Lindblad equation using Kraus operators and a CPTP map
This is a derivation found in [1] but with details from [2]. I am only emphasizing the details that I found interesting, because it is a very short derivation originally and the way it hides some mysteries is very enticing. Most consider a postulate of quantum mechanics that a completly positive trace preserving (CPTP) map, famous for the propriety of mapping physical systems to other physical systems, can be described by Kraus operators, specifically a map from one density matrix to another. Considering said map (we make implicit the Choi-Kraus’ theorem), then $$\hat{\rho}^{\prime}=\sum_{\mu}\hat{K}_{\mu}\hat{\rho}\hat{K}_{\mu}^{\dagger}\qquad\text{with }\quad\sum_{\mu=0}K_{\mu}^{\dagger}K_{\mu}=\mathbb{I}.$$ Lets now check that if $\hat{\rho}$ obeys the proprieties of a density matrix, then so will $\hat{\rho}^{\prime}$. That is, $\hat{\rho}\prime$ is hermitian, positive semi-definite, and unit trace. The unit trace comes from the fact that it is a CPTP map, but to show this noti...