Nonstandard derivation of the Gorini-Kossakowski-Sudarshan-Lindblad master equation of a quantum dynamical semigroup from the Kraus representation

Abstract

We give a new nonstandard proof of the well-known theorem that the generator $L$ of a quantum dynamical semigroup $\exp(tL)$ on a finite-dimensional quantum system has a specific form called a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator (also known as a Lindbladian) and vice versa. The proof starts from the Kraus representation of the quantum channel $\exp (\delta t L)$ for an infinitesimal hyperreal number $\delta t>0$ and then estimates the orders of the traceless components of the Kraus operators. The jump operators naturally arise as the standard parts of the traceless components of the Kraus operators divided by $\sqrt{\delta t}$. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.