State Space Decomposition of Quantum Dynamical Semigroups

TL;DR

This paper explores a novel interpretation of the decomposition of quantum dynamical semigroups into invariant subspaces, applying it to quantum walks and trajectories to analyze their structure and uniqueness.

Contribution

It introduces a new perspective on the Hilbert space decomposition of quantum dynamical semigroups and examines its application and uniqueness in quantum walks and trajectories.

Findings

Decomposition provides insight into invariant subspaces of quantum channels.

Application to quantum walks reveals structural properties.

Analysis of quantum trajectories shows conditions for uniqueness.

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.