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

Yui Kuramochi

Published 2024-06-06Version 1

We give a new nonstandard proof of a 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 then naturally arise as the standard parts of the traceless parts divided by $\sqrt{\delta t}$. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.