The Gorini-Kossakowski-Sudarshan-Lindblad generation theorem,and a generalization to non-stationary evolutions

The Lindblad equation embodies a fundamental paradigm of the quantum theory of open systems, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generation theorem says precisely which superoperators can appear on its right-hand side. These are the generators of completely positive trace-preserving (or nonincreasing) semigroups. A complete exposition of this theorem is given. The finite-dimensional case is handled using a form of Jamiołkowski transform. The treatment requires no previous knowledge of complete positivity and obtains the Choi-Kraus presentation along the way. The (separable) infinite-dimensional case is handled by means of a sequence of finite-dimensional approximations, using the finite-dimensional case as a crucial tool. An extension to time-dependent generator is given.The condition for CP evolution is just that for semigroups applied at each instant, and the Lindblad decomposition can be chosen continuous in time.