Symmetry reductions of Lindblad equations - simple examples and applications

Open quantum dynamics in the Markovian approximation is described by the Lindblad master equation. The Lindbladian dynamics is closed in the Liealgebra Λ = su(n), i.e. it has su(n)symmetry.We say that the Lindblad equation admits a symmetry reduction if it has an invariant vector subspaceΛ0⊂Λwith the Lie algebraic structure. Symmetryreductions restrict dynamics to smaller subspacesthat additionally are Lie algebras.In these notes, trivial reductions relying onthe reducibility of the Hamiltonian and Lindblad operators are described. Examples of nontrivial reductions in the infinite temperature limit and the paritypreserving Majorana reductions are presented. Applications to open spin dynamics are discussed.