An elementary construction of the GKSL master equation for N-level systems

TL;DR

This paper offers a clear, elementary method to construct the GKSL master equation for N-level quantum systems, clarifying the role of each property of quantum dynamical semigroups in generator forms.

Contribution

It provides a detailed, self-contained construction of the GKSL master equation for N-level systems and characterizes generators with partial properties of quantum dynamical semigroups.

Findings

Explicit construction of GKSL master equation for N-level systems

Necessary and sufficient conditions for generator forms

Insights into properties of quantum dynamical semigroups

Abstract

The GKSL master equation for N-level systems provides a necessary and sufficient form for the generator of a quantum dynamical semigroup in the Schrodinger picture where the underlying Hilbert space is $\mathbb{C}^N$. In this paper we provide a detailed, self-contained, and elementary construction of the GKSL master equation for an N-level system. We also provide necessary and sufficient conditions for forms of generators of semigroups which have some, but not all, of the defining properties of quantum dynamical semigroups. We do this in such a way to illuminate how each defining property of a quantum dynamical semigroup contributes to the form of the generators.