Lindbladian

System described by a Master equation of the form $\frac{d}{dt}{\rho }_t=\mathbb{L}[\rho ]=-\frac{i}{\hslash }[\hat{H},\hat{\rho }]+{\sum }_{\alpha =1}^{N^2-1}{L}_{\alpha }({\hat{K}}_{\alpha }\rho {\hat{K}}_{\alpha }^{†}-\frac{1}{2}\{{\hat{K}}_{\alpha }^{†}{\hat{K}}_{\alpha },\hat{\rho }\})$, where

• $\mathbb{L}$ is the Quantum Dynamical Semigroup generator
• $L_{\alpha }:=\sqrt{C_{\alpha }}{\hat{K}}_{\alpha }$ are the Lindblad Operators.
• $C_{\alpha }$ are positive operators. This follows from the Quantum Dynamical Semigroup generated by $\mathbb{L}$ being Completely Positive. I need to write down this proof. Help me ❓

The Lindbladian generalises the Schrodinger Equation to an open quantum system $S$. In open quantum systems the noise introduced by the environment changes the dynamics of the system, requiring the addition of “adjustment” terms. These adjustments make Unitary Operators unfit to describe a open quantum system, requiring us to use more general Operators.