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

TL;DR

This paper presents a novel nonstandard proof that characterizes the generator of a quantum dynamical semigroup as a GKSL (Lindblad) form, using Kraus representation and nonstandard analysis techniques.

Contribution

It introduces a new nonstandard proof linking Kraus operators to the GKSL form of quantum dynamical generators, providing insights into their structure and approximation.

Findings

Established a nonstandard proof of the GKSL form from Kraus representation.

Showed how jump operators emerge as standard parts of traceless Kraus components.

Proved that close completely positive maps have close Kraus operators.

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.