level of energy). A transformation of such a system, after measurements for instance, is modeled by specific operators on matrices called quantum channels, preserving the set of states. These operators can be seen as the sum of tensor products of unit matrices. As for Markov operators, we are interested in the spectral gap of the quantum channel which can be seen as a quantifier of the distance of the operator to a rank one projector, and one way to optimize the gap is to consider Haar distributed unitaries. A proof of the second largest eigenvalue or singular value being optimal in the non-Hermitian case is to use Schwinger-Dyson equations previously used by Hastings in the Hermitian case.
