Completely Positive Trace Preserving Methods for the Lindblad Equation

The Lindblad master equation is a valuable tool in quantum mechanics, which describes the dynamics of open systems. In the scope of our research, it is combined with the one-dimensional Maxwell's equations to form the generalized Maxwell-Bloch equations. Since analytical solutions are not available in the general case, numerical methods have to be employed to solve the Lindblad equation. In this work, we focus on methods that are completely positive trace preserving (CPTP), i.e., that guarantee to preserve the properties of the density matrix. We review existing approaches and compare the most promising candidates in terms of computational performance.