Optimization on matrix manifolds, particularly on complex Stiefel manifolds, plays an important role in various tasks of modern quantum theory. The dynamics of open quantum systems are described by quantum channels, also known as Kraus maps. Properties of the geometric structure of the set of quantum channels are essential for various tasks in diverse areas of quantum theory ranging from quantum information theory to quantum computing and quantum control. This review summarizes the results on representing quantum channels as orbits of a unitary group acting on the complex Stiefel manifold and examines the corresponding quotient space geometry. The implication of this framework for Riemannian optimization is demonstrated with a particular emphasis on the absence of suboptimal extrema for broad classes of quantum control objective functionals.
Russkikh et al. (2025) studied this question.