State Space Decomposition of Quantum Dynamical Semigroups
Nicolas Mousset, Nina H. Amini
TL;DR
The paper revisits the Hilbert-space decomposition of finite-dimensional quantum dynamical semigroups into invariant enclosures, elaborating a second reading aligned with Carbone and Pautrat. It clarifies the orthogonal decomposition of the recurrent subspace, introduces the cut-off semigroup, and provides a uniqueness criterion via a partial isometry when non-uniqueness occurs. The framework is then specialized to 2D systems and to open quantum random walks, linking minimal enclosures with Markov-chain communication classes, and to quantum trajectories, showing how identifiability conditions govern long-time enclosure selection. These insights strengthen understanding of long-time stabilization and enable potential applications in reservoir engineering and feedback control.
Abstract
The mean evolution of an open quantum system in continuous time is described by a time continuous semigroup of quantum channels (completely positive and trace-preserving linear maps). Baumgartner and Narnhofer presented a general decomposition of the underlying Hilbert space into a sum of invariant subspaces, also called enclosures. We propose a new reading of this result, inspired by the work of Carbone and Pautrat. In addition, we apply this decomposition to a class of open quantum random walks and to quantum trajectories, where we study its uniqueness.