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Open quantum systems are, in general, described by a density matrix that isevolving under transformations belonging to a dynamical semigroup. They canobey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. Weexhaustively study the case of a Hilbert space of dimension $2$. First, we findfinal fixed states (called pointers) of an evolution of an open system, and wethen obtain a general solution to the FGKLS equation and confirm that itconverges to a pointer. After this, we check that the solution has physicalmeaning, i.e., it is Hermitian, positive and has trace equal to $1$, and find amoment of time starting from which the FGKLS equation can be used - the rangeof applicability of the semigroup symmetry. Next, we study the behavior of asolution for a weak interaction with an environment and make a distinctionbetween interacting and non-interacting cases. Finally, we prove that therecannot exist oscillating solutions to the FGKLS equation, which would resemblethe behavior of a closed quantum system.

The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems